UNIVERSE AS GEOMETRY Discovery of Polyhedra
Malinee Kulrakumpusiri Design Theory “What is Element Design” summer semester 2015 Prof. Joachim Krausse Master of Arts Integrated Design Anhalt University of Applied science
FOREWORD This is not a geometric booklet!! Don’t judge a book by its topic No, you will not find difficult formulars, such as, Pythagoras, Parabolar, or matrix. I am not going to ask you to calculate any possiblity in mathametical means. I have to inform you this because I know how geometry can scare someone off, including me. In the other hand, this booklet is a small investigative research on special geometric forms that appered for a long long time in our history. Since the first dicovery of such a beautiful geometric forms called , Polyhedra, they take a long journey to expand their fascinating world through times . This booklet aims to investigate the story of the polyhedra from ancient period through applications in modern world. It will show you how these small dicoveries inspired people in various occuptation areas to create big master pieces of the era. At the end, I hope that this booklet could, at least, inspire you to take geometry as a part of your universe and also inspire you in your next master piece project.
Malinee Kulrakumpusiri
INDEX FOREWORD
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UNIVERSE AS A GEOMETRIC PROGRESSION
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PLATO COSMOLOGY
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DISCOVERY OF POLYHEDRA
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INSPIRATION
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SUMMARY
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REFERENCE
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THE UNIVERS GEOMETRIC P
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SE AS A PROGRESSION While many of our first cosmologies were based on myths and legends, it is the Greek philosophical tradition that introduces an intellectual approach based on evidence, reason and debate. Even many of their ideas is not qualify as scientific theories, their reliance on mathematics as a tool to understand the Universe affected us a lot in this day. The formulation of Geometric Cosmology led to the development of the biggest philosophical achievement of humankind.
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GREEK PHILOSOPHER : PLATO (c.427-347 B.C.E.)
THE UNIVERSE AS A GEOMETRIC PROGRESSION
Plato was also one of ancient Greece’s most important patrons of mathematics. Inspired by Pythagoras, he founded his Academy in Athens in 387 BC, where he stressed mathematics as a way of understanding more about reality.
Although Socrates influenced Plato directly as related in the dialogues, the influence of Pythagoras upon Plato also appears to have significant discussion in the philosophical literature.
Plato’s most prominent student was Aristotle, shown here with Plato in Raphael’s School of Athens, Aristotle holding his Ethics and Plato with his Timaeus. Written towards the end of Plato’s life, c. 355 BCE, the Timaeus describes a conversation between Socrates, Plato’s teacher, Critias, Plato’s great grandfather, Hermocrates, a Sicilian statesman and soldier, and Timaeus, Pythagorean, philosopher, scientist, general, contemporary of Plato, and the inventor of the pulley. He was the first to distinguish between harmonic, arithmetic, and geometric progressions. In this book, Timaeus does most the talking, with much homage to Pythagoras and echos of the harmony of the spheres, as he describes the geometric creation of the world.
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“
geometey will draw the soul toward truth and create the spirit of philosophy”
UNIVERSE AS GEOMETRY
“
Plato
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PLATO’S COSMOLOGY Plato deduces the need for the four elements. (Timaeus, 31B-32C) First, we must have fire, to make the world visible, and earth to make it resistant to touch. These are the two extreme elements, fire belonging to heaven and earth to earth. . . it is necessary that nature should be visible and tangible ... and nothing can be visible without fire or tangible without earth ...
PLATO�S COSMOLOGY
But two cannot hold together without a third as a bond . . . But it is impossible for two things to cohere without the intervention of a third ... And the most perfect bond is the connued geometric proportion.... and the most beautiful analogy is when in three numbers, the middle is to the last as the first to the middle, . . . they become the same as to relation to each other. But the primary bodies are solids, and must be represented by solid numbers (cubes). To connect two plane numbers (squares) one mean is enough, but to connect two solid numbers, two means are needed. But if the universe were to have no depth, one medium would suffice to bind all the natures it contains. But the world should be a solid, and solids are never harmonized by one, but always by two mediums.
Timaeus is one of Plato’s dialogues, mostly in the form of a long monologue given by the titular character, written circa 360 BC. The work puts forward speculation on the nature of the physical world and human beings and is followed by the dialogue
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Hence the Divinity placed water and air in the middle of fire and earth, fabricating them in the same ratio to each other; so that fire might be to air as air is to water and that water is to earth. fire/air = air/water = water/earth
UNIVERSE AS GEOMETRY
Thus the ratio is constant between successive elements, giving a geometric progression.
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PLATO WROTE...
“We must proceed to distribute the figures [the solids] we have just described between fire, earth, water, and air. . .”
PLATO”S COSMOLOGY
“Let us assign the cube to earth, for it is the most immobile of the four bodies and most retentive of shape”
“The least mobile of the remaining figures (icosahedron) to water”
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“the intermediate (octahedron) to air”
UNIVERSE AS GEOMETRY
“the most mobile (tetrahedron) to fire”
“There still remained a fifth construction, which the god used for embroidering the constellations on the whole heaven.”
PLATO”S COSMOLOGY
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PLATONIC SOLID • cube • tetrahedron • octahedron • icosahedron • dodecahedron These have come to be known as the Platonic Solids Platonic solids are bounded by regular polygons, all of the same size and shape. All these shapes are highly regular and occur naturally. All Platonic solids have an inscribed sphere tangent to every face and a circumscribed sphere through every vertex. Other polyhedra generally do not.
In the early 20th century, Ernst Haeckel described (Haeckel, 1904) a number of species of Radiolaria, some of whose skeletons are shaped like various regular polyhedra. Examples include Circoporus octahedrus, Circogonia icosahedra, Lithocubus geometricus and Circorrhegma dodecahedra. The shapes of these creatures should be obvious from their names. Many viruses, such as the herpes virus, have the shape of a regular icosahedron. Viral structures are built of repeated identical protein subunits and the icosahedron is the easiest shape to assemble using these subunits. A regular polyhedron is used because it can be built from a single basic unit protein used over and over again; this saves space in the viral genome.
The tetrahedron, cube, and octahedron all occur naturally in crystal structures. These by no means exhaust the numbers of possible forms of crystals.
UNIVERSE AS GEOMETRY
Platonic Solid In Nature
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The Discovery of...
POLYHEDRA Eventhough, platonic solids are considered as The first type of regular solid shapes to be discovered, but the first polyhedra might be dated back before 2500 B.C.
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WHAT IS POLYHEDRA ?
DISCOVERY OF POLYHEDRA
A polyhedra are the three-dimensional version of a polygon: it is a chunk of space with flat walls. In other words, it is a three-dimensional figure made by gluing polygons together. The word is Greek in origin, meaning many-seated. The plural is polyhedra. The polygonal sides of a polyhedron are called its faces. However, they can be polyhedra under a few conditions; the edge must be straight and sharp corner. Of all polyhedra in this world, Platonic solid are very special that they are the most symetrical polyhedra and they are composed of the same face same size. A polyhedron is said to be convex if its surface (comprising its faces, edges and vertices) does not intersect itself and the line segment joining any two points of the polyhedron is contained in the interior or surface. A polyhedron is a Three dimensional example of the more general polytope in any number of dimensions.
Singular form of polyhedra is polyhedron.
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POLYGON
POLYHEDRA
Polygons are two-dimensional shapes. They are made of straight lines, and the shape is “closed”
Polyhedron is a solid in three dimensions forms with flat polygonal faces, straight edges and sharp corners or vertices
edges
vertices
Euler’s Formula To make sure we have counted correctly! This can be written neatly as a little equation: F+V−E=2
UNIVERSE AS GEOMETRY
faces
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TIMELINE 3000 bc.
2000 bc.
1000 bc.
Platonic Solids
DISCOVERY OF POLYHEDRA
Arcamedean Solids 0 bc.
1000 Dürer Solids Kepler Solids 2000
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Pre-Historic Polyhedra appeared in early architectural forms such as cubes and cuboids, with the earliest four-sided pyramids of ancient Egypt
Greek Civilization
Renaissance Johannes Kepler (1571 - 1630) used star polygons, typically pentagrams, to build star polyhedra.The Kepler-Poinsot polyhedra may be constructed from the Platonic solids by a process called stellation. Most stellations are not regular
UNIVERSE AS GEOMETRY
The earlier Greeks were interested primarily in the convex regular polyhedra, which came to be known as the Platonic solids. Pythagoras knew at least three of them, and Theaetetus (circa 417 B. C.) described all five. Eventually, Euclid described their construction in his Elements. Later, Archimedes expanded his study to the convex uniform polyhedra which now bear his name
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HOW MANY POLYHEDRA ARE THERE? Many polyhedra and theory were developed during the contemperary age (1789 until these days). In 20th Century, there are infinite forms of polyhedra existing in this world. However, to scope the amount the polyhedra, they can be classified under many groups, either by the family or from the characteristics that differentiate them. We can also classify Polyhedra by type : • Convex Polyhedron a straight line can only cut the surface at two points
• Concave Polyhedron a straight line can cut the surface at more than two points.
• Regular Polyhedra A regular polyhedron is composed of angles and faces (regular poly gons) that are all equal.
DISCOVERY OF POLYHEDRA
• Irregular Polyhedra An irregular polyhedron is defined by polygons that are composed of elements that are not all equal.
Or group them by their Special Classes : • Platonic Solids : 5 convex solids with identical regular polygon faces and identical vertices
• Archimedean Solids : 13
Johannes Kepler defined classes of polyhedra, discovered the members of the class, and proved that his set was complete. For example, Keplerdiscovered the infinite class of antiprisms.
solids with several types of regular polygon faces and identical vertices
• Johnson Solids : 91 solids with regular polygon faces
• Kepler-Poinsot Solids : 9 • Uniform : 81 (Coxeter-Skilling Polyhedra)
Harold Scott MacDonald Coxeter, one of the greatest geometers in 20th century, was the first to comply the complete list of uniform polyhedra.
UNIVERSE AS GEOMETRY
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Parts of Polyhedra Collection from Max Bruckner
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INSPIRATION & Innovation Through Polyhedra The beauty and interest of the Platonic solids continue to inspire all sorts of people, and not just mathematicians. For a look at how artist and designer inspired by polyhedra, These section will show you some interesting projects.
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JOHANNES KEPLER
INSPIRATION
(December 27, 1571 – November 15, 1630) Kepler was a German mathematician, astronomer, and astrologer. A key figure in the 17th century scientific revolution, he is best known for his laws of planetary motion. In addition to his astronomical accomplishments, he systematized and extended all that was known about polyhedra in his time.
Mysterium Cosmographicum
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By ordering the solids correctly, Kepler found that the spheres could be placed at intervals corresponding (within the accuracy limits of available astronomical observations) to the relative sizes of each planet’s path, assuming the planets circle the Sun. Kepler also found a formula relating the size of each planet’s orbit to the length of its orbital period.
However, Kepler later rejected this formula, because it was not precise enough. In terms of the impact of Mysterium, it can be seen as an important first step in modernizing the theory proposed by Nicolaus Copernicus in his “De Revolutionibus”. Modern astronomy owes much to “Mysterium Cosmographicum”, despite flaws in its main thesis, “since it represents the first step in cleansing the Copernican system of the remnants of the Ptolemaic theory still clinging to it.”
UNIVERSE AS GEOMETRY
Johannes Kepler’s first major astronomical work, Mysterium Cosmographicum (The Cosmographic Mystery), was the first published defense of the Copernican system. He found that each of the five Platonic solids could be uniquely inscribed and circumscribed by spherical orbs; nesting these solids, each encased in a sphere, within one another would produce six layers, corresponding to the six known planets—Mercury, Venus, Earth, Mars, Jupiter, and Saturn.
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Kepler Solids
Kepler obtained them by stellating the regular convex dodecahedron, for the first time treating it as a surface rather than a solid. He noticed that by extending the edges or faces of the convex dodecahedron until they met again, he could obtain star pentagons.
INSPIRATION
Arthur Cayley gave the Kepler–Poinsot polyhedra the names by which they Further, he recognized that these star pentagons are generally known to- are also regular. In this way he constructed the day. Kepler solid are composed of regular concave polygons and were unknown to the ancients.
two stellated dodecahedra. Kepler’s final step was to recognize that these polyhedra fit the definition of regularity, even though they were not convex, as the traditional Platonic solids were.
In 1809, Louis Poinsot rediscovered Kepler’s figures, by assembling star pentagons around each vertex. He also assembled convex polygons around star vertices to discover two more regular stars, the great icosahedron and great dodecahedron.
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Albrecht Dürer Albrecht Durer was a German painter, printmaker, mathematician, and theorist born in Nuremberg. He wrote Four Books of Human Proportion (Vier Bücher von menschlichen Proportion), only the first of which was published during his lifetime (1528), as well as an introductory manual of geometric theory for students (Underweysung der Messung, 1525; 125.97 D932)
UNIVERSE AS GEOMETRY
( May 21, 1471 – April 6, 1528)
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Durer Solid In 1514 the Albrecht Dürer created the copper engraving Melencolia I. It was immediately recognised as a masterpiece, not only because of its remarkably fine and detailed execution, but also because of its unusual symbolism. The distinctive 3D shape (Dürer’s solid) in Melencolia I has been the subject of innumerous analyses and still no one is sure what it is or what it means. Dürer’s solid, also known as the truncated triangular trapezohedron, is the 8-faced solid depicted in an engraving entitled Melancholia I
INSPIRATION
Polyhedra Net In 1825 Dürer published a four volume treatise, Underweysung der Messung, which dealt with, among other things, the construction of various curves, polygons, and other solid bodies. One of the first books to teach the methods of perspective, it was highly regarded throughout the 16th century and presents the earliest known examples of polyhedral nets ; polyhedra unfolded to lie flat for printing.
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Buckminster Fuller Richard Buckminster “Bucky” Fuller was an American architect, systems theorist, author, designer, and inventor. Four hundred years after Durer and Kepler, Buckminster Fuller continued a similar process of experimental observation of structure in 3D. Fuller’s earliest approaches to geometry and the harmony of the sphere drew unconsciously from those who came before, yet introduced the metaphysical concept of ephemeralization, an expansion into the weightlessness of space.
UNIVERSE AS GEOMETRY
(July 12, 1895 – July 1, 1983)
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Geodesic Dome
Expo 67, US Pavillion
INSPIRATION
Fuller invented the Geodesic Dome in the late 1940s to demonstrate some ideas about housing and ``energetic-synergetic geometry’’ which he had developed during WWII. This invention built on his two decade old quest to improve the housing of humanity. It represents a brilliant demonstration of his synergetics principles; and in the right circumstances it could solve some of the pressing housing problems of today (a housing crisis which Fuller predicted back in 1927). A geodesic dome is a triangulation of a Platonic solid or other polyhedron to produce a close approximation to a sphere or hemisphere. Fuller discovered that if a spherical structure was created from triangles, it would have unparalleled strength. Fuller’s original dome was constructed from an icosahedron by adding isosceles triangles about each polyhedron vertex and slightly repositioning the polyhedron vertices. In such domes, neither the polyhedron vertices nor the centers of faces necessarily lie at exactly the same distances from the center.
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Jitterbug
Jitterbug is a ball room dance style, similar to swaing and Lindy hop style.
Buckminster Fuller was successful in modeling quantum effects that brought out Platonic solids in a new light. The octahedron can be twisted further, and the entire system collapses into a “super triangle” consisting of four pairs of triangles, which then can be folded into a tetrahedron. Finally, it is possible to fold it into one single triangle, congruent with the basic surface of the cuboctahedron, but now with eightfold edges. This is the zero phase of the jitterbug and Fuller’s theory was (similar to the Greek’s stoicheia) that everything started with a triangle Plato call this “in the circle of becoming” ; the matter of becoming and perishing. This is supposed to correspond the transformation of the elements as forms. Plato fail to complete “triangle universal module”, but Fuller proves to the world that Plato’s triangle theory is valid.
UNIVERSE AS GEOMETRY
Buckminster Fuller’s (1895-1983) discovery of the “jitterbug” transformation (1948) was his “eureka” experience Characteristically, it was the complete opposite of a “stone”: not a solid body at all, but a fluid motion in which one body dissolves into another, one spatial figure develops from the other, just like in a dance.
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Robert James Moon INSPIRATION
(February 14, 1911 – November 1, 1989) University of Chicago professor of physics and physical chemistry Dr. Robert J. Moon Dr. Moon has applied thegeome trical method of NicoÂlaus of Cusa and Johannes Kepler,for unraveling the nucleus of the atom. Thus, his revolutionary new approach is directly located in the best traditions of mathematical physics, and he brings to it rich insights from his decades of experimental work.
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Moon Model : geometrical model of the atomic nucleus In the 1984 – 1986 period, Moon came up with his proposal for a geometric ordering of protons and neutrons in the atomic nucleus based on nested platonic solids. This “Moon Model”,was inspired by Johannes Kepler’s conception of the solar system, as described in Kepler’s work Mysterium Cosmographicum.
b.
a.
b.
The first shell proposed by Moon is a cube, which reflects the distribution of the eight protons in the nucleus of Oxygen. (b) The next shell would form after adding six new protons, reflecting the distribution of the fourteen protons in the nucleus of Silicon.
The completion of the third shell in the form of an icosahedron reflects the distribution of the twenty-six protons in the nucleus of Iron (26Fe). (b) The vertices of the octahedron are located on six icosahedral faces in such a way that both solids have two mutually parallel faces.
The completion of the fourth shell in the form of a dodecahedron would reflect the distribution of the forty-six protons in the nucleus of Palladium (46Pd). The icosahedral vertices would be able to freely move inside its enveloping dodecahedron.
The completion of the twin icosahedron leads us to the nucleus of Thallium containing eighty-one protons. (b) The last five protons complete the twin dodecahedral structure which reflects the eighty-six protons in the nucleus of the noble gas Radon.
UNIVERSE AS GEOMETRY
a.
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THEN Through history, polyhedra were symbolic of deep religious or philosophical truths. For example, Plato’s association in the Timaeus between the Platonic solids and the elements of fire, earth, air, and water (and the universe) was of great import in the Renaissance. For others, Polyhedra have been closely associated with the world of art. The peak of this relationship was certainly in the Renaissance. For some Renaissance artists, polyhedra simply provided challenging models to demonstrate their mastery of perspective. This was tied to the mastery of geometry necessary for perspective, and suggested a mathematical foundation for rationalizing artistry and understanding sight, just as Renaissance science explored mathematical and visual foundations for understanding the physical world, astronomy, and anatomy.
SUMMARY
AND NOW In modern era, polyhedra do not only simply provide inspiration and a storehouse of forms with various symmetries from which to draw on, but Polyhedra can be adapted to graphic design, product design, or visual programming. For me, everything in this universe is connected. With combination of science, geometry, art, and etc, I believe that creative design can be made easier than a pure knowledge of one area.
FUTURE...?
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REFERENCE • The Collected Dialogues of Plato: Including the Letters Plato , Edith Hamilton (Editor), Huntington Cairns (Editor), Lane Cooper (Translator)
• THE COSMOLOGY OF PLATO’S TIMAEUS PRESENTED AS A SERIES OF POSTULATES http://galileo.phys.virginia.edu/classes/609.ral5q.fall04/LecturePDF/L05-TIMAEUS.pdf
• The PLATONIC SOLIDS https://www.dartmouth.edu/~matc/math5.geometry/unit6/unit6.html, Paul Calter, 1998
• Sacred Solids in the Atomic Nucleus
http://www.sacred-geometry.es/?q=en/content/sacred-solids-atomic-nucleus Author : Jordi Solà-Soler, July 2012
• Durer Polyhedron
http://www.theguardian.com/science/alexs-adventures-in-numberland/2014/dec/03/ durers-polyhedron-5-theories-that-explain-melencolias-crazy-cube
• Your Private Sky: R. Buckminster Fuller : The Art of Design Science By Richard Buckminster Fuller, Joachim Krausse, Claude Lichtenstein
• Buckminster Fuller Institute • Wikipedia • Beautiful pictures from google.com
UNIVERSE AS GEOMETRY
http://bfi.org/