Theoria Portfolio by Daniel Pound

THEORIA

Daniel Pound

Theoria â&#x20AC;&#x153;He who loves practice without theory is like the sailor who boards ship without rudder and compass and never knows where he may cast.â&#x20AC;? Leonardo da Vinci

Contents

Part 1 : Methodology

Part 2 : 585 x 585 x 585

Introduction

Catenary Systems

Mรถbius Strip

Developable Surfaces

Design and Fabrication Process

Minimal Surfaces

Design Process

Ruled Surfaces

Final Iteration 1

Origami

Final Iteration 2

Close Packing

Introduction

Methodology In order to truly understand the processes of designing, it is essential to understand the building blocks that inspire and dictate the way we design. The following section focuses on understanding the basic systems of geometry that are essential to architecture and three dimensional design. Through the use of models and diagrams, the chapter will look at the principles of catenary systems; developable, minimal and ruled surfaces; origami and close packing. This understanding of the active systems and their application in architecture should help guide us in future design, though the knowledge of what forces are at play.

Catenary Systems

Catenary Systems : Principles

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Distance between points

height of catenary

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The word catenary comes from the Latin word catena, which mains ‘chain’. This name comes from the fact that the curvature is that of an idealized hanging chain under its own weight when supported only at its ends. When inverted the shape of a catenary curve is found to be the most efficient shape for a structural arch. Whilst many people believed that shape taken by a catenary represented a parabola, it wasn’t until the work of Hooke, Leibnoz, Huygen and Bernouili in 1691 related it to the function of the hyperbolic cosine function.

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y = a cosh( x / a ) The diagram opposite explores the relationship between the length of chain and height of catenary curve created. The percentage value represents the ratio of length of chain to width between hanging points.

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Precedents: Antonio Gaudi One of the catenary arches greatest champions was Antonio Gaudi with his work in the city of Barcelona. One of his most well known and recognised works is the use of hanging chains whilst designing the Sagrada Familia. The images on the right show two examples of Gaudiâ&#x20AC;&#x2122; hanging chain models that were used to calculate and initialise the structure of the church. Gaudi also utilised catenary arches on a smaller scale in the Guell chapel. This is often considered to be his most successful of projects to be incorporate the catenary due to his ability to calculate such a complex structure from the use of hanging chain models.

guell chapel, Barcelona

Within the Guell Chapel, Gaudi utilised a large number of shallower catenary arches as shown in the image above. These were calculated by hanging multiple chains of shorter length in order to create a vaulted catenary arch. Once this was calculated using a hanging chains, the form was inverted to create the arch.

Sagrada familia, Barcelona

Unlike the chapel, the Sagrada Familia used taller arches to create the structure of the main church area. Like the smaller arches, hanging chains were used to calculate the form, although the variables within the design were hanged. The length of chain in relation to the distance between the hanging points. This allowed for the creation of the taller arches used.

Hanging chain model

Exploratory Models Model 1 The first model looks at the relationship between the length of chain and the distance between hanging points. With the same hanging distance, the three variations look at changing the length of chain that was used, giving the result of deeper curves from the greater ratio.

Model 2 This model looks to create a domed catenary vault through the use of multiple chains of equal length. The biggest problem with the chains that were used is the limiting nature of the size of the links when working at small scales. In order to increase the number of hanging chains used to define the form, a finer and smaller chain would be needed.

Model 3 The final model using string soaked in pva glue was intended to create a catenary arch that could be inverted after it had dried. The nature of the string and lack of weight meant that the string struggled to interpret the catenary curve effectively.

Developable Surfaces

Developable Surfaces : Principles A developable surface is described as a surface with zero gaussian curvature that can be flattened onto a plane without the need for stretching or distorting. When in three dimensions, developable surfaces can also be considered to be ruled surfaces.

Frank Gehry, Guggenheim The Guggenheim Museum in Bilbao by Frank Gehry is considered to be one of the most prominent buildings to be associated with developable surfaces. Gehry uses one of the simplest ways to design and model a building: bending card. This in its simplest form is creating a building through developable surfaces. Whilst they may be broken down into smaller panels, each of the faces of the Guggenheim is itself a developable surface.

The two diagrams below aim to highlight how a three dimensional surface can be flattened out (unrolled) to a plane. These nets can be used in construction to create three dimensional shapes from flat sheet materials.

Developable Surfaces : Models

Exploratory Models The following models look how different folding techniques can be used to create interesting developable surfaces. The far image shows the flat net that is used to create the three dimensional forms in the pictures. Whilst these models are constructed out of paper which is fairly simple to bend, these techniques could be replicated on a larger scale to create a metal facade. Specialist technologies may be needed to fabricate the design such as Robofold. The Robofold technology uses computer programmed robots to bend developable surfaces into three dimensional forms. This method of fabrication opens up the door to more complex faรงades, whilst still maintaining a simple but effective fabrication method that uses flat sheet materials.

Developable Surfaces : 3D Forms

The examples of developable surface up to now have focused on single panels. Whilst these do have three dimensional properties, they are not using multiple pieces to create 3D volumes. The model below explores the process of using multiple developable surfaces to create a 3D volume. The process involved splitting a form into equal surfaces. These surfaces were then unrolled to create the nets of the developable surface ready for cutting. Holes were included along the seams to allow for the stitching together of the pieces. The nature of the plastic and the edge to edge joint meant that it was impossible to join through glueing.

1: A simple three dimensional shape is created using a revolve.

2: Surface is divided into twelve equal segments

3: The twelve surfaces are unrolled ready for cutting and then joining together.

Minimal Surfaces

Minimal Surfaces : Principles A minimal surface is defined as a surface that occupies the minimal area between its physical boundaries. It can also be defined as a surface that has zero mean curvature. Within architecture, minimal surfaces are most commonly used in tensile structures. Early pioneers of minimal surfaces such as Frei Otto often used bubbles and wires structures to simulate the forms created, along with simple ruled surface models.

Model 1 The first model of a hyperbolic paraboloid aims to highlight the how the opposing curvatures of the surface cancel each other out to create a zero mean curvature.

Model 2 The second model looks at the effect on a hyperbolic paraboloid on changing the boundary parameters that are defining the surface. This means that whilst the bounding edges may not be a mirrored image, the minimal surface still remains with a mean curvature of zero.

Model 3 The final model combines the use of a catenary curve and a minimal surface to create a catenoid shape. This is the shape that is assumed by a bubble when stretched between two rings. The parameters of a catenoid can change when the distance between the end rings and the ring diameter is changed.

Exploratory Models

Frei Otto Frei Otto is known as one of the pioneers of the use of minimal surfaces within the tensile structures field. His breakthrough work came in the form of the canopy for the 1972 Olympics in Munich, Germany. In order to investigate the properties of minimal surfaces, Otto utilised the use of bubbles and wires to develop the design of the canopy. The image to the right shows one of his bubble experiments used to understand the forces being applied to the form.

Bubble Models In order to further understand minimal surfaces, explorations with bubbles were undertaken. The images below show the different outcomes. Many of the forms such at the catenoid and hyperbolic paraboloid were easy to achieve, although the helicoil proved harder to gain the minimal surface.

Minimal Surfaces

Ruled Surfaces : Principles A ruled surface can be created by sweeping a straight line along a single or multiple guide curves. The surface created will have a straight line passing through it at any point on the surface. If the point has a straight line passing in two directions it is considered a doubly ruled surface.

The first model of a doubly ruled surface is a catenoid. This is a doubly ruled surface as there is a line flowing in two directions from ever point. The curvature of the profile could increase by increasing the amount the line is offset around the guide curves.

The second model of a paraboloid can it self be considered a doubly ruled surface as well. This form is created by sweeping a line along two guide curves. As these two guide curves are straight and parallel. The same process can be applied in the opposite direction.

Model exploring the principles applied in the Sagrada Familia school roof

Application of Ruled Surfaces: Sagrada Familia School Along side the catenary curves, Gaudi utilised ruled surfaces at the Sagrada Familia when designing the school. The principle was applied to both the roof and walls of the building. Through using wooden rafters and a multiple layers of tiles, Gaudi was able to create a visually interesting building in a simple manner. The diagrams below explain how the systems utilised a straight line along multiple guide curves.

roof structure

Wall structure

The model below aims to represent the principles applied by Gaudi in the roof structure. By changing the parameters of the guide curves a flatter or steeper pitched roof could be achieved.

Origami

X-Form Origami Model

Origami : X - Form Spans The following models look at creating a structure through folding paper to create a tunnel form. The X - Form grid creates a system that relies on the individual triangular planes to create a load bearing structure, holding the individual pieces in place. The following models looks at the relationship between grid size, on the output and the problems that this may bring.

Model 1 The first model uses a net that comprises of equilateral triangles. This results in a shallow recess in the plane indent but a small radius of the tunnels curvature. This also means that there is a large amount of lateral movement in transferring the forces to the ground.

Model 2 The second model shortens the width of the grid and increases the height. This creates tunnel of greater radius and shorter length. The transferring of loads become more vertical than the previous iteration.

Model 3 The final iteration has further increased the width to height ratio of the grid creating the largest radius of the models, but shortest in length. As the ratio of length to width is higher than the others, the spacing between the grid when folding becomes very narrow creating a structure that is less stable than the others.

Precedent : St Paulus Church, Neuss

St Paulus church in Germany by Fritz and Christian Schaller in 1967 utilises the system generated in the origami x - form exploration models to create the design. Using concrete panels as the main structural element the design manages to transfer the loads of the build through the connecting boundaries of each panel. Unlike the models before, each element of the grid (panel size) is unique, allowing for a design that alters in width at certain points. Utilising this idea will allow for a design that is less rigid.

Changing The Parameters

Though adjusting the parameters of the net to create a grid that changes through varying the width of the individual cells, the output was a tunnel that changes in size. At one end the tunnel represents the first model having a short radius, and resembles the final model at the other end with a larger radius. Whilst this model only varies the width of the grid, playing with the height would also allow for the system to be adapted to any form.

Width of grid

Folded Plate Origami Model

Origami : V - Form Spans The origami of V- folds aims at using folded plate structures to create a form. This structural element works on the basis of each folded plate being held in place by the adjacent pieces. The following models aimed to explore the relationship between the number of, and angle of the fold within the system.

Model 1 The First model uses only a single set of folds in the centre to create a structure that appears to lean against the other side. This design is the strongest of the three variations as it is creating a triangular structure with the ground plane.

Model 2 The second model incorporates two folds within the design to create a flat roofed structure. In order to achieve an additional fold, the folding angle had to be adjusted. If the net was stretched horizontally, a lower structure would be created.

Model 3 The final model utilises an additional fold into the structure whilst again maintaining the same strip width. It is clear that from this iteration that the 180 degree tunnel could be increasingly faceted. This however would reduce the strength of the structure created, as there would be a greater number of joints.

Precedent : St Loup Chapel

The St Loup chapel in France by LocalArchitecture utilises the folded plate structure created by the Vform origami. The timber structure uses the connected planes to create a rigid structure, where the forces of the building are both transferred and resisted by the plates. Unlike, the origami models, the design cannot be made out of a single net as each panel is of different width and height.

Changing The Parameters

After looking at the St Loup Chapel, the following model was intended to investigate changing the parameters of the initial models, combining the different net variations into one singular net. The two main parameters that were altered was the width of the strips and the angle of fold that was being used. This allowed for an irregular pattern to the folds, along with the changing of height and volume. The size and rectangular boundary of the net remained the same meaning that all the pieces shared the same flat edge. This became a problem for the model as many of the planes did not sit flush to the ground. This may require a none linear edge to the net to accommodate a flat joint to the ground This model in paper using a set net highlights the possibility of creating structures using a folded plate system. Using this paper method would be useful when making models but scaling up would require a different approach.

Close Packing

Close Packing : Principles Close packing is the principle of gathering the largest volume into the smallest space, in the most efficient way. Many explanation focus on the packing of spheres in triangular prism formation. However, when looking at the packing of volume, bubbles is the most effective way to demonstrate the principles. When two bubbles containing a volume of air meet together, as in the diagram, they take the optimal form. This process produces the most stable and optimised way of joining volumes.

This principle of close packing was utilised at the Eden Centre in Cornwall, to allow for the joining of multiple geodesic domes. Following this principle provided them with the optimal form.

Weaire-Phelan Model In 1887 Lord Kelvin produced research on the close packing of volume, focusing on minimal surface area. His method focused on using repeated bi-truncated cubes, all of the same volume. The Weaire-phelan structure is a close packing system that is considered to be the most effective form, 0.3% more efficient than the Kelvin theory. The build up of images on the right show the two different units building up to create one repeating module of the Weire-Phelan structure.

This module can then be repeated in three dimensions. The Weire-Phelan model believes that this is the form that bubbles of the same volume would take when packed into a space.

Precedent : Water Cube, Beijing

The water cube in Beijing, designed by PTW Architects for the 2008 Olympic games utilises the Weire-Phelan structure for both a visual and structural benefits. In order to represent the water element of the swimming pool design they chose to use the Weire-Phelan model to represent the bubbles. This repeating module allowed them to fill the site with an efficient structure whilst creating a facade that used just 15 different ETFE pillow variations. The section of the building below shows how spaces were carved out of the structure to allow for the internal spaces. 3d section through weire-phelan structure.

585 x 585 x 585

Introduction

585 x 585 x 585 Following the analysis of systems in the first section, the following work will focus on the development of a selected system; developable surfaces. The design will create a developable surface system that fits inside a 585 x 585 x 585 mm box with the intention of exploring the chosen system.

Möbius Strip: Principles

A möbius strip is non orientable surface with one side and one boundary. Mark Burry describes it as “a single, continuous surface, bounded and finite in extent but allowing infinite traversal” The opposing curvatures of the surface have a mean value of zero.

A Möbius strip models

A Möbius strip is made by twisting the ends of a strip of paper through 180 degrees before joining them together. Through this action, the opposite edges become aligned to create a surface with a single closed boundary line.

The principles of a möbius strip remain the same when the surface is rotated an odd number of times. The diagram on the left shows a standard möbius strip with one turn. The image below shows a möbius string that has had the ends turned three times.

M.C. Escher Artist M. C. Escher depicted the infinite traversal quality of a mรถbius strip with the image above. The ants on the plane would continue to traverse the strip, not knowing their orientation.

Topology Topology is the mathematical study of properties that are preserved when an object is deformed, twisted and stretched, but not torn. Following this principle, the Mรถbius strip, Torus ring and Klein bottle all share the same topological identity and can therefore be considered homeomorphic (share same topological properties).

Design Development: Options Option 1: Möbius Taurus Ring

Option 2: Interlocking Rings

The first option looks at turning the möbius strip into a three dimensional form by applying the principle to a topologically identical torus ring. This was achieved by joining the edges of six möbius strips together to create the möbius torus ring.

In order to create a more visually interesting and sculptural piece, further rings were added in an interlocking form to create a design that appeared more complicated. This form also became less orientable, a strong feature of the möbius strip. When developing the unrolled strips of the torus ring, it became apparent that all the strips were the same when the joint/starting point was offset. This allowed for all of the strips to be produced the same way.

Unrolled developable surface strips Single developable surface strip

Model stitched together

Multiple möbius torus rings

Option 3: Trefoil Knot

Option 4: Random Knot

After the interlocking rings, the next move was to create a knotted form that was achieved through a continuous loop. A trefoil knot is the shape taken by the mรถbius strip edges when rotated 1.5 full turns. This surface is also topologically identical to the mรถbius strip and torus ring.

The final option is to generate a random knotted shape. This will remain topologically identical to the other options but will have the effect of have no set orientation.

The nature of the trefoil knot means that the design once again becomes orientable, unlike the interlocking rings, especially in plan view.

The unrolled developable surfaces for this design become more complex than the previous options. They will need splitting into various parts due to the fact that they intersect each other.

Trefoil plan view Unrolled surface nets

Trefoil model

Random knot model

Design Development: Fabrication

Option 1: Paper

Option 2: Finger Jointed Card

The first option that was explored for the fabrication process was the use of paper. This was primarily for initial tests of the process but it also highlighted a vital problem with the method.

The next development looked at using a thicker card that could be laser cut with finger joints to allow for the pieces to be lined up together. This also gave the pieces extra surface area for glueing the pieces together.

Because the pieces were only touching each other at their edges, with no overlap, glueing them together became a problem due to the lack of contact area.

Another problem was that the smooth surface would slide around offering no hold point to the pieces. This meant that tape was needed.

The card worked well with the glue and finger joints in creating the form, although the strength and durability of the card meant that it was getting damaged whilst glueing together. This meant the finished product was not up to the desired standards.

Tape was needed to hold them in place

Finger joints

Option 3: Polypropylene

Option 4: Stitching the Seams

The next iteration of material options looked at using 0.8mm polypropylene as a means of creating the forms.

As with the initial developable surface model, the next step in the process meant that a non adhesive joining method was required. Therefore, along with the finger joints for positioning, holes were included to allow the pieces to be stitched along the seams.

Since the finger joints worked on the previous option, that was carried forward to allow for a better joint. The polypropylene proved to be a little bit too stiff, especially when using wide pieces as in the image below.

The stiffness of the polypropylene along with its resistance to glue meant that it became impossible to join the pieces towards the end. However, the plastic did have the durability and aesthetic qualities that were desired.

The rigidity of the polypropylene meant that pieces could be sewn together to provide tight joints. This would not have been possible whilst using card. This option proved to be the most successful of the approaches and if done correctly, the sewn seams should add an extra level of detail.

Design Development: Process

Initial investigations involved looking at the potential of the mรถbius strip as a sculptural piece

Three rings were interlocked to create a more sculptural form that becomes non-orientable link a mรถbius strip

A Torus Ring was highlighted as a 3d volume that is topologically identical to a mรถbius strip

A trefoil knot was introduced to create an interlocking form that is created from one continuous loop. However, this then became orientable when viewed from above

The mรถbius strip was applied to the Torus ring. This created a 3d form consisting of multiple mรถbius strips

A more random shape was created as a topological variation of the trefoil knot to allow the piece to be non-orientable from all angles

The initial final model was at too small a scale to fill the box to allow the full non orientable effect to be successful

The final model was resized to fill the space within the box

Two of the six sides were coloured different to allow highlight the twist of the mรถbius strip along the path

Design Development: Design Process

A curve in the form of a complex knot was created as the basis for the development of the shape.

The hexagonal profile frames were lofted to create a hexagonal sweep of the guide curve.

The hexagonal frames were applied to the curve and rotated 360 degrees over the course of one loop. This allows for the mรถbius strip to be established

Each of the six faces of the form were unrolled to create the outline of the developable surface.

The pieces were divided in smaller pieces for fabricating. This also erased the problem of a self intersecting surface that occurred when the faces were unrolled.

Finger joints and holes were then introduced to the surface to allow for the pieces to be stitched together when cut out.

Developed Design: First Iteration The first iteration of the final design focused on the random knot option discussed earlier. This created a pentagonal mรถbius surface along the path. The aesthetics of the design were pleasing as the white polypropylene worked well with the black stitching. The deign was more complex than previous iterations which resulted into the strips being broken down into more pieces than previously.

Despite the design looking aesthetically pleasing, there was two main problems when the form was placed in the box. Firstly, the form appeared too small within the volume. Secondly, the fact that all pieces were white meant that it became difficult to highlight the mรถbius strip effect

Developed Design: Final Iteration

Following on from the first iteration of the sculpture, this final iteration addressed the two problems of the first. The design was scaled up in proportion to fill the space in the box. Secondly, the shape was increased to be six sided to allow for two opposing strips to be a different colour. This allows the clear identification of the strips flowing around the sculpture. Through increasing the size of the form and the number of sides to six, the strength of the structure was a great deal weaker than the previous iteration.

Developed Design: Final Iteration

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Daniel Pound Leeds

33321749 School Of Architecture