The Cosmos / Thought Experiments


Beautiful Losers: Plato’s Geometry of Elements

This essay is part of the series Beautiful Losers.

Plato believed that he could describe the Universe using five simple shapes. These shapes, called the Platonic solids, did not originate with Plato. In fact, they go back thousands of years before Plato; you can find stone models (perhaps dice?) of each of the Platonic solids in the Ashmolean Museum at Oxford dating to around 2000 BC, as pictured below. But Plato made these solids central to a vision of the physical world that links ideal to real, and microcosm to macrocosm in an original, and truly remarkable, style.


AN1927.2727-31 Neolithic Carved Sandstone Balls, Copyright Ashmolean Museum, University of Oxford.

Let me explain, first, what the Platonic solids are. To begin, consider something simpler: regular polygons. Regular polygons, by definition, are two-dimensional shapes bounded by sides of equal length, each making the same angles with its neighbors. Equilateral triangles, squares, regular pentagons, and so on are all regular polygons. Platonic solids are the three-dimensional analog of regular polygons, and prove to be far more interesting. Platonic solids are bounded by regular polygons, all of the same size and shape. One can prove mathematically that there are exactly five Platonic solids. Here they are:


The Platonic solids and their proposed identification with fundamental world-elements

The tetrahedron has four triangular faces, the cube six square faces, the octahedron eight triangular faces, the dodecahedron twelve pentagonal faces, and the icosahedron twenty triangular faces. Plato proposed that four of these solids built the Four Elements: sharp-pointed tetrahedra give the sting of Fire, smooth-sliding octahedra give easily-parted Air, droplety icosahedra give Water, and lumpish, packable cubes give Earth. The dodecahedron, at last, is the shape of the Universe as a whole. Later Aristotle emended Plato’s system, suggesting that dodecahedra provide a fifth essence—the space-filling Ether.

Plato’s ideas lent dignity and grandeur to the study of geometry, and greatly stimulated its development. The thirteen and final book of Euclid’s Elements, the grand synthesis of Greek geometry that is the founding text of axiomatic mathematics, culminates with the construction of the five Platonic solids, and a proof that they exhaust the possibilities. Scholars speculate that Euclid planned the Elements with that climax in mind from the start.

From a modern scientific perspective, of course, Plato’s mapping from mathematical ideals to physical reality looks hopelessly wrong. The four (or five) ancient “elements” are not simple substances, nor are they usable building blocks for constructing the material world. Today’s rich and successful analysis of matter involves entirely different concepts. And yet…

In its general approach, and its ambition, Plato’s utterly mistaken theory anticipated the spirit of modern theoretical physics. His program of describing the material world by analyzing (“reducing”) it to a few atomic substances, each with simple properties, existing in great numbers of identical copies, coincides with modern understanding.

Deeper still penetrates his insight that symmetry defines structure. Plato sensed enormous potential in the fact that asking for perfect symmetry leads one to discover a small number of possible structures. Based on that foundation, and a few clues from experience, the outlandish synthesis that his philosophy suggested should be possible, to realize the World as Ideas, might be achievable. And clues were there to be found: Near-coincidence between the number of perfect solids (five) and the number of suspected elements (four); suggestions of how observed qualities might reflect underlying shapes (e.g., the sting of fire from the sharp points of tetrahedra). One must also admire the boldness of genius in seeing an apparent defect in the theory—five solids for four elements—as an opportunity for crowning creation, either with the Universe as a whole (Plato) or with space itself (Aristotle).

Modern physicists, when seeking equations to describe the unfamiliar laws of the microcosm, must make guesses based on fragmentary information. Optimistically—and lacking constructive alternatives—they have turned, as Plato did, to symmetry as their guide. Symmetry of equations is perhaps a less familiar idea than symmetry of shapes, but there is nothing obscure or mystical about it. We say an equation, like a shape, displays symmetry when it allows changes that make no change. So for instance the equation

X = Y

has a nice symmetry to it, because exchanging X for Y changes it into this:

Y = X

and this transformed equation expresses exactly the same content as the original. On the other hand X = Y + 2, say, turns into Y = X + 2, which expresses something else entirely. As this baby example demonstrates, symmetric equations can be rare and special, even when the symmetry involves quite simple transformations.

The equations of interest for physics are considerably richer, of course, and the “changes that make no change” we hope they allow are much more extensive and elaborate. But the central idea inspiration remains, as it was for Plato, the hope that symmetry defines a few interesting structures, and that Nature chooses one (or all!) of those most beautiful possibilities.

Plato’s Beautiful Loser was, in hindsight, a product of premature, and immature, ambition. He tried to leap directly from beautiful mathematics, some imaginative numerology, and primitive, cherry-picked observations to a Theory of Everything. In this his ambition was premature. Also, Plato failed to draw out specific consequences from his ideas, or to test them critically. He sketched an inspiring world-model, but was content to “declare victory” without engaging any serious battles. The mature and challenging form of scientific ambition, which aspires to understand specific features of the world in detail and with precision, emerged only centuries later.

Tell us what you think on Twitter, Facebook, or email.

Frank Wilczek

    Frank Wilczek has received many prizes for his work in physics, including the Nobel Prize of 2004 for work he did as a graduate student at Princeton University, when he was only 21 years old. He is known, among other things, for the discovery of asymptotic freedom, the development of quantum chromodynamics, the invention of axions, and the exploration of new kinds of quantum statistics. Frank is currently the Herman Feshbach professor of physics at MIT. His latest book is A BEAUTIFUL QUESTION: Finding Nature’s Deep Design.

    • Guest

      Thank you for writing this. I really enjoyed learning from you!

    • Spartanxx2032

      Plato’s solids, were they precursors of the discovery of fractal symmetries?

    • Stephen

      I wish to point out that your contention (widely shared by physicists) that Plato was wrong in associating the tetrahedron, octahedron, cube & icosahedron with the physics of matter is itself wrong. Divide their faces into their sectors and then divide into sectors each triangle formed by their centres and by their edges and sides of face sectors and you will discover that 2480 points, lines & triangles other than vertices surround the axes of the 5 Platonic solids, that is, 496 geometrical elements on average in addition to its vertices surround the axis of a Platonic solid, 248 being in each half. This is the regular polyhedral counterpart of E8xE8, one of the two symmetry groups with dimension 496 known to describe superstring interactions that are free of quantum anomalies.

      Far from being a loser, Plato’s association of the five regular polyhedra with the physics of matter turned out to be a winner……

      • Indeed. “Changes that make no change” symmetry also leads us to how dual icosahedron-E8 helps the universe evolve increasing complexity for the least energy via #Universality as multiscale Superposition-Entanglement. So we submit that Plato was wrong in the specifics (especially on elements), but absolutely correct in his elementary intuition that optimal, analogical geometries scribe the arcs of our worlds, however many of them there are. Substantiation? My inquiries, cooperation and consultations with Garrett Lisi, Adrian Bejan and Stuart Hameroff (E8, Constructal and Orch-OR vrs Phi-IIT ToEs respectively), as shared here:

    • Maybe, but your underlying premises seem to be predicated on some species of materialism. The problem with any materialism and consequently any argument so predicated is that when one traces back the assumptions of materialism to its metaphysical appeals one runs into an inconstancy. I don’t know what Plato — and so presumably Socrates — actually knew (or didn’t) and what might have been the source of that knowledge if it existed, but I personally don’t believe it prudent to count it out just yet (and between the two of us I am not the one arguing from a position that is ultimately logically invalid;) I would suggest we give Plato’s theory a couple more centuries (not as arbitrary as it might first appear) to see how it plays out before we call the winner.

    • For centuries Aristotle’s four elements (plus spirit) served to explain the conceptual makeup of the world. It still does for the purposes for which it is appropriate. In fact, you *can* build the world from a platonic solid. The tetrahedron will tile space, and is the model for the molecular physics that describes that process. In fact, when tetrahedra and octahedra come together to tile space, they form a very strong latticework – made up of the octet truss. Chemistry describes molecular bonding by the geometric shapes that the chemical bonds describe. Crystal latticeworks of this kind make up everything from ice to titanium steel.


      blah blah blah i love pie

    • Goodie Bag

      wonderful article, love it 😉

      • fabfab

        soooo fab 😉

    • al

      The great pythagorean achievement was seeing the uniform regular polyhedra as a finite class ( and the proof was finally produced in Euclid 13). In a sense the neolithic collection is a fake: stone balls have been carved in many more or less regular patterns but there is no clue that the five chosen for the picture were in some sense special. (LLoyd DR, 2012 BHMS)
      Aristotle, a student of Plato, added a fifth element so his system included two inverse pairs plus one without an inverse. Today we say that the 5 solids are ‘dual’, in two pairs and one self dual. (There is no hint of Plato knowing any of this: for his system the dodecahedron is without inverse, but by chance he has hit upon the inverse/dual pair cube-octahedron (Plato’s Geometrical alchemy, 2013 []))
      Plato says (Timaeus 31-2) that tetrahedron, octahedron, icosahedron, cube form a nonlinear progression and there is still no good answer what he meant by this (any suggestions?).
      Also: in various dimensions there are analogues – in d=1 an undenumarable infinity, in d=2 a denumarable infinity, in d=3 N=5, in d=4 N=6 and for d>4 N=3 (Schlaffli 1898). So Plato perhaps had
      a hunch that we live in an interesting world.

    • Brian Johnston

      If we consider that earth air fire and water are actually different states energy comes in then it all makes sense. Air is electricity, water is magnetism, fire is pure energy and earth is hadrons and bosons. In physics this would be a quadrapole with the fifth element being the center point as the manifestation of the particular manifestation.

    • PlatoDNA

      Wow! The negative comments towards Plato only seem to be true about this essay. According to my research, Plato is still way ahead of modern science. I have found many clear examples of this throughout his writings. Plato understood the four main fundamental forces of nature, they are his four elements. He describes the earth’s electromagnetic field in detail in his Phaedo, and has many passages that line up with a lot of theories in “modern science” throughout his writings. His elements are not ‘building blocks’, Plato describes them well in Timaeus…

      ”as the several elements never present themselves in the same form, how can anyone have the assurance to assert positively that any of them, whatever it may be, is one thing rather than another? No one can. But much the safest plan is to speak of them as follows. Anything which we see to be continually changing, as, for example fire, we must not call ‘this’ or ‘that’, but rather say that it is ‘of such a nature’, nor let us speak of water as ‘this’, but always as ‘such’, nor must we imply that there is any stability in any of those things which we indicate by the use of the words ‘this’ and ‘that’ supposing ourselves to signify something thereby, for they are too volatile to be detained in any such expressions as ‘this’, or ‘that’, or ‘relative to this’, or any other mode of speaking which represents them as permanent. We ought not to apply ‘this’ to any of them, but rather the word ‘such’, which expresses the similar principle circulating in each and all of them”.

      Also in Plato’s Timaeus, he describes the structure of DNA in perfect detail and a few thousand years before Watson and Crick ever existed.

      “…in imitation of their own creator they borrowed portions of fire and earth and water and air from the world, which were hereafter to be restored—these they took and welded them together, not with the indissoluble chains by which they were themselves bound, but with little pegs too small to be visible, making up out of all the four elements each separate body, and fastening the courses of the immortal soul in a body which was in a state of perpetual influx and efflux. Now these courses, detained as in a vast river, neither overcame nor were overcome, but were hurrying and hurried to and fro, so that the whole animal was moved and progressed, irregularly however and irrationally and anyhow…”

      Modern science generally describes DNA’s structure as chains with four pegs holding them together that make up each separate person, and are too small to be visible. DNA’s ladder like sides are said to breathe in and out, a lot like Plato’s influx and efflux. There is about six feet of DNA in almost every cell of the body, approximately 100 trillion cells in an adult, that’s over 113 billion miles worth, those rivers of DNA are very vast! Science also says that DNA bends and twists a billion times a second matching up with how Plato says they ”were hurrying and hurried to and fro”. Of course there is also the fact that DNA is responsible for our genetics and evolution exactly how Plato describes it “the whole animal was moved and progressed, irregularly however and irrationally and anyhow”! That this paragraph describes DNA perfectly is undeniable.

      But I guess that’s not very specific. What a loser!

      There are many other examples like this that can be found throughout his writings. Plato describing many concepts found in modern science in perfect detail over 2,000 years ago shows that his understanding of nature was many levels above the views of modern science.

    • “Changes that make no change” symmetry also leads us to how dual icosahedron-E8 helps the universe evolve increasing complexity for the least energy via #Universality as multiscale Superposition-Entanglement. So we submit that Plato was wrong in the specifics (especially on elements), but absolutely correct in his elementary intuition that optimal, analogical geometries scribe the arcs of our worlds, however many of them there are. Substantiation? My inquiries, cooperation and consultations with Garrett Lisi, Adrian Bejan and Stuart Hameroff (E8, Constructal and Orch-OR vrs Phi-IIT ToEs respectively), as shared here:

    • Thepaultm

      Hi Frank.
      You obviously failed to mention that CubicWonder is the only 3d fractal system of perfect order that gives us 3 types of DNA geometry. This fractal system is basically the geometry of Plato the 5 platonic solids. It proves the chaos theory is a load of baloney. Before you start calling Plato a loser explain why CubicWonder should be ignored.

    • Dodecahedron, Universe (or Aether (actually a string particle field)) == correct