THE GEOMETRY OF CREATION
Plato's Timaeus and the Regular Polyhedra
by Nicholas Gier and Gail Adele
(Revised October 10, 2003)
University of Idaho, Moscow, Idaho
"Let no one ignorant of geometry enter here"
(Sign over the entrance to Platos Academy)
Ancient Scots, Regular Polyhedra, and their Duals
Even though the discovery of the regular polyhedra is attributed to the Pythagoreans, there is some fascinating evidence that they may have been known in prehistoric Scotland. (This is even more amazing if it is true that the Pythagoreans only knew three of the five regular polyhedra.) In the Ashmolean Museum at Oxford University there are five rounded stones with regularly spaced bumps. The high points of each bump mark the vertices of each of the regular polyhedra. The stone balls also appear to demonstrate the duals of three of the regular polyhedra. (The Greeks apparently did not know this.) For example, if the six faces of the cube become points, they become the six vertices of the octahedron. If the eight faces of the octahedron become points, they become the eight vertices of a cube. Remarkably, groves in the Ashmolean stones indicate each of the duals, including the fact that the tetrahedron is its own dual. Instead of assuming that these ancient Scots were experts in solid geometry, we have hypothesized that they must have discovered all of this by sphere stacking, which will be demonstrate later in this article.
Figure 1. Neolithic Stone Balls (ca. 2,000 BCE) representing the Platonic Solids and their Duals. From left to right: an icosahedron with embedded dodecahedron dual; a tetrahedron with embedded tetrahedron; an octahedron encompassing a cube; a dodecahedron (?) but no dual; and a cube with an embedded octahedron.
Two problems arise, however, when one examines the five stone balls (as explained and represented by Keither Critchlow ) more carefully. First, if there is an the embedded tetrahedron in the one ball (second from left), then it cannot possibly be the dual because its edge length will always be one third of the length of the encompassing tetrahedron. (While the ratio for the tetrahedron's dual is quite elegant, the edge length ratios for the other regular polyhedra are not so elegant: .7071067810 of the edges of the octahedron inscribed in a cube; .4714045206 of the edges of the cube embedded in the octahedron; 1.17082094 of the edges of the icosahedron embedded in the dodecahedron; and .5393446629 of the edges of the dodecahedron inscribed in the icosahedron.) [1a] Second, we do not believe that the stone ball (second from right) that allegedly represents the dodecahedron is a dodecahedron, and even if it were, the ball does not show the dual in the way that the others do. Another way to phrase our objection is to claim that we could not produce this figure by stacking spheres.
Plato's Timeaus and the Regular Polyhedra
Some of the greatest achievements of Plato's Academy came in the areas of mathematics and astronomy. Heraclides Ponticus discovered the axial revolution of the earth and the revolution of Venus and Mercury around the sun. Eudoxus gave us the theory of proportion, the method of exhaustion, and the concentric spheres of the Ptolemaic cosmology. Theaetetus was the inventor of solid geometry and the general theory of incommensurables. He also constructed the regular polyhedra, demonstrated that each of them could be inscribed in a sphere, and proved that there could only be five. (2) Some have suggested that the Timaeus was the official scientific handbook of the Academy, written to show off the brilliant discoveries of Plato's students.
During a trip to Sicily in 367 B.C.E., Plato may have first learned of the regular polyhedra from Archytas, the last of the Pythagoreans.(3) In the Timaeus Plato used the regular polyhedra as the basic elements of the universe. (Here Plato seems to be following Empedocles and not Pythagoras, because the basic elements are brought together or dispersed by Platonic versions of Empedocles' Love and Strife.) Plato was dissatisfied with Democritus' atoms, primarily because their size and shape were left to chance. Instead, Plato's creator god, the Demiurge, gave the basic elements definite geometrical shapes, but just as small and imperceptible as the atoms. The Demiurge chose the first four regular polyhedra for this purpose, because they were the most perfect, the most beautiful, and therefore the best. As earth is most stable and immobile of elements, God chose the cube, with its large base areas, to constitute it. Fire is the least stable and most mobile, so it is made up of regular triangular pyramids (tetrahedra). Water is more stable than air, so it is composed of regular icosahedra. Plato observes (59d) that the bases of icosahedra "give way" more easily than cubes, so that explains why water is able to flow. Perfect octahedra are left to make up the air. Fire, being composed of the smallest solid with the sharpest points and edges, is the most destructive of the elements. "We all feel that fire is sharp, and we may further consider the fineness of the sides, and the sharpness of the angles, and the smallness of the particles, and the swiftness of the motion--all this makes the action of fire violent and sharp, so that it cuts whatever it meets" (61e). It can of course burn the earth, smelt its ores, and boil water into air.
According to Plato, fire bodies destroy air and water by cutting. Air bodies can be cut by fire or crushed by water, giving two fire bodies. Water bodies may be cut by fire or they can, for example, crush twenty fires to give five bodies of air and two bodies of water. Similarly, earth may either be cut by fire, air, and water but may crush them all. In Plato's view the boiling and evaporation of water are described in elegant mathematical terms: One icosahedron of water with twenty triangular sides becomes two octahedra of air (16 faces) and one pyramid of fire (4 faces). As the final transformation of water is to air and finally to the fire of the heavens, one icosahedron of water can ultimately become five pyramids of fire.
Plato believed in only four primary elements (some added ether as a fifth), so where was he to use the beautiful dodecahedron? The Demiurge used it for the whole universe and made patterns of animal figures on each pentagon, an obvious reference to the twelve signs of the Zodiac (55c). In Plato's dialogue Phaedo (110b) Socrates describes the earth as a ball "made of twelve pieces of skin." If these pieces were inflexible and regular polygons, the faces would have to be pentagons and the solid would have to be a dodecahedron. As leather is a flexible material, the earth would appear as a sphere, like the leather "medicine balls" some use in gymnasiums. It is conceivable that Plato and his students knew that if inscribed in the same sphere the dodecahedron has a greater volume than the icosahedron. Also if a cube and octahedron are inscribed in the same sphere, the cube has the larger volume.
Interesting enough, Plato's basic building blocks are not the solids themselves or even their triangular or square faces; rather, the basic "atoms" are two different types of triangles. One is half of the equilateral triangles that make up the faces of the pyramid, octahedron, and icosahedron. This is a right triangle with angles of 30 degrees, 60 degrees, and 90 degrees. The other is half of the square that is a face of the cube. This triangle is a right isosceles triangle. Plato argues that these triangles are also the best and most perfect. Not only is this selection of "geometric atom" odd, but so is Plato's method of constructing the solids from them, details of which I have omitted.(4) Furthermore, if transformation of all the basic elements into one another is Plato's main goal, he does not achieve it with this method. One can interchange the half equilateral triangles to make the faces of pyramids, octahedra, and icosahedra, but the half squares obviously cannot be used to construct anything but cubes. A further problem is that the dodecahedron requires pentagons for its construction. Later ancient mathematicians demonstrated that the pentagon faces could indeed be divided into thirty congruent scalene triangles, but of course different from those comprising the three polyhedra with triangular faces.(5)
A Platonic Microcosmos and Macrocosmos
Einstein and Plato had at least one thought in common: God does not play dice with the world. It looks as if both have been proved wrong by the overwhelming vindication of Heisenberg's indeterminacy principle. The other argument against Plato and Einstein is the interesting proof about efficient packing. Given the alternative between deliberately placing spheres in a packing case and placing one and shaking (and repeating), the latter method succeeds in packing more spheres in the container.
These major points notwithstanding, Paul Friedlšnder still believes that Plato is the father of modern physics in other significant ways. (6) Like modern science Plato combines the idea of elementary particles and mathematical structure. As a brilliant anticipation of modern physics, the "receptacle" of the Timaeus is combination of space, matter, and energy. Classical physics followed the Greek atomists in separating matter and energy, but modern physics has returned to the inseparability first proposed by Plato.
Although Plato's specific geometry is now unscientific, it is still a vindication of his persistent formalism to find that, as Friedlšnder states, "today the molecule of methane gas is visualized spatially as a regular pyramid with four hydrogen atoms at its four corners and a carbon atom in the center. A tetravalent carbon atom is represented by some chemists in the form of a tetrahedron, with the united atoms placed at its four vertices." (7) In the macrocosmos Plato is further supported: lead ore and rock salt are composed of crystalline cubes; octahedral crystals form fluorite ore; garnet crystals are dodecahedra; iron pyrite comes alternatively as cubic, octahedral, and dodecahedral crystals, and silicate crystals appear as tetrahedra.
The most recent discovery in the geometry of creation came in 1985 when it was learned that the viruses responsible for most colds and meningitis are icosahedrons. Michael Rossmann of Purdue University led a team that mapped the three-dimensional structure of these viruses. This knowledge aided biochemists in designing a drug that would render this specifically shaped object inactive. Finally, 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. (8)
1. See Keith Critchlow, Time Stands Still (London: Thames and Hudson, 1979).
1a. We are indebted to John Woll, Professor of Mathematics at Western Washington University, for these calculations. Woll was a bit surprised by the fact the edges of the inscribed icosahedron were actually longer than the edges of the encompassing dodecahedron. As Woll states: "I became reconciled to this when I computed that the centers of two regular pentagons joined along an edge are actually further apart than the length of their common edge."
2. See Carl B. Boyer and Uta C. Merzbach, A History of Mathematics (New York: John Wiley, 2nd ed., 1969), p. 98.
3. Ibid., p. 97. Boyer and Merzbach state this qualification: "A scholium (of uncertain date) to Book XIII of Euclid's Elements reports that only three of the five solids were due to the Pythagoreans, and that it was through Theaetetus that the octahedron and the icosahedron became known" (p. 98). T. L Heath believes that the Pythagoreans surely knew of the latter two, but it was Theaetetus who "was the first write at any length about [them], as he was probably the first to construct all five theoretically and investigate fully their relations to one another and the circumscribing spheres" (A History of Greek Mathematics [Oxford, 1921], Vol. 1, p. 162).
4. See Francis M. Cornford, Plato's Cosmology (Humanities Press, 1952) for a detailed explanation for Plato's seemingly arbitrary approach (pp. 210-239).
5. See ibid., p. 298.
6. Paul Friedlšnder, Plato (Princeton, 1969), vol. 1, chap. 14.
7. Ibid., p. 253.
8. See New Scientist (October, 2003). See www.newscientist.com/news.