Domes and Dome Houses

Bachelor of Architectural Technology and Construction Management

7th semester Dissertation

Domes and Dome Houses

Author: Povilas Mikuta Consultant: Erik Toft

VIA University College, Horsens, Denmark 08.10.2013

HORSENS CAMPUS, DENMARK

ELECTIVE TITLE: Domes and Dome Houses CONSULTANT: Erik Toft AUTHOR: Povilas Mikuta DATE/SIGNATURE: 08.10.13 STUDENT IDENTITY NUMBER: 142850 NUMBER of COPIES:- 1 No. of PAGES:- 41 No. OF CHARACTERS:- 106 954 FONT: Arial 12

All rights reserved – no part of this publication may be reproduced without the prior permission of the author.

NOTE: This dissertation was completed as part of a Bachelor of Architectural Technology and Construction Management degree course – no responsibility is taken for any advice, instruction or conclusion given within!

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Preface This dissertation is part of the final examination for Bachelor of Architectural Technology and Construction Management. It contains construction, functional principles and potential of domes and dome houses. Most of the information was gathered from written sources such as books and articles, to give a better understanding of the subject I have made sketches and models used as illustrations in this dissertation.

Acknowledgments I would like to thank my consultant Erik Toft for helping me to make this dissertation as informative as possible.

Abstract Dome a shape that was so successfully used throughout the history time and again, has been forgotten in the modern times. This became the purpose of this dissertation, to find out - was it a right choice to dismiss this shape in modern architecture and what place can dome take in the world of the future. In order to achieve this, answers to some more fundamental questions had to be found first.

Research started from a look into the history of domes, dome dwellings and their use. How each new dome pushed the boundaries of engineering with one solution more ingenious than the other. The functionality of this form: both as a way to enclose large common areas and as a simple family dwelling. How it performs statically and build ability options. Analysis of major advantages and problems which this structure can cause.

Finally, an insight into its potential and proposals of how dome might influence the future world.

Most of the research was done in the library and on the internet, gathering information from various sources and putting it together in order to achieve objective evaluation. Findings soon showed the number of benefits this shape has. Surprisingly most of them have a lot in common with sustainability issues that are being solved by architects and engineers today; like material savings and heat loss optimization. Finally the future vision, that ranges from fantastical ideas of near science fiction, to grand structures of domed cities by Buckminster Fuller, which soon might become reality. If a climate change will not be stopped one day entire city of Houston might become trapped under the dome.

Key words Dome, dome history, dome construction, geodesic dome, tensegrity, synergetics, Buckminster Fuller.

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Contents Abstract ............................................................................................................................... 2 Key words............................................................................................................................ 2 1. Introduction and problem background .......................................................................... 5 1.1 Background information ............................................................................................. 5 1.2 Professional relevance ............................................................................................... 5 1.3 Problem statement ..................................................................................................... 5 1.4 Delimitation ................................................................................................................ 5 1.5 Theoretical basis and choices for empirical data........................................................ 6 1.6 Research method ....................................................................................................... 6 1.7 Report‘s overall structure ........................................................................................... 6 2. History........................................................................................................................... 7 2.1 Neolithic Pit Dwellings ................................................................................................ 7 2.2 Yurts........................................................................................................................... 8 2.3 Wigwam/Wickiup ........................................................................................................ 9 2.4 Igloo ......................................................................................................................... 10 2.5 Hogan Dwelling ........................................................................................................ 11 2.6 Ab-Anbars ................................................................................................................ 11 2.7 Yakhchals ................................................................................................................ 12 2.8 Trullo ........................................................................................................................ 13 2.9 The Pantheon........................................................................................................... 14 2.10 Hagia Sophia.......................................................................................................... 15 2.11 Brunelleschi’s Dome .............................................................................................. 16 2.12 St. Peter’s Dome .................................................................................................... 17 2.13 St. Paul’s Cathedral Dome ..................................................................................... 17 2.14 Capitol Dome ......................................................................................................... 18 2.15 Montreal Biosphere ................................................................................................ 19 3. Functionality................................................................................................................ 20 3.1 Statics ...................................................................................................................... 20 3.2 Geodesic domes and synergetics ............................................................................ 21 3.3 Tensegrity ................................................................................................................ 23 3.4 Build ability ............................................................................................................... 25 3.5 Surface Area – Square meters – Volume................................................................. 27 Case No. 1 ..................................................................................................................... 27 Case No. 2 ..................................................................................................................... 28 3

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Case No. 3 ..................................................................................................................... 29 3.6 Floor plan layout....................................................................................................... 29 4. Advantages and disadvantages .................................................................................. 31 4.1 Material Savings....................................................................................................... 31 4.2 Ventilation and indoor climate .................................................................................. 31 4.3 Cooling effect ........................................................................................................... 32 4.4 Heat loss optimization .............................................................................................. 33 4.5 Hurricane and Earthquake resistance ...................................................................... 35 4.6 Construction Complexity .......................................................................................... 36 4.7 Leakage ................................................................................................................... 38 4.8 High precision .......................................................................................................... 39 4.9 Lack of experience ................................................................................................... 39 5. Future potential ........................................................................................................... 40 Buckminster Fuller............................................................................................................. 40 5.1 Instant shelter........................................................................................................... 40 5.2 Climate domes ......................................................................................................... 41 5.3 Cloud nine ................................................................................................................ 41 5.4 Garden of Eden Concept ......................................................................................... 42 5.5 Domed Cities............................................................................................................ 43 Potential ............................................................................................................................ 44 5.6 Dome over Houston ................................................................................................. 44 6. Conclusion .................................................................................................................. 44 7. List of references ........................................................................................................ 46 8.

List of illustrations ...................................................................................................... 48

Appendix A ........................................................................................................................ 50 Elara – two storey dome (http://www.monolithic.com).................................................... 50 Janus – two storey dome (http://www.monolithic.com) .................................................. 51 Io 20 – single storey dome (http://www.monolithic.com) ................................................ 52 Hyperion – single storey multiple domes (http://www.monolithic.com) ........................... 52 Appendix B ........................................................................................................................ 53

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1. Introduction and problem background 1.1 Background information This is 7th semester dissertation for the Bachelor of Architectural Technology and Construction Management education. Main part of this report will deal with dome constructions and their potential. I chose to investigate this subject, because I believe it has been unfairly forgotten in the world of modern architecture and construction. Dome as a shape seems to have a lot of qualities that are cornerstones of sustainable construction.

And since sustainability is a big thing today and will become even bigger in the future I decided to investigate and see what kind of potential this form has in the future world.

To achieve this goal I will look in the past to find out what place do they took in history since first application by Romans till Buckminster Fuller – man who truly reinvented this form. I will analyze their working principles to find out how domes perform so well statically and under extreme conditions. Look how it performs as a building is it a comfortable shape for a house or is it just a beautiful, but rather impractical dream.

Investigate what sustainable qualities are within this shape and their potential in the future.

And ultimately I hope all this research will help me to answer question: Is dome a shape of the future?

1.2 Professional relevance During my education I accidentally stumbled upon an article about a company that specializes in construction of dome houses. I was really surprised to see how well those structures perform in real life and of course as a curious person I’m interested to know are things really working that way and why. And as a constructing architect I recognize that this sort of shape has a great potential in the world of sustainable architecture. Therefore I would like to investigate myself and see if it is actually possible to make dome / dome houses thing of a future.

1.3 Problem statement •

Can dome be a shape of the future buildings?

• • • •

How domes have been used in previous centuries?

How are domes functioning?

What are the advantages of domes over traditional forms?

What place can dome take in the future?

1.4 Delimitation In this report I will focus more on technical architecture aspects of domes and will not go into design options as such.

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1.5 Theoretical basis and choices for empirical data Main criteria for choosing data for the dissertation are reliability of the source and how up to date information is. Critical look evaluating relevance and usefulness of the data is essential for objective dissertation.

Most of the data comes from reports, studies and publications related to the domes and their construction. Parts of the report are based on the exercises and calculations that I made by myself to make it as relevant to the dissertation topic as possible.

1.6 Research method Main body of the dissertation is based on secondary analytical research, with small elements consisting out of analytical primary research.

1.7 Report‘s overall structure Report mainly consists out of three parts: I. Introduction with problem formulation II. Research, analysis and findings III. Conclusion Dissertation first of all started with formulating problem statement. Second step was to find research questions which would help to sort out and find relative data to the problem statement. Step three was accumulating information and data for the project, then formulating and stating findings. In the end a conclusion summing up and evaluating the information gathered was drawn.

Fig. 1 Brain storm of the report structure

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2. History To get a full understanding of domes and what place they might take in the future one must search for clues in the past. In this chapter I will cover history and evolution of domes, from very first structures that led to development of the dome till the most advanced geodesic domes.

2.1 Neolithic Pit Dwellings Journey of dome evolution begins with the very first manmade structures. Neolithic pit dwellings seem to be one of the first attempts to achieve shape, similar to one that we know as a dome. The best documented pit dwellings are those of Thompson Indians in southern British Columbia, Canada. During the end of XIX century ethnologist James Teit carried out careful studies documenting construction methods, techniques and traditions related to pit houses.

As the name already suggests, construction of such structure usually started from excavation of a 2 Reconstructed Pit House round pit, 7-12 meters in diameter, 1-2 meter in depth Fig. http://media.canada.com/0784bb05-ab0e-4bd2(although tribal chief’s house may have been twice that aec9-4ccd77e2817b/day2-01.jpg size). Earth walls of the pit were usually sloping outwards to stabilize them. Logs were inserted in the floor construction parallel to the sides of the wall. Connecting edges were notched together to support main rafters which were sunk into the soil at steep angles.

Usually there were four main rafters creating shape more similar to pyramid, but later on, this number increased and therefore shape transformed into one that quite close resembles a dome. Next step was a layer of beams placed between the rafters in concentric circles thus finishing main load bearing structure. Between beams poles with pine needles and grass ware laid out to create first insulating layer. Main choice for material preventing from rainfall was cedar bark which covered insulating layer. Final touch was a layer of earth placed on the top masking whole structure and a log ladder lowered down through the smoke hole.

Eventually grass covered entire exterior and blended it into the landscape.

The only artistic piece of the house was the log ladder often carved with beautiful Fig. 3 Neolithic Pit House Section (by Povilas Mikuta) patterns and figures making it the centre piece of the dwelling.

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Even though Interior of a house had no partitions it was divided into four areas which were defined by the placement of the roof rafters. This came from the Thompson Indian ethnographical view of a world as a circular lodge subdivided into separate spheres.

This type of dwelling shares similar qualities to conventional domes regarding ventilation and indoor climate, storm resistance etc. But because construction was carried out in the old more primitive way, it was not that complex and did not require high precision. Although it can be argued, that by being a traditional structure of Thompson Indians for at least 3500 years this skill of building was perfected generation after generation. (Mills, 2012) 2.2 Yurts Yurt is a traditional dwelling of tribes in central Asia and Mongolia. In its shape yurt is not as similar to dome as some traditional dwellings of Native Americans, but to get an understanding why dome buildings perform as they do it is crucial to overview this dwelling.

Traditionally yurt construction consisted of framework made out of slender willow branches, which were relatively strong and Fig. 4 Traditional Yurt http://www.yurtinfo.org/yurt-faq light. Circular wall frame was made in cross pattern secured by rings in the top and bottom. Roof spanned from walls till compression ring at the top, this ring connected the roof frame and gave a circular opening-vent at the top of the yurt. Eventually sheep wool felt was placed on top, securing the inside from natural elements. This was a perfect choice for the locals since it provided perfect shelter in local climate and was very easy to transport. Frame and wool felt could fit on a couple of camels and whole structure could be put together in a matter of an hour or two. (Oliver, 1987) The circular shape of the yurt is what makes it so perfect for the local climate. By being round yurt has 12% less surface area than its rectangular counterpart. This enables it to save on materials (making it lightweight) and reduce heat loss at the same time. Since it is a traditional dwelling of people who travelled through steppes in central Asia one of the main problems in their way was wind. Plains stretching for hundreds of kilometres allow it to achieve amazing speed. This creates the problem when it comes to lightweight buildings. How do you secure them from being blown away? The answer again lies in the circular shape. Since there are no corners or sharp edges that could catch wind, air can go around structure Fig. 5 Yurt Structure Sketch (by Povilas Mikuta) smoothly without causing any problems. Furthermore, opening at the top of the building gives a good air circulation which creates comfortable indoor climate. So when you come to think about it all four problems that are still bothering building industry today was solved with one solution, round dome like shape. (Kemery, 2006) 8

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2.3 Wigwam/Wickiup When we hear words “Wigwam” or “Indian Tepee” we get an image of cone shaped dwelling. This misleading image is lodged in our brains from many Hollywood movies and cowboy stories. In fact many of Indian houses resembled more of a dome than a cone.

Florentine explorer Verrazano describes wigwam as a hemispherical dwelling covered in mats of various materials that he saw at Narragansett Bay. Main wigwam structure was made from wood poles; greater end of poles were put into the ground and bent to form arches, then tied together using strips of bark or wood. Most common source for poles was walnut, young hickory, basswood or elm. Often poles were split in half to force it bend more easily. Some Fig. 6 Replicated Wigwam http://www.iaismuseum.org/village.shtml tribes preferred using ironwood. While it is still green it bends very easily and do not crack while drying/toughening. To lash poles together Indians used strips of white oak, tough roots or inner bark of basswood.

Sometimes additional line of poles was tied connecting arches and enabling them to transfer load more evenly throughout the structure.

Later wall mats were placed. This activity was usually carried out by women, who gathered bulrush or reeds then sewed them into mats using animal bone needles and split spruce root treads. Bulrush stalks were cut during September and October than placed carefully to dry out and avoid mildew. Mats were placed in such a way so that grain would provide sufficient rain runoff sometimes forming a small eave in order to avoid water getting inside trough small cracks. Occasionally tree bark was used as material for mats, but that made mats fairly heavy so this type of cover was more often used during long winter months.

Fig. 7 Wigwam construction sequence (by Povilas Mikuta)

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Cavity inside the reed or bulrush stalks provided sufficient insulation while keeping it lightweight. According to anthropologist Karen Peterson Indians recognized “the principle of insulation by means of walls enclosing a dead air space”. Moreover such construction was very flexible in terms of mobility. When changing a location mats were taken off the wigwam frame and transported to new location and new frame. It can be said that this was one of the first prefabrication attempts.

Similarly to previously discussed pit dwelling smoke hole was left at the top of wigwam. Sometimes edges of these smoke escape were even fireproofed with a clay lining, some accounts has it that this lining backed hard and black. (Peter Nabokov, 1989) 2.4 Igloo Smiling Eskimo next to a dome shaped igloo house is probably very well known image from the childhood as a symbol of cold parts of the world as well as ice-cream advertisement.

This makes igloo one of the best recognizable dome houses in the world.

In terms of structure it is probably first completely solid dome without any framework supporting it, which makes it similar to some of Fig. 8 Inuit Igloo the first dome structures in the west civilization. http://www.arcticphoto.co.uk/gallery2/arctic/peoples/greenland As in many other cultures dome shaped vault w/qq0608-08.htm 1 for Eskimos symbolizes firmament .

Igloo as a dwelling was born from economical awareness. They were built without any supporting framework, formwork or scaffolding. All that it took was a couple of people and a few knives suited to cut snow blocks. The true magic of igloo lies in the way snow pieces are put together. First row of blocks was placed in a circle (3,5 – 4,5m in diameter) where top part was cut to allow subsequent lines of blocks begin turning into a spiral.

Slanting edges of the blocks forced every new row incline inwards more and more thus creating a solid dome shape. This method ensured that each new block will lean against the previous one avoiding any temporary support until whole structure has been finished.

Close to the base of the igloo an arched entrance was formed to create a trap which prevents cold air from entering interior space. Igloos occasionally were arranged in groups, where a number of domed units shared a common entrance. Each space had its specific function, often these grouped buildings served for uses of whole community capable of holding as much as 60 people. (Mills, 2012) Just like previous dome shaped dwellings igloo had a circular opening at the top for efficient ventilation as well allowing excess of warm to escape and prevent the roof from melting. Additional ventilation opening sometimes were simply poked through walls of dome when need arose in order to allow stale air to get out. Furthermore round shape of a dome eliminated formation of dead and cold air pockets, not mentioning the benefits of heat conservation and efficient air circulation. Often interior of the dome was lined with 1

Firmament – sky conceived as a solid dome.

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animal skins trapping cold air between hides and snow shell, thus preventing the dome from melting and still keeping warm air inside. It is a masterful insulation for the dwelling, when outside temperature is around -40oC temperature inside could reach 15oC or sometimes even more.

Light inside was provided by opening at the top of the dome or an opening above the entrance. Window was made from sewed seal intestine lining or a block of clear ice formed in a sealskin bag and curved to fit the shape of a dome. Sometimes a window was cut in the side of an igloo to capture the last rays of the sun. (Oliver, 1987) 2.5 Hogan Dwelling Hogan is a traditional Navajo Indian house, closely resembling wigwams of Indians living on the east coast of North America. One of the radical differences is that Hogan is more of a permanent dwelling, not designed to be easily relocated when needed. In some way it seems to be a bit of a middle ground between Neolithic pit house and wigwam, it combines qualities from both structures.

Hogan’s main support is a few timber Fig. 9 Navajo Hogan http://navajorug.com/the-navajorafters forming a shape close to arch and hogan-shelter-and-center-of-their-world/ supporting interconnected pieces of wood laid in concentric circles and forming domed roof. Usually this wooden substructure was covered in clay and tree bark which provided quite good insulation layer, nice indoor climate, protection from nature’s elements and most importantly glued the timber together creating stable structure that is able to transmit the loads exactly like a dome, providing unobstructed space inside. Unlike pit dwelling most common hogan did not have a smoke hole at the top, transforming dome roof into completely solid structure.

Hogan’s are one the first attempts to create complex reciprocal frame structures2.

Plan of the house looks as a single reciprocal frame structure supported by another frame reciprocal frame structure and so on until it completes full self supporting dome. (Larsen, 2008), (Peter Nabokov, 1989) 2.6 Ab-Anbars Ab-Anbar is word commonly used in Iran for an underground water storage facility.

System of Ab-Anbars was developed for arid regions of Iran to ensure proper water supply for the locals. Today these structures have lost their strategic importance to water pipe lines which always ensure flow of fresh water, but nevertheless they still are amazing buildings and are considered to be important heritage.

2

Reciprocal frame - a class of self-supporting structure made of three or more beams and which requires no centre support to create roofs, bridges or similar structures.

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Speaking about domes it is important to look at these buildings in order to understand what potential lies in a shape of a dome. Specifically - ventilation methods used.

Main demands that ab-anbar structure had to meet was totally waterproof, capable of containing large amounts of water and provide proper ventilation (keeping water fresh). As a self supporting structure dome easily creates a very large unobstructed volume which is perfect if you need to store Fig. 11 Ab-anbar in Qazvin large amount of water. Water tightness is http://www.trekearth.com/gallery/Middle_East/Iran/East/Yazd/Yaz d/photo1006911.htm achieved with plaster, so called sarooj. This mix of clay, lime, ash and seed’s pod makes plastering watertight as well as preventing it from cracking. Water stored in ab-anbar was kept cool by underground container, thick walls and ingenious ventilation system. What makes this ventilation system special is combination of natural air flows in the dome and ventilation towers (badgir). Badgir’s catch Fig. 10 Working principle of Ab-anbar (by Povilas the breeze on the top and through ventilation Mikuta) chambers directs it into a dome where most often it escaped through ventilation opening at the top of a dome. System like this made sure that air inside the storage chamber was circulating, helping to keep water fresh and more importantly cool. Some storage facilities managed to use this combination of technologies to keep water just a few degrees above freezing even in hottest summer. (Yazdi, 2007) 2.7 Yakhchals Iranian’s used qualities of a dome not only for storing fresh water, but also ice. After exploiting exceptional qualities of a dome in Ab-Anbars, Iranians made some modifications to make it suitable for ice storage during burning hot summer months.

Keeping water cool in the desert is one thing, but to keep ice and in such a way that it does not involve any mechanical technologies or conductors sounds almost impossible. And only a humble dome made this possible.

Construction of these ice storage tanks Fig. 12 Yakhchal in Yazd Province http://www.eartharchitecture.org/index.php?/archives/1045was very similar to Ab-Anbars, just on a bit Yakhchal-Ancient-Refrigerators.html bigger scale. Yakhchals had a big waterproofed pit and a cone shaped dome. Unlike AbAnbars these structures had a deep well in the bottom of the pit so that melted ice could 12

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seep into the ground and the dome was made in a step manner. To provide sufficient stability bottom of the dome was made much thicker than top. Often difference between top and bottom wall thickness was 90%. This solution have put centre of mass much lower, making construction more stable and reducing amount of labour force when structure was getting higher. Step like shape provided good access for maintenance of the dome. Unlike water storage cisterns Yakhchals did not have any ventilation towers. All necessary air flow was gained through natural shape of a dome. Upper layers of air would get warm and lower layers, closer to ice, remained cool. Using laws of physics, fresh and cold air goes down and warm air goes up and escapes through an opening at the top, thus storing ice safely for months to come. (Bahareh Hosseini, 2012) 2.8 Trullo One of the most imaginative adaptations of a dome can be found in southern Italy. Trullo – traditional countryside dwelling, exceptionally built from local materials (tufa or limestone) in the region of Puglia, Italy.

These dwellings have a very unique construction which mostly relies on friction between stones. Dome is built in two layers. First layer consists of wedge shaped stone pieces, each of them is individually adjusted to make sure that it fits perfectly with other stones, creating strong friction and in this way allowing to build the dome without using any kind of mortar at the same time keeping its structural stability. Each layer leans towards the centre more and more creating a dome shape. This is achieved by carving stones to look similar to parallelogram, this way joints stay in horizontal position. Second layer of stones Fig. 13 Trulli mostly consisted of limestone tiles, which were as well http://www.understandingitaly.com/pugliacontent/trulli.html carved to fit round shape of the dome and tiled outwards to establish efficient rainwater runoff. It is thought that second layer of stone followed straight after the first one to create narrow scaffolding on which masons could stand and work on upper rings. This was possible because of so called ring tension. Normally dome is thought of as a three dimensional, rotated arch, but in this case each row of stones as well can act as a horizontal arch, capable of taking not only vertical, but also horizontal loads.

That is why it could be built without big supporting structure or additional scaffolding making it very fast to build. This speed became very economical in more ways than one. In older days taxes in Italy were collected only if the house is finished (meaning roof is on).

So if locals heard that a tax collector is coming to town they would quickly demolish trullo dome to avoid taxes. Once collector had left the town, roofs would be put back on in a very short period of time. (Campbell, 2001), (Oliver, 1987)

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2.9 The Pantheon Pantheon is the first great building of western civilization that marked a start of tradition to build grand dome structures all over the world.

Most likely source of inspiration for domes was other classical architectural component widely used by Romans to build bridges and aqueducts known as an arch. Dome is nothing but an arch rotated 360o around its axis. What makes Pantheon so remarkable among other Roman domes is its span. It is first dome which span reached 43,3 meters (some scientists argue that this number might be closer to 50m), most of other domes at Fig. 14 The Pantheon in Rome that time were at least twice smaller and so Pantheon http://www.historywiz.com/galleries/pantheon. htm held record of largest spanning dome for more than a thousand years.

Pantheons dome is remarkable as much by its structure as by magical interior space that it creates. Concrete – a popular material for buildings among Romans was chosen for this dome not by accident. As material concrete has great flexibility, forming pretty much any shape and keeping remarkable strength. Moreover to reduce unnecessary loads, weight of the aggregate was reduced in the higher levels of the dome. Dome have a very distinct way to spread forces, top of the dome is facing compression while lower half deals with tension. This tension, forces bottom of the dome outwards which might risk in the collapse of the whole structure. To counteract this outward thrust Romans placed seven concrete rings providing support and so redirecting dome load on to heavy walls below.

Each ring lowers the diameter of the opening following dome proportions; it resembles a series of machine washers placed on top of each other.

Then dome transforms into round smooth shape with opening (oculus at the top). To make this illuminating opening possible a compression ring (5,9m diameter) was formed around it to secure distribution of compression forces much like in the Fig. 15 Possible centrings used to support the previously discussed yurts. dome during construction (by Povilas Mikuta) Pantheon was troubling engineers and architects for centuries. Many specialists recognized that there could be failure points in the ancient dome where enormous stress might lead to catastrophe. But this has not happened for centuries. So is there any reason to worry? An Italian superintendent of 14

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monuments Terenzio documented many cracks in the lower part of the dome this lead to a further study of went wrong. Further Mark and Hutchinson study showed that cracks continue upward approximately 57o from horizontal. This is a point where compression turns into tension stress – a weak point in the unreinforced concrete dome. Carks are appearing generally above the openings in the upper cylindrical wall. At this point local tensile stress in the bottom of the dome is largest. To understand what kind of tensile stress Pantheon is actually under a computer driven analysis was performed. This is some of the data used in the analysis: bottom dome weight 1600kg/m3, upper dome weight 1350kg/m3 and 1750kg/m3 for the walls (again weight of the aggregate decreased with height). Interestingly study has shown that if 2200kg/m3 concrete would have been used in construction, Pantheon stress in the dome would have been 80% higher. We can conclude that with normal weight concrete Pantheon would have never been built. This shows the ingenuity of Romans their deep knowledge about forces acting in the dome and cautious use of material. (Moore, 1995) 2.10 Hagia Sophia Hagia Sophia built in 537 A.D. might not have dome greater than the one of Pantheon, but it is essential step in the history of dome building. This building gave birth to some architectural solutions regarding dome building as well as other structural issues that have been used for centuries after the building was finished.

Some solutions still have not changed Fig. 16 Hagia Sophia http://istanbulvisions.com/hagia_sophia.htm even in modern times.

During Nika Revolt in 532 old Constantinople basilica was burnt to the ground and emperor Justinian commissioned a construction of a new church bigger and greater than its predecessor also featuring a great dome. This is where first problem occurred.

Churches that are built according to Greek orthodox traditions have a square space in the centre of it, but how do you place a round dome on a square base. Solution was chosen to build four piers one in each corner and connect them with great arches (each spans more than 30 meters). It allows a flow of people through the church and follows the traditions of church building in the Byzantine Empire. This build up meant that dome is resting only on four points - tops of the arches, problem occurs that in the unsupported parts (corners) dome will start cracking and eventually will collapse on its own weight. Here is the place where first time in history architects and engineers used pendentives to support the dome.

This architectural component not only gives an elegant look when pendentives mimicking dome shape creates seamless connection, but it still is the best and most comfortable way to place dome on a square or rectangular base. Originally on top of arches and pendentives a cylinder was placed which in turn supported great dome. Unfortunately an earthquake on 14 of December 557 exposed a weak link in this domes design. Connection between dome and cylinder was a dangerous solution in place where earthquakes are quite common. Speaking from engineerical perspective if finished building is standing this means that it will stand unless something changes. In case of Hagia Sophia change was 15

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an earthquake. Bottom of the dome as it was already described has a tendency to push outwards so when it is lifted up on a cylinder rather than connected straight to a main load bearing structure counteraction force is severely reduced and creates a weak link which, as a history shows, cannot take force movements caused by an earthquake. (Kostof, 1995) 2.11 Brunelleschi’s Dome New milestone in the history of dome building was set by Filippo Brunelleschi. It was a new engineering feat when it comes to dome building and marked the new time in architecture.

Dome was supposed to soar over 44 meter wide central church space and it was a standard solution to support the whole dome with wooden centrings, (just like building arches) during the construction process. But the dome itself started 52 meters above the ground, therefore supporting such structure would have been an engineerical feat in itself, some all the trees in Tuscany would have been necessary to support it. Many architects unsuccessfully tried to come up with a solution.

Then after long years of waiting Filippo Brunelleschi 18 Dome of Santa Maria del Fiore came up with a daring plan - a double skin dome capable Fig. http://www.brunelleschisdome.com/ of supporting itself during the construction process.

Some masonry domes have been built without centring previously, on the basis of rings. Each ring placed on top of each other was narrower and narrower and acted as a horizontal arch in this way supporting itself throughout the construction. However dome of Santa Maria del Fiore was built on octagonal base, not a round one, therefore this solution of support seemed to be not viable. What Filippo Brunelleschi did was realizing that inner dome 200mm in its widest and 150mm in its narrowest was able to accommodate approximately 75mm thickness circular vault in its middle. Rowland Mainstone claims that this is a proof of how dome could Fig. 17 Structure of Brunelleschi‘s Dome (by Povilas have supported itself. By being built like a Mikuta) round one, just cutting parts outside and inside to form octagonal vaults. Where bricks ended up outside of this circular ring, a herring-bone brick pattern was used to stabilize them. This is one part of the story. Outer dome due to its small thickness could not accommodate a circular vault in its thickness. So a different solution was implied here – nine horizontal masonry arches were built on the inside of the outer dome (hiding in the cavity between the domes) allowing the outer shell to be self supporting as well. To 16

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counter act the outward thrust at the bottom of the dome tin plated sandstone and oak chains were placed tightly encircling the bottom of the dome.

This type of construction became an inspiration and a riddle to many domes that had been built later and fascinated architects for centuries. Even after careful studies questions were the construction took place like this or Brunelleschi had more secrets remains. (Kostof, 1995), (King, 2000) 2.12 St. Peter’s Dome St. Peter’s dome, designed by Michelangelo is one the key pieces in the skyline of Rome. During long years of church construction many architects tried to leave their mark on the basilica. Original design by Donato Bramante featured a Pantheon like solid construction dome. However after Michelangelo took over as a chief architect, he proposed a completely different design of high soaring double skin dome similar to the one in Florence. Even though diameter of the dome was smaller than the one in Florence, real technological breakthrough lies within domes’ shell.

For many years St. Peter’s dome puzzled engineers and architects. Dome is rests on a drum, which strange 19 St. Peter‘s Dome enough is far from solid, number of windows in the shell Fig. http://entertainment.howstuffworks.com/arts/art makes drum quite weak. Some engineers even stated work/st-peters-basilica5.htm that this connection is a main weak point of the construction, there is no flying buttresses supporting outward thrust of the dome. Despite of all these concerning signs dome still remains standing. The secret lies in the great innovation. During the construction five huge metal chains were casted in the concrete of a dome to hold outward thrust. This feat of engineering can be interpreted as a first use of reinforced concrete. Technology so advanced that it was more than two hundred years ahead of its time. Recent studies of the dome using geo-radar confirms the technology used. However in XVIII century dome started showing signs of cracking and hasten repairs took place. Two new circumferential iron chains were added on the outside of the dome to secure it in place, like iron rings holding the barrel. Even though there were many speculations why dome was cracking and that there might be more damage to it than it is possible to see with a naked eye, repairs stabilized the dome and it stood safely for a few hundred years till present days. I think it is safe to say that dome of Saint Peter’s started a new page not only in dome building but in the history of architecture and construction. (Press, 2011), (Mainstone, 1999) 2.13 St. Paul’s Cathedral Dome After great fire of London in 1666 main Cathedral of London had burned to the ground and Sir Christopher Wren was commissioned to build a new Cathedral. To represent ideas of enlightenment age cathedral was built honouring classical style and this meant that main feature should be a great dome complementing the skyline of London. Since new St. 17

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Paul’s was built on top of the remains of a prior church, this meant that new construction will rest on old foundation. When construction of new grand dome (second largest in the world at that time) began, it was discovered that foundation is resting on different types of soil and thus heavy dome structure started to settle unevenly. To solve this problem Sir Christopher Wren came up with extraordinary solution – a triple dome. The middle dome acted as a main load bearer inspired by gothic builders. One of the best recognizable features of gothic architecture is pointed arch, this structure is incredibly strong, requires less material than roman arch and therefore it is a lot lighter. But this sort of shape was more similar to “Witches hat” rather than perfect hemisphere of a dome. To overcome this Wren came up with best architectural trickery there is. A light wooden dome covered in lead was built around it hiding the witches’ hat and giving the illusion of perfectly round dome. Since the pointed dome would also look ridiculous from the inside - perspective raising too fast, a smaller hemispherical dome was built to fit in with the interior proportions. An oculus opening at the top of the inner dome gives a glimpse of the pointed dome tricking the eye into believing that what you see is the top of the inner dome.

(Kostof, 1995)

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Fig. 20 St. Paul‘s Cathedral http://www.explorestpauls.net/oct03/textMM/DomeConstructionN.htm

1 2 3

Fig. 21 Dome‘s Structure (by Povilas Mikuta) 1 – Thin Outter Dome 2 – Main load bearing dome „Witches Hat“ 3 – Round Inner Dome

2.14 Capitol Dome Domes of St. Peters and Santa Maria del Fiore draw a limit to how big masonry domes can be. To build bigger, greater domes required some radical and new technologies. Change was brought around in 1855 by architect from Philadelphia Thomas Ustick Walter. He made a proposal of a new dome for United States Capitol entirely constructed out of cast iron. Looking from the distance one is easily tricked into thinking that dome is made of stone, but all the ornate decorations are casted iron pieces shaped to look like delicate stone carving.

Capitol was the first iron dome of that size, 29 meters in diameter and 88 meters high. Like its predecessors Capitol dome is double skin construction.

First inner dome rises up to 55 meters. Behind it 36 curved iron ribs hold exterior dome. Even though inner dome is

Fig. 22 U.S. Capitol Dome http://www.digitalimages.net/Gallery/Scenic/AsstScenic/Was hDC/washdc.html

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self supporting it is still fixed to the ribs. Casted iron trusses and corbels support dome structure and transfer forces to masonry walls. Being made of iron it was thought to be a dome that will not crack, but time proved otherwise. In 2012 Capitol dome had 1300 known cracks. In fact dome is in such bad condition that it desperately need repairs. Some of the cracks are so severe that water seeps straight down and began harmful corrosion process.

Nevertheless casted iron proved to be a universal material, it allows us to create almost any size and shape pieces, that can be put together like a jigsaw puzzle. This means that construction can be erected much faster than its masonry counterparts.

(Steinhauer, 2012)

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Fig. 23 Section of Capitol‘s dome (by Povilas Mikuta)

2.15 Montreal Biosphere Last stop and the most remarkable one in the history of dome building is 1967 World Fair in Montreal. United States pavilion designed by Buckminster Fuller became a key piece of world fair marking the new age of geodesic domes. Based on theory of synergetics Montreal Biosphere is a geodesic dome. Shape of the dome is based on class 1, frequency 16 icosahedron made from interconnected steel rod triangles forming a round structure. Cover of the dome Fig. 24 Montreal Biosphere http://unusualwas made out of acrylic plastic panels fixed architecture.com/montreal-biosphere-canada/ to the three dimensional space framework.

This type of light construction allowed building a great clear span sphere that dwarfs any dome build before it. Diameter of the biosphere is 76 meters and it soars to a dizzying height of 62 meters. To manage indoor temperature a system of shades was installed. It was truly a daunting structure of its time even though it suffered from some basic weaknesses domes faced at that time, specifically leakage. Original structure did not last long, in 1976 during renovation works an acrylic skin caught fire from welding and burned away. Load bearing geodesic steel structure remained intact, but nevertheless dome was closed to the public for 14 years until it was bought by Canadian Department of Environment, renovated and turned into museum. (Baldwin, 1996), (Khan, 2009)

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3. Functionality 3.1 Statics To be able to build and use domes in the most advantageous way possible, first of all one has to analyze its statical capabilities and force movements. As mentioned earlier dome was inspired by classical building component - arch. Dome is nothing else if not an arch rotated 360o around its Z axis and forms rigid, clear span round structure. This means that forces distributed throughout dome are acting in a very similar way as they do on arches.

Arch is based on the force of compression acting throughout the arch and transferring the load to load bearing structure, like walls, columns and piers. In some masonry arches more load means more compression, stones are pushed together much more closely and therefore structure becomes incredibly rigid. Same moment when arch is exposed to vertical load a horizontal reaction appears near the bottom of the construction trying to spread it apart. This horizontal thrust is closely related to the shape of the arch.

High arches or pointed arches are more stable since vertical load is transferred more directly therefore horizontal thrust is reduced. When arch gets shallower horizontal thrust will increase proportionally. To stabilize the structure abutments or flying buttresses were usually added on the sides of an arch to counteract the horizontal action.

Forces in the dome are acting very similarly. Throughout the dome vertical forces (loads) are acting exactly the same as in arches.

When dome is exposed to vertical loads two types of forces appear in the structure. Along the whole length of dome’s generatrix3 meridian direction forces act similarly as in arch. Along this direction there is no bending moments in the dome because in addition to meridian forces dome is also taking ring forces. Ring forces act Fig. 25 Forces in the Dome (by Povilas Mikuta) perpendicularly to meridian ones and create a compression in the top part of the dome and tension in the bottom part. Therefore bottom of the dome spreads just like an arch and needs to be stabilized, otherwise risk of cracks and possible collapse is very real. As history shows these ring forces (or so called “Hoop stress”) caused most of the troubles to the architects and builders in the previous centuries.

Now let’s go a bit deeper into the maths: Resultant of vertical load - F Meridian forces - NΦ Ring forces - Nθ In any point of horizontal section of a dome

Fig. 26 Mathematical Sketch (by Povilas Mikuta)

3

Generatrix – in geometry generatrix is a cruve that, when rotated about an axis, produces a solid figure (en.wikionary.org/wiki/generatrix)

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NÎŚ and Nθ are constant. So from balance formula ∑ = => = ∙ 2 Radius of horizontal section of the dome =

This mean that meridian force in any point of the dome is:

= /2

If there is an even load on sphere surface w (e.g. N/m2):

= / 1 +

Meridian force in any horizontal section: = −

1 +

1 +

(Razaitis, 2004)

3.2 Geodesic domes and synergetics Domes made out of rods or struts are one of the most economical ways to enclose large areas and save on materials. Domes’ framework can be made out of steel, aluminium or timber giving it a lot of design possibilities.

First, traditional spherical domes were made out of arches connected by concentric 1 circles that ensure its stability. This sort of system is very hard to solve statically and evaluate common interaction between different frame members.

Improved dome, with diagonals which divide space between arches and circles into triangle fields is called a Schwedlers’ dome.

2 Greatest Schwedler’s type dome is built in Charlotte, USA its diameter reaches 101.2 meters, height 34 meters.

This construction was simplified when net dome was designed. Net dome differs from previous two by absence of arches.

They are changed by rods forming triangles 3 which connect concentric circles. However 27 Dome Types (by Povilas Mikuta) there is a downside. Net dome requires a lot Fig. 1 – Traditioanl Sperical Dome of different framework pieces which in turn 2 – Schwedlers Dome makes construction more complicated and 3 – Net Dome expensive.

Biggest difference in this field was made by Buckminster Fuller who came up with geodesic dome. Basis for this shape was icosahedron placed in a round sphere. This way 21

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it is possible to build dome consisting out of 20 equilateral triangle faces. Angle between triangle sides is equal to 72o. At this point every triangle can be subdivided into even smaller equilateral triangles. When icosahedron faces are projected into surface of the sphere, triangles end up a bit curved therefore we get a triangle where sum of corners is equal to 216o. This number does not really fit with normal geometry where sum of triangle corners is always equal to 180o. What Fuller did was Fig. 28 Spherical Triangle realizing that main net of triangles creating sphere can be made (by Povilas Mikuta) by placing six usual (flat) equilateral triangles around one point creating hexagon. Since hexagons cannot fill whole round surface of a sphere twelve pentagons were also included and along with main hexagons create a perfect round structure. This invention leads to standardization of dome construction elements, connection methods, details allowing building domes faster and easier than ever before by making them perfect subject for prefabrication. (Razaitis, 2004) Now we arrive at the doorstep of synergetics. To be fair in the beginning of the research process I had no idea about the extent of this subject. It is subject for completely separate and much bigger report or even a book. Two volumes published by Buckminster Fuller about synergetics sums up to 1300 pages. But in order to comprehend principles of geodesics and wide dome possibilities for the future it is necessary to at least take a short look at this topic.

Synergetics is a combination of words synergy4 and energetic. Synergy – is described as behaviour of the whole - not investigating and stipulating the performance of separate parts or any other subunits of a component. (J. Baldwin 1996 p.68) Energetic is a reference to motion in geometry, since every little thing in the universe is in some sort of motion. This means, that statical thinking in Cartesian coordinates X Y and Z are far from accurate. There is no motion, no time included. So, energetic geometry is 4D, it has an additional dimension included – time/ frequency. Time takes a part in everything, because all actions have duration.

Fig. 29 Synergetics in Geometry (by Povilas “Shortest distance between two points is TIME”.

Mikuta) In addition to whole system of synergetic is based on economy, because all elements in nature are created in the most economical way and since triangle is the most economical shape of geometry it became basis for synergetics. 90 degree coordinate system becomes obsolete, because of its statical quality and therefore synergetics employ much more dynamical - 60 degrees coordinate system. (Just like corners of equilateral triangle are all equal to 60 degrees).

4

Synergy - interaction of multiple elements in a system to produce an effect different from or greater than the sum of their individual effects. (en.wikipedia.org/wiki/Synergy)

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Perhaps the easiest way to communicate what synergetics is all about is examples.

I couldn’t help it, and borrowed some examples specified by Buckminster Fuller, that illustrates what it is all about. One of the examples is an alloy made of chrome nickel and steel. Its tensile strength is way higher than the sum of the tensile strengths of its components. This means that combination of such elements has a much bigger bonus than one would normally expect. Fuller’s theory said that if synergetics are working in chemistry it should work in geometry as well and geodesic domes were a proof.

Working principle of synergetics can be explained by an example involving six identical struts. From six struts it is possible to make two equilateral triangles, but if we arrange these triangles synergetically we get a tetrahedron. So instead of two triangles we get a 3D shape made of four triangles and containing volume. Struts and their number remains the same, just their relationship is changed and this gives an incredible Fig. 30 Radiolaria http://en.wikipedia.org/wiki/Radiola efficiency. ria Nature is the most creative user of geodesic patterns. Some vital parts of anatomy like eye ball and reproductive system parts are based on geodesic patterns. Numbers of tiny deep sea creatures, like zooplankton, appears as if it should be crushed by huge water pressure. However when geodesic pattern is employed these small creatures can function under very dangerous circumstances without breaking a sweat. (Baldwin, 1996), (Fuller, 1997) 3.3 Tensegrity Tesegrity is again a brain child of Buckminster Fuller. Fuller liked to say that “Universe is islands of compression in a sea of tension”. Examples start from broad picture of space. Planets are the islands of compression in a sea of gravity. Moon always stays same distance from earth because of gravitational force, just like it would be connected to earth by a tensional cable keeping that exact distance. Even smallest particles in the universe are made based on that exact same principle. If one would take a look at atoms and molecules closely enough it would be obvious that they are arranged in Fig. 31 Tensegrity model invented by Kenneth Snelson (by Povilas Mikuta) the same gravitational model, relatively spaced away from one another. “It is all energy, ordered by angle and frequency”. (J. Baldwin 1996 p.75) The reason why this theory was formed has a simple structural answer. Tension is more efficient way to transfer loads than compression. Structural component taking direct tension are more effective then ones taking direct compression forces, which in turn are more effective than components taking bending actions. Reason why compression and bending structures like columns and beams are less efficient is that their failure usually 23

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appears in a form of deformation like buckling. These deformations generally appear way before load bearing capacity of the material is reached. On the contrary tension structures generally fail when bearing capacity of the material or joints is over exceeded. (O. Popovic Larsen, A. Tyas 2003 p.46) Fuller recognized this efficiency and suggested that mankind should follow nature and as well use most economical ways to do things. Thus quest for tensional forms which takes continuous tension rather than compression began. Proof that this theory is real and can be employed creating physical shapes was presented by one of Fuller’s students - a sculptor Kenneth Snelson. He created an icosahedron from six struts being held together by cables. It is interesting that most economical way to keep structure together (tensegrity) from the very first attempt resembled a geodesic pattern, again one of the most efficient ways of using material. It seems that geodesics and tensegrity are working together intuitively.

I took the liberty of reproducing a copy of first tensegrity model and the advantages of it were quite fast to see. The rubber threads position struts and by the help of tension keeps the shape of icosahedron. This model stands out by its resistance to deformations. If take any two struts and squeeze them together whole structure will deform symmetrically and will maintain relationships between different elements. Same thing happens if any two struts are spread apart, and the same moment you struts are released whole structure goes back to its original shape.

However even though mankind finally learned about tensegrity and started investigating its working principles it seems that nature has already beaten us in the race of creating structures based on these principles. Natural world has once Fig. 33 Tensegrity Spine more proven itself to be most creative inventor. As it so happens http://www.biotensegrity.com/tense grity_truss.php nature always takes most economical way, in this case it is employing tension as a main force in many forms. Lately tests have shown that some biological cells exhibit similar tensegrity qualities like the model mentioned before, especially the ones relating to deformation. Many biological structures can take a great deal of deformation and then come back to its original state. Moreover, recent studies are implying that even human spine shows signs of tensegrity. If old fashioned model of spine acting similarly like an architectural column would be true, humans could barely lift a thing without breaking a bone or tearing muscles. Actually spine is acting more like an extraordinary beam made out of semi-rigid structural components connected by flexible connective tissue. If one of the semi-rigid elements Fig. 32 Tensegrity dome frame (by Povilas Mikuta)

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(in case of spine it is vertebrae) is pushed in a different position it affects the whole structure which deforms to accommodate that action and will be able to go back to its original position afterwards. (Levin, 2010) One of the most remarkable implications of tensegrity construction came in a form of Georgia Dome. It is a largest cable supported dome in the world, with the surface area of 34 800 square meters. The basis for the dome starts from humble circus tent. Struts are taking compressive load of the roof and transferring them to the poles. In the case of the circus tent poles are usually reaching the ground and acting as columns, but in the case of tensegrity dome columns are cut in order to make unobstructed open space underneath the dome. At this step tension cables are attached to the bottom of the poles and pulls equally in all directions, thus stabilizing the construction and transferring roof load to heavy load bearing walls of the stadium. (Baldwin, 1996) 3.4 Build ability One of the most important issues with any building is build ability. This subject largely depends on materials chosen for the construction. Domes and dome houses are no exception.

Generally we can divide domes into two categories regarding the build ability: ones made out of structural framing covered with cladding and monolithic domes.

Basic material for constructing monolithic domes has not changed from ancient Roman times. Concrete was and still is material most suitable for monolithic domes. Of course composition of the concrete has changed a bit during the centuries, but idea still remains the same. With conventional reinforcement, old fashioned monolithic dome design can reach unprecedented strength. However this type of domes presents quite big self weight which in turn increases the outward thrust at the bottom of the dome. These forces most often are counteracted by placing a ring beam around the bottom edge of the dome.

Alternatively a continuous abutment might be placed around the base dealing with the load. Statically monolithic dome seems to be a bit more straight forward, but it comes at a price. To construct such a structure a complex formwork is required to support concrete before it is stable enough to take its own weight. Since most of prefabricated formwork systems are only suitable for rectangular structures, the task of building formwork for such structure will be a job for carpenter. Another possibility for concrete dome is prefabricated elements. The first prefabricated concrete dome was built in Los Angeles in 1963. Due to its unique construction the price of the building was relatively high, but construction itself was finished in a record time – 16 weeks. This is a proof how well geodesic domes fit prefabrication. Principally dome is constructed from prefabricated, if necessary prestressed concrete panels, which are connected and fixed with bolts. Problematic part of such construction is joints. Usually to ensure stable water runoff elements are constructed in a manner of overlapping joints. Additionally elastic sealants and flashings are placed to thoroughly cover the joints. Nevertheless any elastic joint is a part most likely to be damaged so constant inspection and maintenance of such building would be essential.

(Baldwin, 1996)

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Frame domes have much more freedom when selecting materials. Most popular choice for frame is steel, alternatively aluminium or timber can also be used. Depending on the shape and steepness of the dome timber members can be supplemented with stainless steel cables. This method is very good for economy because it allows lighter structure and force flow towards more suitable material. Frames can be supported in many ways starting with column and beam systems and ending with simple walls. Just like in the case of monolithic domes a ring beam has to tie the base of the dome to counteract its outward thrust. However since frame structures are generally lighter the thrust will not be as big as in monolithic construction.

Depending on the material chosen to cover a dome, node connectors on top of the frame structure might be necessary. Node connector is fixed on top of the frame and hermetically seals connection between separate triangle cover pieces. If a single material is used to cover the dome a system of brackets and cleats might be used economically to secure the cladding. This enables cladding to be fixed directly to the frame structural components. Another type of node is used when connecting the structural frame members.

Generally one shape of node is used when erecting the frame, possible different type of nodes can be used at point where frame ahs to be fixed to load bearing walls or columns.

One of the most famous steel and glass space frame systems “Mero” have been created in 1940s and with continuous improvement is still used today. Traditionally a spherical node with holes to accommodate structural members was used when building geodesic domes. Although in the recent years “Mero” has improved this system by making nodes polygon or square shapes. This improvement allows more freedom when accommodating different angles thus enabling construction not only domes, but as well fascinating space frame structures. All these systems are more often used when building public spaces, a system used to build domestic dome dwellings is a bit less sophisticated and resembles traditional wooden house construction.

Usual choice of structural frame material for these houses is wood. Thick wooden pieces form a rigid shell which is secured with sheets of plywood from the outside, just like normal wooden wall. Insulation is placed in the spaces formed by the frame followed by vapour barrier and interior lining. Next step varies from the company designing the dome. a) Often distance strips or other type of spacers are placed on top the plywood to form a ventilated airspace, which prevents moisture damage. Then follows layer of plywood and two layers of bitumen felt. Common mistake in the practice is stopping waterproofing layer too high. Unlike regular building almost all surface of the dome should be considered as a roof. Further construction depends on the type of exterior finish chosen. b) Some frame solution accommodates openings in the frame. In this case insulation is placed in such a way that air cavity would form itself close to the plywood allowing air to travel from the base of the dome and escape at the top. When this solution is used waterproofing two layers of bitumen felt or other type of membrane is placed directly on the first layer of plywood (windboard).

After this step is completed a further construction of exterior finish can take place in a regular manner just like it would be an ordinary roof. (Natural Space Domes, 2005), (Baldwin, 1996) 26

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3.5 Surface Area – Square meters – Volume Case No. 1 I will start the example by picking a radius of dome and then go into calculations. So let’s say that radius of our dome is 5m. With the help of some simple mathematical formulas volume and surface area of the dome can be calculated fairly easily.

Surface area of the half sphere (dome) is calculated: = 2 R – radius of a dome (5m)

!"# $%&'()#

= 2 ∙ ∙ 5 = 157,

4 4 - = 0 => - = ∙ ∙ 50 6 6 = 261,67 ,0 Base of a dome is equal to: 2($# = => 2($# = ∙ 5 = 78,5 ,

Fig. 34 Dome Case No.1 (by Povilas Mikuta)

Because of the sloping sides of the wall the very edge of it might be hard to use so I have narrowed the diameter of the dome to 9,488m, area where ceiling height is higher than 1,5m.

Area of usable dome: 2($# %$(24# = => 2($# %$(24# = ∙ 4,744 = 70,67 , Total surface area: $%&'()# = 157 + 78,5 = 235,5 , Now we can take a conventional house resembling square box with an area of ~78,5m2 and calculate surface area to compare the figures. In today’s architecture a square box is the most efficient shape so comparing these figures can be quite interesting.

If we presume that one side of the box would be 8,86m this would give us an area close to our sample dome: $7%(&# = 8,86 ∙ 8,86 = 78,499 , . Assume that box is approximately 3,2m in height and with a flat roof. With roof construction thickness it will leave a proper ceiling height inside. So surface area of the building would be:

Fig. 35 Square House Case No.1 (by Povilas Mikuta)

$9 #$ = 8,86 ∙ 3,2 ∙ 4 = 113,408 , $%&'()# = 113,408 + 78,499 ∙ 2 = 270,406 , 27

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- = 78,499 ∙ 3,2 = 251,2 ,0 Comparing these two cases advantage of the dome construction is clear to see.

Dome has around 13% less surface area losing heat and at the same time encloses around 4,2% more volume. However one must keep in mind that usable dome floor area is a bit smaller than the total one.

Case No. 2 This time I will choose a dome diameter of 7m, this should give a possibility to explore options for a two storey building.

Dome diameter - 7m.

Distance between ground floor surface and 1st floor surface – 3m.

2 ∙ ∙ 7 = 307,72 ,

!"# $%&'()# =

4 - = ∙ ∙ 70 = 718,01 ,0 6 2($# = ∙ 7 = 153,86 , 2($# %$(24# = ∙ 6,8 = 145,2 , r – radius of 1st floor

Fig. 36 Dome Case No.2 (by Povilas Mikuta)

7 = 3 + : => : = √40 = 6,32 , <$= >4!!& = ∙ 6,32 = 125,42 , <$= >4!!& ?$(24# = ∙ 5,3 = 88,2 , Total area: 153,86 + 125,42 = 279,28 , Total usable area: 145,2 + 88,2 = 233,4 , Total surface area: $%&'()# = 307,72 + 153,86 = 461,58 , If we take 11,82m as side dimension of the building we should end up with similar total area spread in two storeys. Assumption is that height of the building is 6m. = 11,82 ∙ 11,82 ∙ 2 = 279,42 , $9

11,82 ∙ 6 ∙ 4 = 283,68 , $%&'()# = 283,68 + 139,71 ∙ 2 = 563,1 ,

#$ =

Fig. 37 Square House Case No.2 (by Povilas Mikuta)

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- = 139,71 ∙ 6 = 838,26 ,0 This example shows something quite interesting. Surface area of the cube is around 21% bigger than the one of the dome, but the volume of the cube is 16% bigger.

This happens because of the dome shape itself, were space around the sides is not in full height, therefore one can argue that square meters of these buildings is not directly comparable. This leads us to the next experiment comparing structures with similar usable area.

Case No. 3 As a basis case number two dome and figures will be used.

Total usable area: 145,2 + 88,2 = 233,4 , Total surface area: $%&'()# = 307,72 + 153,86 = 461,58 , @

Total volume: - = A ∙ ∙ 70 = 718,01 ,0 If assuming that side of new square building is 10,8m it will give a square meter area spread in two floors close to usable square meter area of the dome. Height of the box remains the same 6m. = 10,8 ∙ 10,8 ∙ 2 = 233,28 , $9 #$ = 10,8 ∙ 6 ∙ 4 = 259,2 , $%&'()# = 259,2 + 233,28 = 492,48 , - = 116,64 ∙ 6 = 699,84 ,0 This case comes closer to a traditional geometry Fig. 38 Square House Case No.3 (by theorem, that dome encloses biggest possible volume Povilas Mikuta) with smallest surface area. Surface area of the dome 6,3% smaller and volume is 2,6% bigger. Then again total dome floor area is much bigger and it is just a matter of using it in a smart way.

On the other hand all these numbers do not show the whole picture. It is not always the most economical box that is the complete shape of the building, pitched and hyped roofs as well have an impact of the economy. If the building envelope is enclosed by floor, exterior walls and flat ceiling, than heat loss area matches the ones calculated in the examples, but roofs shape itself might affect the total area of the building. This means that traditional building shapes might need much more materials than a dome, where heat loss area and material area always matches.

3.6 Floor plan layout Dome was and still is an amazing architectural form. As history shows in the western culture domes were most often used as a message of grandeur, enclosing vast open spaces. But when it comes to building a structure for housing people for providing 29

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workspaces, it presents some fundamental challenges. Proposing a floor plan for a building of such shape can turn into very difficult task. Nevertheless dome houses have small, but sure group of enthusiasts, who are convinced that this issue can be tackled in a professional manner. It is obvious that curving outer walls of the dome causes the trouble when planning a layout of a dome house. Often rooms are planned around the connecting space in the centre of the dome which acts as an access point to all rooms and possibly staircase leading to the top floor. Centre space sometimes blends with an open plan livingroom mimicking standard modern design.

Another a little bit bizarre situation is created by domesâ&#x20AC;&#x2122; round perimeter. Usually internal partition walls are angled towards the centre of dome basically eliminating 90 degree corners. Because of this most of the furniture is lined along the middle portions of internal walls, leaving a lot of hardly usable space. Occasionally one, in rare cases two, 90 corners are formed by the internal walls making furnishing a bit easier, but it is just a couple rooms that might have this quality. As calculation of usable area revealed, dome shell in the ground floor is not curving inwards very fast. Only 10-15cm distance inwards from the shell creates an area where ceiling height is less than 1,5m. Also there is a possibility of lifting dome a bit higher on a cylinder, allowing base of the dome to go straight up and only later start to curve like a dome making things a little bit easier. On the other hand cylinder solves only one problem, provides a bit more space of suitable height, but by still exterior wall is being curved horizontally and does not allow a usual furniture layout. Even with such drawbacks some dome houses still prove that furnishing of a ground floor can be carried out without compromising comfortable living conditions.

Similar problems with layout occur in the first floor of the house, if one is constructed. Although with ever rising height of the dome steepness of the dome shell increases dramatically, thus making space along the shell on the 1st floor much more difficult to utilize properly. Traditionally 1st floor is more private, accommodating bedrooms and bathrooms. Where furnishing bedroom would not be that difficult and different from ground floor spaces, bathroom can turn to be some challenges. Planning of the installation shaft in such building would be of utmost importance. Because of the sloping shell walls, most convenient spot to place the shaft is closer to the centre of dome where it can go up in a straight line. Ventilated pipes will be a concern as well. Since dome walls are curved, hiding a ventilated pipe in the construction will require many pipe connections to be made in order to reach the top of the dome for proper ventilation.

Natural air flow in the dome is one of its greatest advantages, but in order for it to function properly an opening from bottom to the top in the centre of the dome is necessary.

When it is not hard to achieve this in a single storey dome, a two storey dome is a bit more difficult. An opening in the middle of the 1st floor is necessary. This takes away most easily adaptable part of the floor area. Some designs exploit this for placing a staircase and a small access balcony on the 1st floor providing access to the rooms. While in other cases a ventilation channel in the middle of the dome is created and hidden in the construction of interior walls. For sample plans see appendix A. (Natural Space Domes, 2005), (Econodome, n.d.)

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4. Advantages and disadvantages 4.1 Material Savings As previously established in the section dealing with surface area and floor area relation, dome is the most economical shape that can be used in architecture. Generally total surface area (envelope) of the dome is around 12% - 14% smaller when compared with a rectangle box. However when it comes to material savings dome have much more strong points.

Box shape buildings with flat roofs are a tradition for southern regions where rainfall is fairly limited and evaporates quickly, but in other parts of the world where locals have to account for heavy rainfall and snow, flat roof still is not a feasible solution. So in many parts of the world pitched and hyped roofs are adopted as traditional forms. Even with contemporary technologies of water proofing these roof shapes are still quite popular, especially for smaller buildings. Therefore an actual advantage on material savings can be a tricky thing to express in numbers because it is strongly related to any other possible shape of the construction.

However there is one aspect of dome structure which gives a decisive benefit over almost any other form of building. Throughout its whole shell dome is a self supporting structure, eliminating any need for supporting interior structures, saving vast amounts of material. Since materials used for structural support are quite expensive, this is not only an environmental advantage saving material and CO2 emissions, but money and time spent on the building site is reduced effectively. Although the actual benefit may vary depending on whatever there is an upper foor(s) or not, since floor partition itself might need some structural support on the inside of the dome.

Unfortunately there is a serious downside when building a dome house using common materials like wood, gypsum sheets etc. All of these products come prefabricated in rectangular pieces, while dome is most often planned in triangles. This provides a problem when trying to cover dome shell with these materials. When triangle pieces are cut out of rectangular sheets it creates a lot of waste, cut-offs are in shapes that cannot be utilized in a further construction. That is the reason for consideration of what materials to use. For example fibreglass can be ordered in any shape necessary. If using some sort of metal covering for the building, some waste material will occur, but certainly not that much plus it can be recycled completely. This should be taken into consideration very carefully by the designers. There is no point to choose dome as material saving structure if in the end you will produce much more waste. (Baldwin, 1996) 4.2 Ventilation and indoor climate Good indoor ventilation is incredibly important issue when evaluating buildings.

Dome houses have two different air flow models for winter and summer.

During the summer months dome shell is relatively warm therefore air near the dome walls gets warmer and rises up, mixes at the top of the dome and starts setting down, creating circular air flow.

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In winter dome shell is relatively cool and therefore air circulation is just simply going other way around. Warmest air in the centre of the dome goes up, hits the shell and starts setting down by cooling along the exterior walls of the dome.

Fig. 40 Dome ventilation winter mode (by Povilas Mikuta) This creates a very efficient air flow for both winter and summer modes. Additional advantages lay within smooth shape of the dome.

Hence traditional rectangular shaped buildings and rooms has some serious drawbacks. First issue is “dead areas” around the room corners. Air always move in the easiest way possible that is why Fig. 39 Dome ventilation summer mode (by Povilas Mikuta) areas close to sharp or 90 degree corners experience very slow air change or none at all. In dome buildings this problematic subject can be reduced significantly by utilizing smooth shape of the dome. If this subject is well considered during the design phase, no obstructions are placed blocking air circulation, dome house will provide superior natural ventilation effect than any conventional rectangular building.

Second issue – when air gets warm it becomes lighter (simple physics) and therefore the warmest air in the room gathers near the ceiling, where it has no use for persons inside the building. Thus a lot of energy is wasted on heating air which does not really make you feel warmer. This effect is proportional to the height of the ceiling – bigger room height means bigger temperature difference between ceiling and floor surfaces.

Even though domes have very good air circulation quality, still it would be nice for air to start setting down before it cools too much. If there is no upper floors natural air circulation can be effectively accelerated in the dome house by installing a ceiling fan. This option is strongly recommended when radiant floor heating is used, because air flows are affected along whole floor area, heating the air and forcing it up. Additionally there is an option used in “Natural Space Domes” kit houses, an air intake duct installed at the top and connected to an air handling unit maintaining air circulation. All of these systems have to be evaluated according to the height of the dome. (Natural Space Domes, 2005), (Baldwin, 1996) 4.3 Cooling effect This phenomenon was first observed by Buckminster Fuller. When experimenting with one of his projects he noticed that inside thin skin steel dome air temperature was quite cool despite the high temperature and sun outside. A few tests with smoke were conducted and surprising result revealed itself. Interior air was being sucked out of the dome through the air vents at the base of the dome while large amounts of fresh air were 32

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introduced through a relatively small opening at the very top.

This seemed as if air in the dome is acting against the laws of nature. Fullers’ conclusion stated that white painted dome and light coloured surroundings reflected sun heat causing air to lift which in turn creates low pressure zone at the base of the dome. This effect forces the air Fig. 41 Dome cooling effect principle (Summer mode) by Povilas Mikuta in the dome to flow through the base air vents lowering pressure inside the structure. Like stated previously as hot air goes up it gradually cools down, this means that air circulating outside the top of the dome is cooler than in lower levels. Hence low interior air pressure drags cool outdoor air through an opening at the top. Opening acts like a funnel speeding an air flow. As the air enters the dome it expands rapidly and cools the dome in process similar to the one taking place in refrigerator’s cooling coils. Tests have shown that this phenomenon makes interior temperature at least 15% lower than ambient temperature outdoors. This works in both dry and humid climates, insulated and uninsulated constructions. Example of such construction was demonstrated in Kumasi, Ghana in 1943. A thin aluminium dome was made according to the principles of self cooling without any fans, coolers or any other machinery. Despite the fact that temperature outside was incredibly high, there was no air conditioner, people visiting the dome complained that it was too cool inside! (J. Baldwin 1996 p. 116) 4.4 Heat loss optimization Many sources state that dome buildings have 30% less surface area than conventional buildings and therefore energy consumption for heating and cooling drops down by a third. Well this is just partially true, there is more tricks of the trade behind this statement.

As calculated in section about surface area and square meter relation, 30% ratio is only true when comparing summed area of exterior walls and roof leaving out the floor slab. But when it comes to heat loss this is not a full picture. Any building also looses heat through ground slab so it should be included in the comparison as well. Depending on the arrangement of the building – number of storeys and ceiling height, total surface area ratio can vary between 15% - 25% in favour of the dome. Better utilized dome volume with upper floors and mezzanine platforms will produce more favourable ratio.

Surface area ratio is not the only advantage of dome shape. Simple geometry states that 2D circle encloses largest possible area in smallest perimeter, this means that foundation perimeter in the dome is severely smaller than in its rectangular counterparts which in turn reduces linear loss around the foundation.

Another great advantage of energy savings can be achieved by building framed geodesic dome. For example wooden frame geodesic dome contains around 40 – 50% less framing material in roof and walls than conventional wood frame building. And even 33

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though wooden stud in the wall is not usually considered as a thermal bridge its λ-value will never go close to the value of construction containing large amount of insulation. This way approximately additional 5% of heat loss can be avoided. (Natural Space Domes 2005 p.7) However this argument depends on the frequency of the dome, higher frequency means more structural framing members which then increases heat loss.

Furthermore smooth dome shape contributes to energy efficiency in a unique way, which most architects and engineers do not even dare to guess. Conducting his experiments on Dymaxion projects Buckminster Fuller realised the importance of aerodynamics in architecture. In other words “building’s heat loss is in direct proportion to its aerodynamic drag”. (J.Baldwin 1996 p.112) Every building has a small invisible layer of air clinging on the construction from the outside, this air mass contributes to buildings energy use. But winds hitting vertical house walls destroy this additional layer of insulation.

Dome with its unique smooth shape does not experience high wind loads and vortexes, it just simply let the wind flow around it, thus minimally disturbing the heat holding boundary.

Some scientists compare this to human and animal behaviour as well. When it is cold, dogs instinctively hunch down into fuzzy ball, minimizing their surface area and aerodynamic drag. Additionally walls exposed to direct winds experience high pressure zone, while walls downwind are exposed to low pressure vortexes. Low pressure destroys heat retaining layer and sucks out warm indoor air through the smallest openings or construction flaws. Air loss inside is replaced by cold outdoor air being pushed trough similar construction flaws on the high pressure side. Today technology and construction practices are quite far advanced, but it is still naive to hope that there will never be any mistakes in the construction. Moreover with time passing, naturally more problems will reveal themselves. And still aerodynamics in architecture is a bit of a grey area. Engineers calculate wind loads and similar values to make sure that building can withstand it, but no considerations of aerodynamics influence on energy use is really being made.

These issues are giving us a bit clearer picture of what is actually going on with dome building and why are they so energy efficient. To prove dome energy efficiency some test studies have been carried out.

An anonymous dome house owner conducted an experiment comparing temperature inside and outside the dome. In order to get an accurate baseline figure no other activities were taking place inside the house (like cooking, running equipment, anything that generates heat). While test was being conducted dome was barely finished, some insulation was still missing. Data was taken throughout the week 7th – 12th of March 2002. Week started sunny then turned overcast and wet and finished sunny again. On average dome temperature varied by 7,5 degrees, while outside temperature varied by 20 degrees. Test conclusion was quite interesting, in a cold weather solar gain was able to increase dome temperature by 5 degrees or more. However when weather turns overcast temperature inside the dome starts dropping slowly, and consistently. Ironically if the weather gets a bit warmer, but there is no sun, insulation keeps the air inside of the dome cooler than outside. Moreover additional test showed how easy it might be to increase temperature inside the home. If cooker was set on slow cook for a few hours temperature inside the dome increases by 12 degrees. During the winter of 2005 – 2006 a 400watt halogen heater was only needed to warm the interior if the weather was overcast. The 34

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most picturesque example was in the end of the post about the test: “I have just checked my temp gauge tonight and it reads 22,8 degrees inside the dome and 3,5 degrees outside, that’s a difference of 19,3 degrees, the only heating in the dome is me, my two dogs and the heat from this computer!“ (http://www.geo-dome.co.uk/article.asp?uname=temptest) Even though this is not an official scientific test it gives and interesting insight into the domes’ efficiency and lies foundation for further tests and possible thesis.

More reliable studies have been carried out by U.S. Department of Energy in the state of Florida. Study compared a dome produced by American Ingenuity Company, featuring concrete – steel element with 7 inches (approx. 178mm) of EPS insulation with conventional house and energy efficient Styrofoam house designed by Dow Chemicals.

Test showed that dome building far surpassed both counterparts. In the summer time dome energy savings reached more than 36%, in winter it exceeded 42%. With thicker insulation layer of northern climate buildings we can make a conclusion that savings would be approximately the same. Most likely a bit bigger savings in the winter and a bit less in the summer time, giving the idea that air conditioning is not as necessary in the north.

Blow door test showed that dome is 56% tighter than conventional house and 29% than Styrofoam, Dow Chemical building. (http://www.aidomes.com/us-department-of-energystudy/250/article) Blow door test partially proves that aerodynamic theory on spherical shaped buildings is actually real and important factor.

4.5 Hurricane and Earthquake resistance During their history domes have proven themselves to be able to withstand the most brutal conditions nature has to offer. The biggest advantage of geodesic domes has been proven to be their strength.

In fact geodesic dome is strongest structure per kilogram of material employed.

Previously established aerodynamics of a dome also plays crucial part when it comes to high winds of hurricanes and tornados.

Fig. 42 Dome after withstanding Hurricane Ivan Actual examples have proven how http://domeofahome.com/dome-information/hurricane/ durable dome can be. In 2004 hurricane Ivan hit the United States causing damage for 18 billion dollars. Pensacola beach in Florida was caught in the direct way of the hurricane. While high winds and floods destroyed much of the beach houses, almost levelling them to the ground the dome house escaped structurally undamaged. In fact during the worst part of the hurricane MSNBC news crew was staying in the house along with the owners reporting about the situation on the beach.

Finally when broadcasting equipment was damaged so much that it ended up inoperable most of the MSNBC crew fell asleep! In a category 4 hurricane that was an astonishing revelation on sense of safety inside the dome. When storm has lifted and it was possible to inspect the damage it was clear to see that dome escaped virtually undamaged comparing to other buildings in the area that had been completely blown off the foundation. Some damage was made by the flood in the ground floor, and parts around staircase have been 35

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torn off the building, a few leaks have appeared through the windows due to strong winds, but other than that not even floating debris hitting dome walls in full force like medieval battering rams were not able to damage structural components of the building. With some main repairs it was easy to make dome home liveable once again. (http://www.monolithic.com/stories/feature-home-doah) Geodesic domes have proven their robustness under the worst conditions nature has to offer. An aluminium geodesic dome was a main feature of American South Pole research station. It stood there for more than 25 years exposed to cold high winds and enormous snow loads.

Structure was successfully adopted by U.S. Military over forty years ago as a basis for defensive early warning radar system, mostly deployed throughout the northern Alaska were no other structural shape was able to withstand such harsh conditions. In fact inexperienced Innuit workers were able to erect defence radar dome under terrible conditions in the time period of fourteen hours while similar domes designed by Buckminster Fuller were troubling seemingly professional workers in the United States.

After the end of cold war Distant Early Warning radar line became unnecessary and therefore abandoned, even though it did its job from 1957 till 1996. (Lajeunesse, 2008) In 1964 similar system was adopted by the Japanese constructing weather radar system on the very top of Mount Fuji. At the time it was worldâ&#x20AC;&#x2122;s highest weather radar 3776m above the sea level. Radar was exposed to rough winds and had to be assembled extremely quickly because summer period up in that height is very short geodesic dome once again was chosen as the strongest and the most economical option. (Baldwin, 1996)

Fig. 44 Mount Fuji weather radar

Fig. 43 DEW Radar line

http://www.mitsubishielectric.com/brief/sp/history/1960s/index. html

http://www.journal.forces.gc.ca/vo8/no2/lajeuneseng.asp

4.6 Construction Complexity Is the dome a complex structure to build? Well this highly depends on the conditions were it is being build, climate and choice of materials. Ironically history shows that dome like dwellings made by Indians and Eskimos in North America, some Asian tribes are very simple, fast to build and meets its functional demands, in other words it is far from being a complex structure. However modern adaptations of big spanning masonry domes, geodesic dome houses increased construction complexity and troubled the builders. Different construction types of domes carry different problems to be solved.

Fig. 45 3 dimensional space frame of a dome http://www.geodesicsunlimited.com/face-variations.htm

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Monolithic domes have a bit more straight forward approach casting reinforced concrete layer shaped as a dome later insulating it and applying external and internal finishes. It might sound easy, but creating a form work shaped like a dome is a difficult job requiring experience and sometimes innovation. Standard solution of putting up a timber formwork would be extremely complex job. Basically you need to build a complex form to support a complex structure while concrete heals. Some companies like “Monolithic”

1

2

3

Fig. 46 Geodesic dome patterns http://www.geo-dome.co.uk/article.asp?uname=calculation 1 – 2V Dome; 2 – 3V Dome; 3 – 4V Dome

improved such dome construction technique by using Airform, which eliminates the need for complex formwork. On the other hand, concrete casting and insulating technologies had to be adapted to meet the conditions of using new kind of formwork and ironically increases the complexity of construction in a different way.

Geodesic dome homes produce whole new range of difficulties. It seems as a shape consisting of many similar triangles is fairly simple, but first look often is deceiving.

Geodesic pattern is a system of a few types of triangles interconnected in a specific way at specific angles. The number of these different triangles as well depends on the frequency of the dome. Higher the frequency of the dome more it resembles perfect sphere.

For example 2V dome would consist out of two types of triangles, two types of struts. 3V dome – two types of triangles made out of three types of struts. 4V dome most complex of all it will require six different triangles made out of six different struts. It clearly shows how complex such structure can be. Bear in mind that nods/hubs connecting between the struts as well might need to be customized to be able to fasten struts coming at different angles. More complex the structure more different hubs it will require. Depending on the size of the construction, complexity might increase even more. The bigger geodesic dome gets the stronger it becomes (per kilogram of material employed). However, huge span domes might buckle or punch in at the points of Fig. 47 Different struts of 4V dome http://www.geo-dome.co.uk/4v_tool.asp connection. Therefore dome must be stabilized and secured. This is usually achieved by adding a third dimension to the structure of the dome.

In Kaiser aluminium dome in Honolulu it was stabilized by folding dome she ets into 3D shapes. If this option is not available, a dome structural frame can be arranged in three dimensions achieving same effect as with 3D folded sheets. It just means that additional type of struts will be necessary making complex structure of space frame.

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Summarizing the complexity of the domes I must say that a lot of the problems mentioned here can be effectively solved by prefabrication. However designing and detailing the connections between elements will be a different challenge. Dome has been around for some time, but standard details of construction still do not really exist, most of the companies develop their own different solutions. Reason why it is this way lies in the attitude towards domes. It is still being seen as a housing of science fiction therefore it stays behind the sidelines of architecture and construction. Articles and other documentation investigating problems with existing dome constructions, which solutions are successful and which are not is quite hard to find. I guess this suspense of unknown really deters many architects and engineers from building dome shape constructions.

4.7 Leakage In the past half of century domes gain the bad reputation of being leaky. To be honest, reputation was deserved at the time when geodesic dome was introduced as a new shape. All kind of different domes made out of wood and plywood, steel and fibreglass, large and small were all leaking. First it was thought that unusual building shape was too much for the builders while as well many of the first domes were self designed. So when you add these two together it is a recipe for trouble. However even domes designed by experienced constructors and build by professional builders were also leaking. Ford Rotunda, Montreal Biosphere both severely suffered from leakage, and both burned during attempts to seal the leaky points with bitumen. This suggested that there was more to the picture than meets the eye. Eventually reason â&#x20AC;&#x153;whyâ&#x20AC;? it is happening was figured out. Today it is not a new thing to account for expansion and contraction due to heat and cold when designing construction, but surprisingly if you go back in time just a few decades it was not a worry for designers. Geodesic expansion and contraction due to sun heat was the main reason for the leaky domes. Domes usually distribute loads and stress evenly, but only in ideal conditions. In real case a certain part of dome which is exposed to direct sun light, causes expansion of the materials. When some parts of the dome expands and contracts it violates integrity of the construction to distribute the loads evenly. Eventually when dome still tries to do its job distributing the load something will fail.

Gaps might appear between panels, sealing material might be squeezed out of the connections creating a pathway for water. Even completely solid skin suffers from this problem. Shell out of welded metal, aluminium or fibreglass suffers from distortion.

Sometimes in severe conditions deformation can even be visible with the naked eye, how deformation follows the sun angle. This repeated exposure causes failure in seals around windows, doors, as well as any other connections. A constant maintenance and inspection had to be carried out in order to make sure that domes functions properly. Additionally distinct dome shape provides troubles for roofing contractors. Often traditional roof materials used in modern construction has to be adapted in a specific way to fit domes, but most of the builders are not necessarily familiar with the ways how roofing material has to be shaped and fitted together, thus creating an open way for leaking. For example shingles are placed in different angles; closer to the bottom it is more vertical at the top it is nearly flat, at his point high wind pressure can cause the water to be pushed under the singles and into the construction. This problem is the trickiest at the points where many triangles meet. (Baldwin, 1996) 38

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Luckily when problems were identified solutions started to appear. Today it is more than possible to manufacture a dome that is not leaking and providing long and comfortable housing.

4.8 High precision In order to achieve completely tight and waterproof domes high precision is the key.

The best way to achieve that is prefabrication. When you think about it is a natural approach towards this type of structures. As mentioned in previous section geodesic domes either based on panels (prefab concrete or wooden elements) or structural framing (aluminium, steel struts) can simply be standardized and divided into four or five different elements which mass produced will form the whole structure.

Such approach allows a construction of really high precision with strictly controlled tolerances.

Precisely cut building parts make connection of different components a lot easier leaving very small amount of room for human error. Connections with modern silicone seals are capable of taking care of the movements that could not have been solved by high precision prefabrication. Some of the solutions like overlapping connections of concrete element domes do not even need to rely on such silicone seals.

Further standardization takes place with Fig. 48 Kaiser Aluminium dome in Honolulu http://www.morleymarkson.com/morleymarkson_w connection pieces as well. For example: when building ebsite/Kaiser_Aluminum_Dome.html Ford Rotunda only one type and size of rivets and bolts were chosen to prevent many misunderstandings on the building site.

Prefabrication can give even more than a high precision building. Construction time can be significantly decreased also. Good example is Kaiser Aluminium dome in Honolulu Hawaii built in 1957. Dome was prefabricated and flown in from California. Just twenty hours after first elements were unpacked whole 44m diameter structure was finished. After two hours more audience was able to listen to Hawaiian Symphony Orchestra in the structure providing superior acoustical performance. (http://www.morleymarkson.com/morleymarkson_website/Kaiser_Aluminum_Dome.html) 4.9 Lack of experience Learning by trying, learning from mistakes was an important part of human history in any field. Dome is not an exception as well. Depending on the intended function and lack in design experience the advantages that make dome beneficial can turn against the inhabitants of such building.

First of all acoustical effect of the dome is exceptional, well managed it can do wonders for theatres and concert halls. However if you chose dome as a space for home or private space preventing this acoustical phenomenon is quite difficult. Surfaces reducing reverberation (echoes) must be placed carefully and designed to interrupt sound waves from being transferred from room to room. Otherwise even a coin toss can be heard all across the dome.

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Air current patterns in domes contribute to good indoor climate and ventilation. On the other hand these even air patterns will distribute heat, cold and any smell evenly as well. In order to avoid these displeasures well performing kitchen hood, bathroom vents are of the greatest importance. Fire and smoke will also be an issue; natural air flows can transfer fire and smoke from one part of the building to another very fast. Therefore escape routes, smoke vents, fire resistant surfaces must be placed carefully. Important issue like this should be taken into consideration especially when applying for building and planning permit, since it might be hard to convince about the safety of the structure.

Nevertheless outside of the building can act very well in the case of fire nearby. Smooth shape of the building spreads the radiant heat and sparks coming from any nearby fire, thus ensuring that it will require higher temperature for the building to catch fire.

Additional issue requiring consideration and experience in design process is extensions. Adding another floor to expand the dome is usually impossible, option of enlarging is also not going to be easy. Dome does not favour expansions. Trying to connect dormer or a garage can be a difficult challenge. Such things should be considered from the very beginning and structure of the dome should be prepared for possible future expansions. Especially connections of dormers and rectangular structure could turn into a real headache. Dome can carry a risk of leaking, but when dormer or other similar structure is attached to the building later it is almost certain that connection point will give in sooner rather than later. Best option for expansion is to either build an additional dome nearby or simply push several domes together just touching one another at fairly small area. Keep in mind that additions and extensions can affect the natural airflow on the exterior of the dome and ruin some of the beneficial effects of it. (Baldwin, 1996)

5. Future potential Buckminster Fuller Being the inventor of geodesic dome, Richard Buckminster Fuller was one of the first people to realize the potential of this structure as well as suggesting some of the groundbreaking ways of using a dome.

5.1 Instant shelter One of the suggestions exploiting the potential of geodesic pattern as an instant housing was quickly adopted.

Interestingly both hippies and U.S. Military were fascinated by this idea, it is hard to imagine two social groups who could be more different, but sharing one common idea. Concept of instant housing was most developed by Fuller along with U.S. Military. In fact it was adopted by Marine Corps on more than one occasion.

Some of the models consisted of prefabricated foil coated cardboard, others of steel frame holding tent canvas. The reason why dome was perfect for this idea was its lightness and huge

Fig. 49 Geodesic dome being towed by a helicopter http://ottodesignvoyager.tumblr.co m/post/41794330301/helicopterlifting-a-geodesic-dome-in-raleigh1954

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volume. It was not long until it was realised that such construction can be fully assembled in the factory and shipped or flown to its intended location afterwards, a technique used by U.S Marines. Aerodynamics of a dome made it suitable for being towed by helicopter without endangering the pilot or people aboard the aircraft. Such type of housing was perfect for emergency situations caused by natural or other disasters. However after experimenting with this instant housing idea military favoured construction out of prefab containers and geodesic dome was abandoned.

Nevertheless company The North Face was not so quick to abandon this idea.

Inspired by the stability of the dome company developed many tents based on geodesic pattern. Continuous improvement and research earned The North Face a reputation of first class tent manufacturer. Models like VE 25 are often taken to the expeditions were terrible weather conditions are expected. (Baldwin, 1996) 5.2 Climate domes Next step of dome adaptability was so called “Garden of Eden” idea. Entire ecosystems being safely housed under the dome vault. First attempts to adapt this theory took place in St. Louis, 1960 October 1st. A new geodesic dome housing tropical rainforests opened its doors to the visitors. Dome spanning more than 54 meters, houses more than 2800 species of plants. According to the standards of when it was made (1960) and Fig. 50 Climatron in St. Louis, U.S.A. http://www.karlhartig.com/dome/6climatron/climatron.html later renovated (1990) it was state of the art technology, however today it would not be considered to be the most energy efficient structure, but then again every in the most genius design has to be constantly updated to be at the very top of efficiency, even though this is not the most important fact. Climatron is just a hint of a much bigger idea, which truly sounds as a plot for Sci-Fi movie. If ever mankind is hoping to colonize moon, other planets and eventually space we will need to take entire environments and bio systems with us. Naturally they will have to be protected from inhospitable environment of outer space. Climatron is testament to what is needed to make this possible. Dome – strong, material efficient, light, capable of spanning large distances structure, could do the job. (Missouri Botanical Garden, n.d.)

5.3 Cloud nine Cloud Nine – a fantastic proposal by Fuller for a geodesic sphere floating in mid air.

Sphere would form a completely independent community consisting of a few thousand people. Incredibly sphere would float in the air all by itself. Surface to volume calculation shows that structural weight of 1,6km diameter sphere would be one-thousandth of the total air mass inside. Sunlight and people inside of the dome would slowly heat up this enormous air mass and the moment air inside of the dome gets one degree warmer than surrounding air, whole structure would go up just like a hot air balloon. Again when 41

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geodesic sphere or a dome gets bigger it gets stronger, therefore 1,6km diameter sphere would be able to support itself, a several thousand people, their belonging and still float in the air. Such sky cities could be anchored to the mountains where inhabitants could travel to the ground and back up.

Maintaining certain temperature would suspend the sphere at a certain altitude allowing it to drift from one place to another like migrating birds.

This idea was meant for far future as a Fig. 51 Cloud Nine by Buckminter Fuller http://www.theecologist.org/reviews/events/271315/radical_nat way for people to live and move around the ure_art_exhibition.html earth minimizing depletion of the resources. Today with ever growing concerns about the well being of the planet, issues of sustainability might force us to reconsider such fictional ideas on a real basis. (Baldwin, 1996) 5.4 Garden of Eden Concept Similarly to Climatron this concept is an entire biological system housed under a dome. It is living outdoors indoors, surrounded by plants, water ponds, but always comfortable despite the weather outside. This was an idea of a future sustainable living.

Human and kitchen waste is purified in a natural way becoming a food for plants, miniature wetlands cleaning waste water in a natureâ&#x20AC;&#x2122;s way. Plants would clean the air making you feel just like in the forest. This was a grand vision for the future. The problem with such vision is the dome skin itself. To work as intended it was designed to be clear. However choice of material is a bit tricky.

Glass would be expensive when covering large areas, while cutting glass into triangles will be wasteful as well, which contradicts with the whole sustainable concept. Plastics suitable for the task usually have too short lifetime. Insulating such transparent structure has been proven difficult as well (Double glazing starts collecting moisture and algae in the cavity eventual ruining the dome shell).

On the other hand with some 52 Garden of Eden, sketch by Buckminster Fuller changes such dream could be Fig. http://www.buckminster.info/Index/D/Domes-Garden.htm brought to life, just in a little bit different interpretation. Thin dome could be made out of new ETFE plastic sheets solving many of the mentioned problems, and housing whole ecosystem of plants and wetlands.

However it will not be able to insulate the dome. This suggests that an insulated house within the dome would be necessary to create liveable conditions all year around. Air mass in the dome must be taken in the consideration. 30,5 meter diameter dome contains 42

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approximately 3,2 tons of air, implying that it would take long time to warm or cool such air mass, allowing to minimize the need for insulation of the interior house. Moreover savings can be made on roofing and other exterior materials of the house, because technically house is not placed outdoors. Additionally wetlands or ponds formed inside of the dome would act as air-conditioning units, storing solar gain of the summer and releasing it during the winter and cooling the air during the summer, thus reducing the need for insulation even more. Such changes could actually make this idea commercially viable much sooner.

Although as Fuller expected such bio systems would be extremely delicate and in order to stay healthy in there, one must know exactly how to run such dwelling, thus specific instructional material introducing doâ&#x20AC;&#x2122;s and donâ&#x20AC;&#x2122;ts in such house would be necessary for inhabitants. (Baldwin, 1996) 5.5 Domed Cities Domed city became a peak moment of domesâ&#x20AC;&#x2122; adaptability. It looks like this idea has evolved by accident, when Buckminster Fuller took a postcard of New York City in his hands and scratched a dome above a midtown Manhattan. At first it might look as way too crazy, but as usual Fuller did not dismiss even the craziest ideas without investigating them. First of all dome like this, spanning over midtown Manhattan could solve many of the climate problems. Dome unique capability of Fig. 53 Dome over Manhattan http://www.bfi.org/slideshowacting as a cooling machine would images/dome-over-manhattan-1960 eliminate discomfort caused by tremendous heat radiating from surrounding buildings causing a phenomenon called Urban Heat Island. During the cold winter months huge mass of air trapped in the dome would be a perfect insulating layer reducing the need to heat energy inefficient buildings inside. Fuller even calculated that such geodesic dome 3km in diameter would have a surface area of approximately 5% the total surface area of the buildings enclosed by it, implying tremendous thermal advantage. Rough cost estimates showed that eliminated need of snow removal in New York would alone pay for the dome in 10 years, making this proposal not only thermally advantageous, but economical as well. However to make this work, most of the businesses and especially transport should be as less polluting as possible, and implicating such drastic change on a population size like Manhattan is almost impossible and will require a lot of time to prepare people for life under the dome. (http://www.ecoredux.com/archive_project03_01.html), (Sadao, n.d.), (Baldwin, 1996)

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Potential 5.6 Dome over Houston With an idea of domed Manhattan never coming true the idea of domes over cities was forgotten. Until 2009, when worsening conditions in the city of Houston, Texas forced the architects and engineers to revisit the idea of moving entire cities indoors. Almost one hundred days a year Houston is exposed to heat higher than 30 degree Celsius, and high humidity, makes it feel even worse. In order to make life conditions inside the buildings bearable, air conditioning is used at an incredible scale. This extensive use of electrical power leads Houston to become the number one CO2 producing city in the North America very fast. To solve these problems a dome with surface area exceeding 1 950 000 square meters was proposed. Dome of this size naturally creates some additional problems that have to be addressed carefully. Dome surface would consist out of 447 000 geodesic panels, but usual materials like glass or Plexiglas would make such structure way too expensive and way too heavy to ever be built. Luckily recent breakthroughs in polymer industry created ETFE (Ethylene tetrafluoroethylene) a material perfect for this structure.

ETFE membrane is 99% lighter than glass, this allows smallest surface to weight ration that people could ever dream about. Proposed panel design consists of three layer of ETFE trapping air in between and shaped like a pillow. Since Houston is frequently exposed to the hurricanes it raises the question can a light construction like that survive tremendous storms. Surprisingly enough there are couple defensive lines protecting the structure. Three dimensional steel space frame can withstand loads higher than any possible wind speed possible on earth. ETFE panels can hold off wind speeds of up to 290km/h. Moreover it is a triple panel, this means that until at least one layer is intact whole structure would not suffer any damage and even if a panel fail, space frame will be able to distribute extra load safely. To ensure that life in the city can go as usual while construction is taking place a dome would be supported by cables attached to dirigibles.

These aircraft structures can stay at single point for long period of time and helium gas fill would ensure that fate of Hindenburg will never repeat itself. Visualization of the project can be found in Appendix B. (http://dsc.discovery.com/tv-shows/other-shows/videos/megaengineering-a-magic-dome-material.htm) Even though decades passed after first proposal made by Buckminster Fuller ideas like these still sound impossible. On the other hand slowly evolving projects like Garden of Eden in United Kingdom signifies that these ideas are alive and taking shape in the world.

Moreover concerns about the climate change and sustainable living might speed up these developments. With painfully slow decisions to counteract global warming, temperature just keeps rising. Eventually cities using domes to control the interior climate in order to cool it down might turn out to be not a fantastical idea, but bare necessity to survive.

6. Conclusion Domes were very important part of architectural history. This shape was used to enclose grand spaces, make political and religious statements. A lot of ingenious engineering breakthroughs, like first reinforced concrete use (St. Peterâ&#x20AC;&#x2122;s dome), building 44

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without centring (Brunelleschiâ&#x20AC;&#x2122;s dome) have been invented with construction of the domes.

Unfortunately many of them somehow were lost in the course of history and left to be rediscovered only centuries later. Thousands of humble dwellings were built across the world exploiting natural cooling airflows and beneficial indoor climate of a dome, quality of minimizing the heat loss and saving construction materials. Still after long history of building domes and dome dwellings this shape and its benefits were unfairly forgotten and gave its way for modern architecture. Dome was not a form that easily accommodated itself in the ideas of modern architecture, such structure was proved to be difficult to plan and standardize. It was Richard Buckminster Fuller who rediscovered the potential of the dome and its place in the future. By creating a geodesic dome he managed to split the dome into many standardized pieces making it perfect for prefabrication and fast construction. He described in detail the benefits of these constructions, benefits that people have used long time ago: dome economics over rectangular houses in terms of material use and stability, ability to keep warmth in the winter and cool down in the summer only by the use of natural means etc. However with the lack of experience and competence many of these advantages can turn into a fatal construction flaw. It took decades to find solution to such fundamental problems like leaking joints. Other issues like planning a layout of a round structure still causes huge head ache for the architects and only way to deal with it is practice.

Despite these small challenges I do believe that dome will become the shape of the future. With sustainability becoming bigger and bigger issue dome might become the answer we are looking for. Its natural qualities and benefits address most of the problems that are being faced today. I think idea expressed by cradle to cradle creators Michael Braungart and William McDonough might help me to illustrate the situation: â&#x20AC;&#x153;Less bad, is no good!â&#x20AC;?. Dome in the sense of its natural airflows and material saving qualities will always surpass conventional rectangular architecture. So what we are trying to do now is to make unsustainable forms of architecture into sustainable ones, rather than taking one that is already sustainable and improving it further. This is why, in my opinion, dome is a perfect candidate to be the shape of the future, because even the best modern houses and technological breakthroughs cannot deny benefits which naturally lies in the shape of the dome.

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7. List of references Anonymous, n.d. Geo-Dome. [Online] Available at: http://www.geodome.co.uk/article.asp?uname=temptest [Accessed 12 September 2013].

Bahareh Hosseini, A.N., 2012. An Overview of Iranian Ice Repositories, An Example of Traditional Indigenous Architecture. Tehran: Department of Construction, Shahid Beheshti University.

Baldwin, J., 1996. Bucky Works Buckminster Fuller's Ideas for Today. 1st ed. New York: John Wiley and Sons, Inc. Campbell, T., 2001. Trulli Amazing Dry Stone Work. Stonexus, pp.48-52.

Channel, D., n.d. Discovery Channel. [Online] Available at: http://dsc.discovery.com/tv-shows/othershows/videos/mega-engineering-a-magic-dome-material.htm [Accessed 12 September 2013].

Econodome, n.d. Econodomes. [Online] Available at: http://www.econodome.com/floorplans.htm [Accessed 11 September 2013].

Fuller, R.B., 1997. Buckminster Fuller Institute. [Online] Available at: http://www.rwgrayprojects.com/synergetics/toc/toc.html [Accessed 7 September 2013].

Yazdi, J.T., 2007. Ab-Anbar, The Ancient Underground Water Houses of Khorasan. Tehran: International History Seminar on Irrigation and Drainage.

Kemery, B., 2006. Yurts Living in the Round. 1st ed. Layton: Gibbs Smith Publisher.

Khan, H.-, 2009. International Style Modernist Architecture from 1925 to 1965. Cologne: Taschen.

King, R., 2000. Brunelleschi's Dome The Story of the Cathedral in Florence. London: Pimlico.

Kostof, S., 1995. A History of Architecture Settings and Rituals. 2nd ed. New York: Oxford University Press.

Lajeunesse, A., 2008. Canadian Military Journal. [Online] Available at: http://www.journal.forces.gc.ca/vo8/no2/lajeunes-eng.asp [Accessed 16 September 2013].

Larsen, O.P., 2008. Reciprocal Frame Architecture. 1st ed. Architectural Press.

Levin, S.M., 2010. Biotensegrity. [Online] Available at: http://www.biotensegrity.com/tensegrity_truss.php [Accessed 9 September 2013].

Mainstone, R.J., 1999. The Dome of St Peter's: Structural Aspects of its Design and Construction, and Inquiries into its Stability. AA Files, (No. 39), pp.21-39.

Mills, H.K.a.E., 2012. The Canadian Encylopedia. [Online] Available at: http://www.thecanadianencyclopedia.com/articles/architectural-history-early-first-nations [Accessed 30 August 2013].

Missouri Botanical Garden, n.d. Missouri Botanical Garden. [Online] Available at: http://www.missouribotanicalgarden.org/gardens-gardening/our-garden/gardensconservatories/conservatories/climatron.aspx [Accessed 14 September 2013].

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Moore, D., 1995. Romanconcrete. [Online] Available at: http://www.romanconcrete.com/docs/chapt01/chapt01.htm [Accessed 1 Septermber 2013].

Natural Space Domes, 2005. Natural Space Domes. [Online] Available at: http://www.naturalspacesdomes.com/all_about_domes.htm [Accessed 10 September 2013].

O. Popovic Larsen, A.T., 2003. Conceptual Structural Design: Bridging the gap between architects and engineers. Kent: Thomas Telford Publishing.

Oliver, P., 1987. Dwellings The House across the World. Oxford: Phaidon Press Limited.

Peter Nabokov, R.E., 1989. Native American Architecture. New York: Oxford University Press Inc. Press, T.A., 2011. Thespec. [Online] Available at: http://www.thespec.com/news-story/2205436-study-ofst-peter-s-basilica-dome-reveals-hidden-construction-techniq/ [Accessed 1 September 2013].

Razaitis, V., 2004. Pastatu Konstravimo Pagrindai (Basics of Building Construction). Vilnius: Vilniaus Dailes Akademijos Leidykla.

Sadao, B.F.a.S., n.d. Ecoredux. [Online] Available at: http://www.ecoredux.com/archive_project03_01.html [Accessed 12 September 2013].

Sigler, V., 2008. Monolithic. [Online] Available at: http://www.monolithic.com/stories/feature-home-doah [Accessed 15 September 2013].

Steinhauer, J., 2012. The New York Times. [Online] Available at: http://www.nytimes.com/2012/08/25/us/politics/capitol-dome-is-imperiled-by-cracks-and-a-partisandivide.html?_r=1& [Accessed 1 September 2013].

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8. List of illustrations Fig. 1 Brain storm of the report structure Fig. 2 Reconstructed Pit House Fig. 3 Neolithic Pit House Section (by Povilas Mikuta) Fig. 4 Traditional Yurt http://www.yurtinfo.org/yurt-faq Fig. 5 Yurt Structure Sketch (by Povilas Mikuta) Fig. 6 Replicated Wigwam Fig. 7 Wigwam construction sequence (by Povilas Mikuta) Fig. 8 Inuit Igloo http://www.arcticphoto.co.uk/gallery2/arctic/peoples/greenlandw/qq0608-08.htm Fig. 9 Navajo Hogan http://navajorug.com/the-navajo-hogan-shelter-and-center-oftheir-world/ Fig. 10 Ab-anbar in Qazvin http://www.trekearth.com/gallery/Middle_East/Iran/East/Yazd/Yazd/photo1006911.h tm Fig. 11 Working principle of Ab-anbar (by Povilas Mikuta) Fig. 12 Yakhchal in Yazd Province http://www.eartharchitecture.org/index.php?/archives/1045-Yakhchal-AncientRefrigerators.html Fig. 13 Trulli http://www.understandingitaly.com/puglia-content/trulli.html Fig. 14 The Pantheon in Rome http://www.historywiz.com/galleries/pantheon.htm Fig. 15 Possible centrings used to support the dome during construction (by Povilas Mikuta) Fig. 16 Hagia Sophia http://istanbulvisions.com/hagia_sophia.htm Fig. 17 Dome of Santa Maria del Fiore http://www.brunelleschisdome.com/ Fig. 18 Structure of Brunelleschi‘s Dome (by Povilas Mikuta) Fig. 19 St. Peter‘s Dome http://entertainment.howstuffworks.com/arts/artwork/stpeters-basilica5.htm Fig. 20 St. Paul‘s Cathedral http://www.explorestpauls.net/oct03/textMM/DomeConstructionN.htm Fig. 21 Dome‘s Structure (by Povilas Mikuta) Fig. 22 U.S. Capitol Dome http://www.digitalimages.net/Gallery/Scenic/AsstScenic/WashDC/washdc.html Fig. 23 Section of Capitol‘s dome (by Povilas Mikuta) Fig. 24 Montreal Biosphere http://unusual-architecture.com/montreal-biospherecanada/ Fig. 25 Forces in the Dome (by Povilas Mikuta) Fig. 26 Mathematical Sketch (by Povilas Mikuta) Fig. 27 Dome Types (by Povilas Mikuta) Fig. 28 Spherical Triangle Fig. 29 Synergetics in Geometry (by Povilas Mikuta) Fig. 30 Radiolaria http://en.wikipedia.org/wiki/Radiolaria Fig. 31 Tensegrity model invented by Kenneth Snelson (by Povilas Mikuta) Fig. 33 48

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Fig. 33 Tensegrity dome frame (by Povilas Mikuta) Fig. 34 Dome Case No.1 (by Povilas Mikuta) Fig. 35 Square House Case No.1 (by Povilas Mikuta) Fig. 36 Dome Case No.2 (by Povilas Mikuta) Fig. 37 Square House Case No.2 (by Povilas Mikuta) Fig. 38 Square House Case No.3 (by Povilas Mikuta) Fig. 39 Dome ventilation winter mode (by Povilas Mikuta) Fig. 40 Dome ventilation summer mode (by Povilas Mikuta) Fig. 41 Dome cooling effect principle (Summer mode) by Povilas Mikuta Fig. 42 Dome after withstanding Hurricane Ivan http://domeofahome.com/domeinformation/hurricane/ Fig. 43 Mount Fuji weather radar http://www.mitsubishielectric.com/brief/sp/history/1960s/index.html Fig. 44 DEW Radar line http://www.journal.forces.gc.ca/vo8/no2/lajeunes-eng.asp Fig. 45 3 dimensional space frame of a dome http://www.geodesicsunlimited.com/face-variations.htm Fig. 46 Geodesic dome patterns http://www.geodome.co.uk/article.asp?uname=calculation Fig. 47 Different struts of 4V dome http://www.geo-dome.co.uk/4v_tool.asp Fig. 48 Kaiser Aluminium dome in Honolulu http://www.morleymarkson.com/morleymarkson_website/Kaiser_Aluminum_Dome. html Fig. 49 Geodesic dome being towed by a helicopter http://ottodesignvoyager.tumblr.com/post/41794330301/helicopter-lifting-ageodesic-dome-in-raleigh-1954 Fig. 51 Cloud Nine by Buckminter Fuller http://www.theecologist.org/reviews/events/271315/radical_nature_art_exhibition.ht ml Fig. 52 Garden of Eden, sketch by Buckminster Fuller http://www.buckminster.info/Index/D/Domes-Garden.htm Fig. 53 Dome over Manhattan http://www.bfi.org/slideshow-images/dome-overmanhattan-1960

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Appendix A Elara â&#x20AC;&#x201C; two storey dome (http://www.monolithic.com)

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Janus â&#x20AC;&#x201C; two storey dome (http://www.monolithic.com)

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Io 20 â&#x20AC;&#x201C; single storey dome (http://www.monolithic.com)

Hyperion â&#x20AC;&#x201C; single storey multiple domes (http://www.monolithic.com)

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Appendix B (http://cdn.leganerd.com/wp-content/uploads/LEGANERD_037771.jpg)

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