In Plato's Timaeus there are four elements that make everything, but these elements are in turn made up of triangles. But what are the triangles....
Is a triangle -- according to Plato/Socrates -- a "thing", is it physical?
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No; it is a geometrical figure.– Mauro ALLEGRANZASep 12, 2017 at 6:41
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See Plato's Timaeus: "In accordance with the requirements for the construction of the body of the universe, the Craftsman begins by fashioning each of the four kinds “to be as perfect and excellent as possible…” He selects as the basic corpuscles (sômata, “bodies”) four of the five regular solids: the tetrahedron for fire, the octahedron for air, the icosahedron for water, and the cube for earth. (The remaining regular solid, the dodecahedron, is “used for the universe as a whole,” since it approaches most nearly the shape of a sphere.)– Mauro ALLEGRANZASep 12, 2017 at 6:45
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... The faces of the first three of these are composed of equilateral triangles, and each face is itself composed of six elemental (scalene) half equilateral right-angled triangles".– Mauro ALLEGRANZASep 12, 2017 at 6:45
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1See Lloyd's Chemistry of Platonic Triangles:"Plato’s geometrical theory of what we now call chemistry, set out in the Timaeus, uses triangles, his stoicheia, as the fundamental units with which he constructs his four elements" represented by Platonic solids.– ConifoldSep 12, 2017 at 20:29
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1 Answer
Plato was making a point about how geometry was at the bottom of a natural explanation of the physical world. This is likely due to the strong Pythagorean influence on him. I don't think that this influence is explicitly mentioned in his dialogues, but it's definitely there. Besides, Diogenes Laertes in his summa of Greek philosophers reports he was known to have kept company with known Pythagorean philosophers.
It might be of interest to know that triangles are one way to investigate the geometric properties of manifolds. It's an important aspect of algebraic topology in whats called - to throw a bit of jargon about - homology & cohomology.
It also might be of interest that in one approach to quantum gravity networks of triangles are immersed into slices of spacetime - which traditionally speaking, has a manifold structure - and then that manifold subtracted to leave only that network of triangles. Their attitude is that they somehow want to discretise spacetime.
Perhaps, that too was in the mind of Plato. We don't know. However we do know that one of his followers and sparring partners, Aristotle did declare infinite divisibility an impossibility...
So maybe he knew something after all - or maybe it is just a startling coincidence ...
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I actually have taken a course on algebraic topology and de Rham cohomology. Really cool answer! This also seems relevant: physics.stackexchange.com/questions/219710/…. I'm don't really know particle physics, so I can't follow that link I shared... but it does look interesting. Jun 13, 2018 at 19:58
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@Clclstdnt: Thats great - I mean the course on algebraic topology! It's nice too to know that my answer was appreciated. I wasn't expecting someone to come along here knowing what algebraic topology was or I'd have thrown in a bit more jargon - like simplices! Jun 15, 2018 at 20:20