A Timeless History
Plato matched each of the five to an element in the Timaeus: fire to the tetrahedron, earth to the cube, air to the octahedron, water to the icosahedron — and the cosmos to the dodecahedron. Euclid built each one in Book XIII of the Elements and closed the book by proving the list is complete. Leonardo drew all five, solid and hollow, for Pacioli in 1509. Kepler nested them inside one another in 1596 to explain the solar system, failed, and found the laws of planetary motion instead.
SourcePlato, Timaeus (c. 360 BC) · Encyclopaedia Britannica — britannica.com · Euclid, Elements, Book XIII · Mathematical Association of America, Convergence — maa.org · Luca Pacioli, De divina proportione (Venice, 1509), plates by Leonardo da Vinci · Johannes Kepler, Mysterium Cosmographicum (Tübingen, 1596) · Raphael, The School of Athens (1509–11), Musei Vaticani
Science In Action
Pyrite grows natural cubes so sharp they look machine-cut — geology does geometry on its own • Some plankton build glassy skeletons shaped like the icosahedron — Haeckel drew them from the deep sea in 1887 • Shave the icosahedron's twelve corners and you get the football — and carbon-60, the 1996 Nobel Prize molecule • Why only five? At one corner, triangles stop at five, squares and pentagons at three — try it in your hands
SourceEncyclopaedia Britannica — britannica.com, “pyrite” · The Nobel Prize — nobelprize.org, Chemistry 1996 · Ernst Haeckel, Report on the Radiolaria, Challenger expedition (1887)
Arts
Leonardo drew each of these five twice for Pacioli — solid, then hollow, the far side showing through the near side — the first time anyone had illustrated solids that way. Woodworkers were ahead of him: the Gubbio studiolo of about 1478, now in the Met, is panelled in polyhedra cut from veneer.
SourceLuca Pacioli, De divina proportione (Venice, 1509), plates by Leonardo da Vinci · MAA Convergence, “Mathematical Treasure: Luca Pacioli's Divina Proportione” · The Metropolitan Museum of Art — metmuseum.org, Studiolo from the Ducal Palace in Gubbio, 39.153
Applications Today
The five turn up wherever nature builds from identical parts: carbon bonds as a tetrahedron, salt crystallises as a cube, diamond as an octahedron, virus shells as an icosahedron. Plato was wrong about why, and uncannily right about where.
SourceRoyal Society of Chemistry — rsc.org · Smithsonian National Museum of Natural History — naturalhistory.si.edu · PNAS — pnas.org
One more thingEuclid's Elements ends with this: all five solids constructed, and a proof that there are no others. The last word of Greek mathematics is this set.
SourceEuclid, Elements, Book XIII · Mathematical Association of America, Convergence — maa.org
Everything perfectly regular in three dimensions is in your hands. There is nothing else.
The School of Athens (Scuola di Atene) — detail: Plato, holding the Timaeus, and Aristotle — Raphael (Raffaello Sanzio) — 1509–1511 — Stanza della Segnatura, Apostolic Palace, Vatican — Musei Vaticani; scan: The Yorck Project (2002), via Wikimedia Commons — File:Raffael 058.jpg (6045 × 4671)
