Theorem of the Day (February 24, 2026) : Theaetetus’ Theorem on the Platonic Solids
Source : Theorem of the Day / Robin Whitty
pdf : https://www.theoremoftheday.org/GeometryAndTrigonometry/Platonic/TotDPlatonic.pdf
notes : https://www.theoremoftheday.org/Resources/TheoremNotes.htm#132
@paysmaths @Theoremoftheday I think it is worth noting that there are two different definitions of semi-regular/archimedean solids.
- Requiring the same configuration at each vertex gives you 14 solids.
- Requiring that there is a symmetry carrying any vertex to any other gives you 13.
The rhombicuboctahedron and the pseudo-rhombicuboctahedron (see attached images) have the same vertex configurations, but only the first one satisfies the symmetry condition.
Archimedes found 13 solids, according to Pappus' ‘Collection’ (our only source for his work on this).
For details of the long-standing confusion of the two definitions and the resulting classes, see B. Grünbaum. ‘An enduring error’. In: Elemente der Mathematik. 64, no.3 (2009), pp.89–101. DOI: 10.4171/EM/120
Paysages Mathématiques
Alan J. Cain