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Now we generally think of the Platonic solids as being the regular convex polyhedra. And while the Ancient Greeks were aware of this solids as being particularly special, I don't believe that it is likely that they thought of them in the same way.

There are two main reasons I suspect this:

  1. The modern definition of regular polytope, "A polytope whose symmetry group acts transitively on its flags" involves some concepts which seem a little to synthetic for the Ancient Greeks.
  2. There are lots of other properties held by the Platonic solids as a set that make them unique among the solids.
    • They are the only regular finite acoptic polyhedra (having no self-intersection or skew elements). I doubt they even considered infinite or non-acoptic polyhedra to be polyhedra, so convexity would likely have been a redundant condition.
    • They are the only finite weakly-regular (isohedral, isotoxal and isogonal) acoptic polyhedra.
    • Every equivelar (every vertex having the same degree and every face having the same number of edges) finite acoptic polyhedron is a distortion of a Platonic solid.

There are more ways to slightly modify or combine the above properties to get new classifiers that specify the Platonic solids. Additionally there are plenty more properties that hold for the Platonic solids specifically, but which are unlikely to have been discovered by the Ancient Greeks.

What did the Ancient Greeks think the Platonic solids were? What property exactly did they consider to set that set apart?

I'd appreciate answers that are grounded in ancient sources, but I also understand that these sources can be vague by modern standards, so some interpretation might be necessary.

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    $\begingroup$ Euclid says, near the end of the proof of XIII.18, "I say next that no other figure, besides the said five figures, can be constructed which is contained by equilateral and equiangular figures equal to one another." A common hypothesis is that this is due to Theaetetus. $\endgroup$
    – Michael E2
    Dec 2, 2023 at 16:48
  • $\begingroup$ Greeks did not think of space as made of points and define solids as sets of them. Nor did they consider a class of polyhedra or polytopes of which platonic solids would be special cases with properties that "set them apart". Instead, they gave a direct construction of vertices and edges of some cases starting from a "given" sphere and its diameter, mostly using plane projections. Then proved that faces are regular polygons and found some ratios of sides. $\endgroup$
    – Conifold
    Dec 3, 2023 at 8:56

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