 
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 selfdesignated "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
(525350 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 earthcube, firetetrahedron, airoctahedron,
watericosahedron, (and universedodecahedron)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 wellknown tale in the history of mathematics tells how Archimedes
discovered a method to determine the volume of a nonuniform 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 nonuniform 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 threedimensional semiregular 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 spacefilling. Kepler introduced a new family of stellated
polyhedra to the expanding repertoire of threedimensional 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 wellknown past time of Kepler's was the close
packing of spheres which led to the first discoveries of spacefilling
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 SixCornered 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 spherepacking
ideas into investigations with the closely packed seeds of pomegranates
and the threedimensional 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 'spacefilling 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 threedimensional 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 closepacked 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 threedimensional
figure which Kepler had also noted was the closest way to pack solid
bodies in three dimensions. From his On
the SixCornered 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 "octahedrontetrahedron" 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 semimechanical 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 (14711528), 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 threedimensional
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 fivefold
symmetry. His tilings included the
pentagon (fivesided figure), decagon
(tensided figure), enneagon
(ninesided 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 semiregular 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 quasiperiodic tilings
to understand the fivefold symmetrical xray
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.
Table
of Contents
[I] [II]
[III] [IV]
[V] [VI]
[VII] [VIII]
References
copyright 1997, Bonnie DeVarco
