II. The Architecture of Life

The Ancient Greeks - Polyhedra, the Golden Mean and the Harmony of the Spheres

The earth on which we live is a sphere in motion. This is a simple concept, taken for granted by all of us. Throughout millennia, mathematics, the physical sciences and their subsequent technologies emerged through close observations of the earth and the celestial hemisphere. It seems odd that our fundamental precepts of science are still based on the concept of a flat earth, even though the earth was known, at least as far back as the days of the ancient Greeks (and most likely well before), to be a sphere, spinning and moving through space. This spherical view of the earth has been covered up, lost, rediscovered and rearticulated over and over throughout history. This view is still being understood in all of its complexity today while the cubic frame of reference continues to dominate the mathematical system on which all of our present sciences are based.

Fuller's synergetic geometry regarded the earth as a sphere in the most essential ways. He sought to frame a system that did not depend primarily on a cubic frame of reference. To do this he looked to nature to find the essential dynamics of her design on the micro, macro and medio scale. As a self-designated "comprehensivist" he set out to bridge the disciplines by basing his discoveries largely on what could be engaged by his own senses. His life experience in the Navy, as a designer, an engineer, an architect, and machinist continued to tell him that he needed a spherical rather than cubic frame of reference. Historically, the basic concepts surrounding a spherical world view often merged in the intersection between seemingly disparate bodies of experience, the sciences and the humanities. Using a spherical frame of reference, music, philosophy, art and the imagination no longer need to be seen in opposition to a scientific, abstract world of fixed rules. A simple look at just the past 2500 years points to this significant fact. 

Mathematics, art and music emerged from the same sources at different periods in history. Each can easily trace its origin back to the contemplation of the spherical world, the celestial horizon, or the curves of life. In the days of the early Greeks, architecture and music were based on harmony and proportion. Geometry emerged as the mathematics of form. Pythagoras (525-350 BC), whose philosophies and mathematics are thought to have been influenced by his early travels in Egypt and Babylonia, proposed that number was the essence of all things. He associated numbers with colors, virtues and form. The Pythagoreans discovered the earliest relationships between two and three dimensions, and wedded arithmetic to geometry by showing numbers as geometric figures.

The Pythagorean system was the first to apply the combinatorial approach to fill space on the plane with elaborate tessellations of triangles, squares and hexagons. This early system was also the first to identify the five regular polyhedral solids in three dimensions. These polyhedra were identified with the physical world: the earth-cube, fire-tetrahedron, air-octahedron, water-icosahedron, (and universe-dodecahedron)4. We have referred to these five regular forms as the "Platonic solids" ever since Plato recorded and documented them in the dialogue, Timaeus a hundred years after Pythagoras. All of the five Platonic solids were the only polyhedra which exhibited similar faces and could be circumscribed by the sphere.

Geometry historian Keith Critchlow has shown convincing evidence that these five regular solids were known to the neolithic peoples of Britain over 1000 years before Plato,5a and renowned mathematician H.S.M. Coxeter noted in his classic text, Regular Polytopes that "To ask who first constructed them is almost as futile as to ask who first used fire."5b Whomever discovered the platonic solids first may never be known but it is agreed that the ancient Greeks were the first to record their use and systematize these forms so they could be applied to the better understanding of the world around them.

The Pythagorean school (which included women as well as men) was identified by the pentagon, containing in its very structure a "divine" geometric quality. The school developed the first formalized theories of music and harmony by introducing the golden section (also known as phi, the golden mean, golden ratio or divine proportion) and applying it to both musical instrumentation and architecture. Phi, 1.618 or golden section is the division of a line segment so that the ratio of the smaller part to the larger part is equal to the ratio of the larger part to the entire segment. Golden section is known to have been used by the Egyptians and others long before Pythagoras yet it was the Pythagorean school that established the relationship between number ratios and sound frequencies and taught that harmonious sounds were emitted by the heavenly bodies as they circumscribed their celestial orbits. Pythagoras believed that the earth was spherical and that the sun, moon and planets all had separate motions.6

Archimedes, the Greek father of experimental science (225 B.C.), took geometry to a new level of sophistication by introducing the concept of volume. A well-known tale in the history of mathematics tells how Archimedes discovered a method to determine the volume of a non-uniform body. His accidental discovery took place when he was asked to solve whether the crown of King Hiero of Syracuse was 100% gold or a combination of silver (which weighed less) and gold. Contemplating the problem while taking a bath, Archimedes was struck with the answer when water spilled from the tub as his body entered. He figured he could distinguish the volume of his body by measuring the displacement of fluid when it is submerged in a full container of water. If he measured the displaced fluid, it should contain the same volume as his own non-uniform body. This approach worked as well for Hiero's crown and of course saved him his job with the King which was no small matter. Upon his accidental discovery, Archimedes ran naked through the city after his bath shouting, "Eureka!" (I have found it!), coining a term that has encapsulated the experiential epiphanies of scientists for centuries after.

The 13 polyhedral solids Archimedes discovered introduced a new series of three-dimensional semi-regular polyhedra to geometry, the Archimedean solids. By cutting off, or "truncating" the corners of the cube, tetrahedron and other regular solids, new planes could be formed and a new family of solids exhibiting symmetric variety was possible. The Archimedean solids did display symmetry, yet their topology displayed more than one shape of face. Seventeen centuries before Magellan sailed around the earth, Archimedes also saw the earth as a sphere. By using triangulation to measure the arcs between two points on the earth based on their respective relationships to the sun's zenith point at high noon, Erastosthenes, his contemporary and friend was the first to prove the curve of the earth's surface.

Johannes Kepler - Sphere Packing and the Music of the Spheres

 During the Renaissance and early Scientific Revolution of the 16th and 17th centuries, after more than a thousand years of what Boorstin termed, "scholarly amnesia" imposed by the religious imperialism of Christianity which sought to wipe out the religious aspects of the Neoplatonic thought, scholars returned to the Greek Classics. These early concepts of the ancient Greeks -- the relevance of the spherical, moving earth, the importance of phi and the significance of the Platonic solids -- were revisited once again.

In the 16th and 17th century, Johannes Kepler was one of the first scientists to revive the concept of the sacred nature of polyhedra. He explored tessellations to fill space on the plane and was the first to use sphere packing to understand space-filling. Kepler introduced a new family of stellated polyhedra to the expanding repertoire of three-dimensional symmetrical figures. He was also the first to sketch the complete family of Archimedean solids. Through his mathematical explorations, Kepler established a geometric cosmogony to understand the Earth and the Celestial Hemisphere.

Kepler's writings about the planetary orbits described their paths in relationship to each other as the "harmony of the spheres." He was sure of the archetypal significance of the five regular polyhedra and applied them to the macro world of the heavens by introducing a celestial "nested" geometry. He returned to the five Platonic solids, the cube, tetrahedron, octahedron, icosahedron and dodecahedron, as the forms inscribing the numerical relationships between the orbits of the planets. Kepler's nested polyhedra circumscribed the sphere shellings and described the ratios between each planet's orbit. Through this model of the planetary motions, he attempted to unite the arcs of spheres with the angles of the family of regular polyhedra.

Kepler remained convinced of the significance of nested polyhedra for his lifetime, even to the point of suggesting that expanding circumspheres could accomodate the elliptical derivations of a perfect spherical orbit that Tycho Brahe's extensive mathematical calculations contained. Even though the ratios fit into his nested polyhedra system for the five planets known in his day, Kepler could not prove this provocative theory. Still, throughout history Kepler's nested polyhedra has become one of the most widely reproduced early conceptualizations showing the geometry of our solar system.

Known mostly for his great contributions to astronomy, Kepler also introduced the concept of periodicity and coined the term "period" which denotes the time it takes for a planet to make one complete trip around the sun. Another not quite so well-known past time of Kepler's was the close packing of spheres which led to the first discoveries of space-filling polyhedra. In order to understand the beautiful patterns of snowflake crystals, he postulated that they were made of minute spheres of ice. In a short treatise, "On the Six-Cornered Snowflake," given as a gift to a friend in 1611, Kepler tried to describe why snowflakes have regular hexagonal shapes, while most flowers are regular pentagonal. He took his sphere-packing ideas into investigations with the closely packed seeds of pomegranates and the three-dimensional hexagonal lattice of the honeycomb. According to mathematician, Keith Devlin:

"Kepler saw a fundamental, mathematical connection between the formation of a snowflake in the atmosphere, the construction of a honeycomb by a family of bees and the growth of a pomegranate. According to his theory, the regular, symmetric patterns that arise in each case can be described and explained in terms of 'space-filling geometric figures,' such as his own discovery, the rhombic dodecahedron."7.

Kepler's speculation on the most efficient way to closely pack spheres with repeating regularity in three dimensions became known as the "Kepler Conjecture" and is a mathematical problem still being explored to this day. Kepler's studies of sphere packing were also integral to his understanding and articulation of periodicity, the regularly repeating motions of the planetary orbits on the macro scale and polyhedral shapes which recur in a three-dimensional lattice on the medio scale.8 

Besides Kepler's more well known return to the Pythagorean concepts of the music of the spheres and the sacred, archetypal nature of the polyhedra, he returned to a much more basic Pythagorean concept with his closest packing arrangements of spheres. He revisited the Tetractys, representing ten, the "perfect" or "divine" number, the number signifying the "Macrocosm." The tetractys was considered by Pythagoras the "form of the macrocosm." Many centuries after Pythagoras, medieval theologians also compared nature and the macrocosm to the number ten. In his book, An Adventure in Multidimensional Space : The Art and Geometry of Polygons, Polyhedra, and Polytopes, Japanese geometrician Koji Miyazaki saw Kepler's view of nature based on closest packing and the tetractys as a clue to the structural and experimental observation of the microscopic world of molecules and atoms. He links Fuller to Kepler because the origins of Fuller's synergetic geometry can also be traced to his experiments in the closest packing of spheres. He writes:

"According to Fuller's Synergetics, the whole universe can be understood as a closest packing of minute equiradial spheres of masses of energy. In particular, one regular tetrahedral portion derived from five close-packed layers is thought to be a fundamental form in nature. Like the tetractys of Pythagoras, it has a heart in the center. " 9a

Fuller also understood the significance of a hierarchy of polyhedral froms that nest and share a common nucleus as these close packed layers grow outwardly in all directions. Fuller compared his own "Cosmic Hierarchy" to Kepler's nested polyhedra.

"Intuitively hypersensitive and seeking to explain the solar system'sinterplanetary behaviors, Johannes Kepler evolved a concentric model of some of the Platonic geometries but, apparently frustrated by the identification of volumetric unity exclusively with the cube, failed to discover the rational cosmic hierarchy -- it became the extaordinary experience of synergetics to reveal this in its first written disclosure in 1944.9b

As we shall see later, Fuller's form which shows the equilibrium state of energetic processes, the cuboctahedron or vector equilibrium, was discovered by his closest packing experiments of 12 spheres around one. Fuller's vector equilibrium of closest packed spheres formed a with 12 spheres around a center sphere could be defined the three-dimensional figure which Kepler had also noted was the closest way to pack solid bodies in three dimensions. From his On the Six-Cornered Snowflake, Kepler noted the difference between cubic closest packing of soft spherical pellets and tetrahedronal closest packing. He concluded:

"In the former mode any pellet is touched by four neighbours in the same plane, and by one above and one below, and so on throughout, each touched by six others. The arrangement will be cubic, and the pellets, when subjected to pressure, will become cubes. But this will not be the tightest packing. In the second mode not only is every pellet touched by its four neighbours in the same plane, but also by four in the plane above and by four below, and so throughout one will be touched by twelve, and under pressure spherical pellets will become rhomboid. This arrangement will be the tightest possible, so that in no other arrangement could more pellets be stuffed into the same container."9c

Kepler's reference to the octahedron and "pyramid" refers to what Fuller later termed "octahedron-tetrahedron" or "octet truss." Kepler's note that "one will be touched by twelve" anticipated Fuller's 12 around 1 by 400 years. But Fuller was unaware of such a correspondence with Kepler's experiments in close packing.

Durer and da Vinci - Perspective and the Geometry of Curves

 During the Renaissance, great artists also returned to the early concepts of the ancient Greeks -- the spherical, moving earth, the importance of phi, and the significance of the platonic solids. The golden proportion and its applications were discussed in detail by Luca Pacioli (1509) in his ebullient mathematical treatise, "De Divina Proportione" Leonardo da Vinci, who illustrated this treatise, depicted within it the first images of polyhedra, such the dodecahedron, as open "cage" structures. The 16th century Renaissance masters, da Vinci and Albrecht Durer were both strongly influenced by Pacioli in their geometric explorations of perspective. These geometries were applied to art, architecture and to the earliest lettering styles, still models for many of our contemporary typefaces.10

 Looking closely at the historical period of the Renaissance, it is easy to see how the disciplines in their infancy were not yet delineated. Allegory was a linguistic tool of alchemy, the protochemistry of the time. Geometry was a tool for discovering and depicting the micro, macro and medio worlds. The compass, straightedge and ruler -- tools of the navigator, astronomer and mathematician -- were also the tools of the artist and philosopher. In his book, Album of Science - From Leonardo to Lavoisier,   I. Bernard Cohen writes:

The variety of technical problems faced by artists brought them into contact with numerous branches of science. Anatomy was associated with physiology, pigments with chemistry. The study of stresses and strains in architectural design was related to physics, and accurate renditions of the heavens drew on the new astronomy and cosmology...

Much attention was paid to the geometry of man as well, and the subject of human proportions was related to geometric figures. Inspired by remarks of the Roman architect, Vitruvius, a number of artists, including Durer and Leonardo, sought to illustrate a geometry of human proportions. The study of motion of human beings led to an analysis of curves that are formally identical to the curves used in classical astronomy to describe the planetary orbits, or epicycles.11

As Leonardo da Vinci and Albrecht Durer sought ways to accurately render three dimensions on a two dimensional plane, they drew from the basics of geometry to introduce perspective and proportion, essential tools used to depict life as accurately and beautifully as possible. This intersection of science and the arts gave birth to the discovery of perspective and became a major contribution the the Scientific Revolution. Finally, the full dimensionality of nature could be depicted on a flat surface bringing a new sense of realism to the study of the human body and the accurate visual depiction of the varied families of plants and animals for botony, biology and medicine. Cohen writes:

"Living creatures, buildings, landscapes, and scientific objects could be rendered in a manner that conveyed a realism lacking in earlier pictures. Various devices were created to produce drawings in perspective in a semi-mechanical way, chiefly by using a grid of lines at right angles to one another through which an artist viewed a scene or object. He then transferred the image to a piece of paper divided by a similar grid. . ."12

Albrecht Durer (1471-1528), one of the first masters of perspective, felt that geometry was the right foundation for all painting and decided to "teach its rudiments and principles to all youngsters eager for art." Indeed, in the 16th century his four volumes of Unterweisung were meant to be understood by artists and artisans rather than mathematicians. Through this treatise, Durer introduced a type of "workshop geometry"13 to other artisans of the day. Unterweisung is marked by its almost complete lack of mathematic proofs even though Durer consulted continually with geometricians. He felt artists, who created with their hands required a treatise that could provide an experimental, experiential approach, one that could be drawn on by using simple tools. It would be more accessible than an abstract treatise filled with mathematical calculations. Durer’s investigations into depicting animate forms such as the human figure and plant life led him to write "The Theory of Curves" as part of this series of treatises on art which influenced generations of artists. Even without calculations and abstract logic, Durer's Unterweisung influenced scientists like Kepler and Newton as well as the artists for whom he wrote the treatise.

Durer developed a "grid" through which a three-dimensional scene could be depicted with accuracy on a flat plane. A linear grid with its verticle and horizontal lines, however was far too simple. Durer then created a newer method of constructing polygons into tilings which filled space on the plane, similar to the tilings that were seen in Islamic decoration. Durer was familiar with Islamic tilings based on regularly repeating forms of the square and triangle. Since Durer’s art problem was depicting the curves of animate objects, the right angles and triangles of ordinary grids were too rigid to contain the unique curves of a leaf, a blade of grass, human breasts or buttocks. So he developed a new series of grids or "tessellations" giving special emphasis to "quadratura circuli" and the construction of polygon tilings which carried the special property of five-fold symmetry.  His tilings included the pentagon (five-sided figure), decagon (ten-sided figure), enneagon (nine-sided figure) and others.14 He studied polyhedra by using "nets." These nets or tessellations could be cut and folded along prescribed lines and joined at their edges to form the regular and semi-regular polyhedral solids.

Durer’s method, different than geometers of the day was to create these polygons not by mathematical calculation but by a working method whereby the "opening of the compass remains unchanged." Durer's fivefold angular figures closely approximated the arcs of curves but were drawn without calculations. He would superimpose these "tracery patterns" atop his artistic depiction of natural forms to create the proper classical curves of animate life. His fivefold tessellations allowed more freedom in depicting such curves. Durer eventually abandoned the superposition of the compass because such classical perfection was not found in the organic undulation of nature. Interestingly, modern crystallographers have returned to Durer’s quasi-periodic tilings to understand the fivefold symmetrical x-ray diffraction patterns of recently discovered quasicrystals, crystals which have defied the known rules of crystallography. Although brought about by the need to depict the curves of nature, Durer was able to create a series of tessellations that depicted the fivefold symmetry of structures on the molecular level more than four centuries before they were found to exist.

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