(BEING CONTINUED FROM 21/06/11)
The Dual of a Solid
There are two important relationships between the dodecahedron and the icosahedron. First, the mid-points of the faces of the dodecahedron define the points on an icosahedron and the mid-points of the faces of an icosahedron define a dodecahedron. The same is true of the cube and the octahedron. If we try it with a tetrahedron, we just get another tetrahedron. Each is called thedual of the other solid where the number of edges in each pair is the same, but the number of faces of one is the number of points of the other, and vice-versa. Plato probably did not know this. Plato describes the basic elements of the “polyhedral atoms”:
Now is the time to explain what was before obscurely said. There was an error in imagining that all the four elements might be generated by and into one another; this, I say, was an erroneous supposition, for there are generated from the triangles which we have selected four kinds–three from the one which has the sides unequal, the fourth alone framed out of the isosceles triangle. Hence they cannot all be resolved into one another, a great number of small bodies being combined into a few large ones, or the converse. But three of them can be thus resolved and compounded, for they all spring from one, and when the greater bodies are broken up, many small bodies will spring up out of them and take their own proper figures. Or, again, when many small bodies are dissolved into their triangles, by their total number, they can form one large mass of another kind. So much for their passage into one another. I have now to speak of their several kinds, and show out of what combinations of numbers each of them was formed. The first will be the simplest and smallest construction, and its element is that triangle which has its hypotenuse twice the lesser side. When two such triangles are joined at the diagonal, and this is repeated three times, and the triangles rest their diagonals and shorter sides on the same point as a center, a single equilateral triangle is formed out of six triangles, and four equilateral triangles, if put together, make out of every three plane angles one solid angle, being that which is nearest to the most obtuse of plane angles. And out of the combination of these four angles arises the first solid form which distributes into equal and similar parts the whole circle in which it is inscribed. The second species of solid is formed out of the same triangles, which unite as eight equilateral triangles and form one solid angle out of four plane angles, and out of six such angles the second body is completed. And the third body is made up of one hundred and twenty triangular elements, forming twelve solid angles, each of them included in five plane equilateral triangles, having altogether twenty bases, each of which is an equilateral triangle. The one element [that is, the triangle which has its hypotenuse twice the lesser side], having generated these figures, generated no more, but the isosceles triangle produced the fourth elementary figure, which is compounded of four such triangles, joining their right angles in a center, and forming one equilateral quadrangle. Six of these united form eight solid angles, each of which is made by the combination of three plane right angles; the figure of the body thus composed is a cube, having six plane quadrangular equilateral bases. There was yet a fifth combination which God used in the delineation of the universe with figures of animals.
Plato: Timaeus (54b-55c) p 1180
The last statement sounds strange and the meaning is not so clear. Plato also says “The Earth, if to look at it from above, is similar to the ball consisting of 12skin’s pieces”
I have found different explanations by others:
The French geologist de Bimon and Poincare considered that the form of the Earth represents by itself the deformed dodecahedron. The Russian geologist Kislitsin also used in his researches the idea about the dodecahedral form of the Earth according to which 400-500 millions years ago the geo-sphere of the dodecahedral form was turn into the geo-icosahedron. As the result the geo-dodecahedron appeared to be inscribed into the frame of the icosahedron.
Others consider these animals to represent the zodiac cycle (since zoo means animal in Greek).
Another explanation maybe is that the pentagon is associated with the golden section and the corresponding ratio is observed in various biological systems.
Wherefore it is clear that the very ratios of the planetary intervals from the sun have not been taken from the regular solids alone. For the Creator, who is the very source of geometry and, as Plato wrote, ‘practices eternal geometry,’ does not stray from his own archetype.” Thus, God, the eternal geometer must have given us the Platonic solids on behalf of the planetary orbit structure. They were made for each other. Johannes Kepler, “Harmonies of the World”, translated by Charles Glenn Wallis, Great Books of the Western World, Vol. 16, (Encyclopaedia Britannica, 1952), pp. 1017-18. The Science of the Harmony of the World (1619)
Kepler was also influenced by Plato’s Ideas and he used Plato’s regular solids to describe planetary motion as shown in a Figure above. He assigned the cube to Saturn, the tetrahedron to Jupiter, the dodecahedron to Mars, the icosahedron to Venus, and the octahedron to Mercury.
Luca Pacioli (1445-1517), inventor of the double bookkeeping method, in a stamp shown with a dodecahedron. (Italian Stamp, 1994 Michel 2319)
Pacioli devotes the second part of his book De Divina Proportione, published around 1509, to the Platonic solids. He writes:
As God brought into being the celestial virtue, the fifth essence, and through it created the four solids . . . earth, air, water, and fire … so our sacred proportion gave shape to heaven itself, in assigning to it the dodecahedron . . . the solid of twelve pentagons, which cannot be constructed without our sacred proportion. As the aged Plato described in his Timaeus.
It is interesting to note that nature likes some of these geometrical shapes for example in crystals. We find basic crystal units in the form of a cube, octahedron in NaCL and CaF2 respectively.
Back to Heisenberg:
…But the resemblance of the modern views to those of Plato and the Pythagoreans can be carried somewhat further. The elementary particles in Plato’s Timaeus are finally not substance but mathematical forms. “All things are numbers” is a sentence attributed to Pythagoras. The only mathematical forms available at that time were such geometric forms as the regular solids or the triangles which form their surface. In modern quantum theory there can be no doubt that the elementary particles will finally also be mathematical forms but of a much more complicated nature. The Greek philosophers thought of static forms and found them in the regular solids. Modern science, however, has from its beginning in the sixteenth and seventeenth centuries started from the dynamic problem. The constant element in physics since Newton is not a configuration or a geometrical form, but a dynamic law. The equation of motion holds at all times, it is in this sense eternal, whereas the geometrical forms, like the orbits, are changing. Therefore, the mathematical forms that represent the elementary particles will be solutions of some eternal law of motion for matter. This is a problem which has not yet been solved. Heisenberg, Physics and Philosophy: The Revolution in Modern Science
Finally there is also a interesting comment by Nicholas Gier and Gail Adele:
… the most amazing vindication of Plato has come from recent surveys of the universe that indicate that the universe may indeed be a dodecahedron, whose reflecting pentagonal faces give the illusion of an infinite universe when in fact it is finite. See New Scientist (October, 2003). See www.newscientist.com/news
What is time and did time exist before the Universe was created? Plato’s answer is that: time is an image of eternity:
When the father and creator saw the creature which he had made moving and living, the created image of the eternal gods, he rejoiced, and in his joy determined to make the copy still more like the original, and as this was an eternal living being, he sought to make the universe eternal, so far as might be. Now the nature of the ideal being was everlasting, but to bestow this attribute in its fullness upon a creature was impossible. Wherefore he resolved to have a moving image of eternity, and when he set in order the heaven, he made this image eternal but moving according to number, while eternity itself rests in unity, and this image we call time. For there were no days and nights and months and years before the heaven was created, but when he constructed the heaven he created them also. They are all parts of time, and the past and future are created species of time, which we unconsciously but wrongly transfer to eternal being, for we say that it ‘was,’ or ‘is,’ or ‘will be,’ but the truth is that ‘is’ alone is properly attributed to it, and that ‘was’ and ‘will be’ are only to be spoken of becoming in time, for they are motions, but that which is immovably the same forever cannot become older or younger by time, nor can it be said that it came into being in the past, or has come into being now, or will come into being in the future, nor is it subject at all to any of those states which affect moving and sensible things and of which generation is the cause. These are the forms of time, which imitates eternity and revolves according to a law of number. Moreover, when we say that what has become is become and what becomes is becoming, and that what will become is about to become and that the nonexistent is nonexistent – all these are inaccurate modes of expression. But perhaps this whole subject will be more suitably discussed on some other occasion. Plato, Timaeus 37c-38b
Comments by others
… Plato tells how Timaeus of Locri thought of the Universe as being enveloped by a gigantic dodecahedron while the other four solids represent the “elements” of fire, air, earth, and water. Euclid’s monumental treatise, the Elements , begins with the equilateral triangle, and culminates in the five Platonic solids, which are again the subject of the extra books XIV and XV (added a few centuries later). Sir D’Arcy W. Thompson once remarked that Euclid never dreamed of writing an Elementary Geometry. What Euclid really did was to write a very excellent account of the regular solids, for the use of Initiates.
Sculpture, University Library Gallery, Baltimore, 1981.http://www.design.upenn.edu/rf/aboutrf.html
I can’t think of anything more perfect than a tetrahedron. If someone came here from outer space, I’d hand them a tetrahedron, and they would understand. I’m sure people discover these things over and over again in different cultures. They are essential parts of the universe; they are parts of order, and, as such, they represent our order.
Art of the Tetrahedron
Some might consider the tetrahedron a rather humble geometric figure.
Any four points in space that are not all on the same plane mark the corners of four triangles. The triangles in turn are the faces of a tetrahedron. It�s the simplest of all polyhedra�solids bounded by polygons.
If each face is an equilateral triangle, the result is a regular tetrahedron, one of the five Platonic solids.
To sculptor Arthur Silverman of New Orleans, however, tetrahedra are very special. He has been investigating variations of tetrahedral forms for more than 20 years in sculptures displayed in public spaces in New Orleans and other cities from Florida to California.
“The tetrahedron is very exciting visually,” Silverman insists. “It�s very difficult to anticipate what you are going to see.”
Until the age of 50, Silverman had been a highly successful surgeon, practicing medicine with considerable enthusiasm and skill. Then he encountered an ailing colleague near death, who advised Silverman that if there was anything he really might want to do, then he ought to do it right away, before the chance slips away.
That encounter changed Silverman�s life. He returned to interests that had captured his attention when he was a teenager, had visited museums to gaze at statues, and had tried carving wood. Studying medicine at Tulane University, he met a sculpture teacher, who invited him to classes and taught him how to look.
In the midst of these early explorations, Silverman discovered the wonders of the tetrahedron, a form to which he returned with a passion many years later.
So, what can you do with (or to) a tetrahedron?
You can elongate a tetrahedron, stretching several edges to create a slim, stainless-steel tower, 60 feet high, then pair it with an identical tower to a produce an elegant structure that seems to soar in formation into the sky. Such a sculpture stands in the middle of a plaza fountain in New Orleans.
You can join tetrahedra together to form an aluminum cascade. The water wall at the Lee County Sports Complex in Ft. Myers, Fla., spotlights such geometric playfulness and activity. Or you can stack them symmetrically to produce a solemn memorial to Martin Luther King Jr. in Baton Rouge, La.
You can slice tetrahedra. A vast, interior wall at the Equitable Center in New Orleans is covered with aluminum tiles based on such a cross section.
You can divide tetrahedra, then rejoin them in various ways. You can look at what�s left when tetrahedra are cut out of a column or from inside a cube. You can elongate a tetrahedron and turn it inside out. You can stand it on edge or balance it on a vertex. The possibilities seem unlimited.
Silverman has produced more than 300 sculptures based on the tetrahedron. “When I get an idea, I play with it as long as I can,” he admits. The sculptor fabricates nearly all his pieces in his studio by welding together metal plates.
This is art that conveys no political, social, or historical message, Silverman remarks. “The sculpture is strictly a visual experience.”
One of the most intriguing of Silverman�s creations is an ensemble of sculptures he calls “Attitudes,” which are spread across a grassy area at the Elysian Fields Sculpture Park in New Orleans. The pieces are identical, but they have startlingly different orientations. When you walk around and look at them, “it�s hard to believe they are all the same structure,” Silverman remarks. “Every time you move, you see something different.”
To Nat Friedman, a mathematician and sculptor at the State University of New York in Albany, Silverman�s creation represents a hypersculpture�a way of seeing a three-dimensional form from many different viewpoints at once.
To see a two-dimensional painting in its full glory, you have to step away from it in the third dimension, Friedman says. To see a three-dimensional sculpture in its totality, you need a way to slip into the fourth dimension. Friedman calls this process hyperseeing. A hypersculpture, which consists of a set of several related sculptures, is one way to approximate that experience.
Various mathematical properties of the tetrahedron are described at http://mathworld.wolfram.com/Tetrahedron.html.
ii.6; Dox.334 Pythagoras: The universe is made from five solid figures which are also called mathematical; of these he says that earth has risen from the cube, fire from the pyramid, air from the octahedron, and water from the icosahedron,and the sphere of the All from the dodecahedron.
We should of course not forget that I presented only selections from Timaeus. I found a introduction of a translation of Timaeus interesting:
“Of all the writings of Plato the Timaeus is the most obscure and repulsive to the modern reader, and has nevertheless had the greatest influence over the ancient and mediaeval world. The obscurity arises in the infancy of physical science, out of the confusion of theological, mathematical, and physiological notions, out of the desire to conceive the whole of nature without any adequate knowledge of the parts, and from a greater perception of similarities which lie on the surface than of differences which are hidden from view. To bring sense under the control of reason; to find some way through the mist or labyrinth of appearances, either the highway of mathematics, or more devious paths suggested by the analogy of man with the world, and of the world with man; to see that all things have a cause and are tending towards an end—this is the spirit of the ancient physical philosopher. He has no notion of trying an experiment and is hardly capable of observing the curiosities of nature which are ‘tumbling out at his feet,’ or of interpreting even the most obvious of them. He is driven back from the nearer to the more distant, from particulars to generalities, from the earth to the stars. He lifts up his eyes to the heavens and seeks to guide by their motions his erring footsteps. But we neither appreciate the conditions of knowledge to which he was subjected, nor have the ideas which fastened upon his imagination the same hold upon us. For he is hanging between matter and mind; he is under the dominion at the same time both of sense and of abstractions; his impressions are taken almost at random from the outside of nature; he sees the light, but not the objects which are revealed by the light; and he brings into juxtaposition things which to us appear wide as the poles asunder, because he finds nothing between them. He passes abruptly from persons to ideas and numbers, and from ideas and numbers to persons,—from the heavens to man, from astronomy to physiology; he confuses, or rather does not distinguish, subject and object, first and final causes, and is dreaming of geometrical figures lost in a flux of sense. He contrasts the perfect movements of the heavenly bodies with the imperfect representation of them (Rep.), and he does not always require strict accuracy even in applications of number and figure (Rep.). His mind lingers around the forms of mythology, which he uses as symbols or translates into figures of speech. He has no implements of observation, such as the telescope or microscope; the great science of chemistry is a blank to him. It is only by an effort that the modern thinker can breathe the atmosphere of the ancient philosopher, or understand how, under such unequal conditions, he seems in many instances, by a sort of inspiration, to have anticipated the truth.”
Plato’s Molecule , Prinzbach, H., et al. Gas-phase production and photoelectron spectroscopy of the smallest fullerene, C20. Nature 407, (7 Sept)60, 2000.
Friedman, N. 1997. Reunification and hyperseeing. Ylem Newsletter(November/December).
Gardner, M. 1961. The five Platonic solids. In The 2nd Scientific American Book of Mathematical Puzzles and Diversions. New York: Simon and Schuster.
______. 1965. Tetrahedrons in nature and architecture, and puzzles involving this simplest polyhedron. Scientific American 212(February):112.
Peterson, I. 2001. Fragments of Infinity: A Kaleidoscope of Math and Art. Wiley.
Silverman, A. 1997. Tetrahedral variations. Ylem Newsletter (November/December).