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Some families of polyhedra

Some families of polyhedra. Connecting technical definitions and funky pictures. Platonic Solids – what’s so special?. What if we drop some of these conditions? . E.g. Question: What does this give us?

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Some families of polyhedra

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  1. Some families of polyhedra Connecting technical definitions and funky pictures

  2. Platonic Solids – what’s so special? What if we drop some of these conditions?

  3. E.g. Question: What does this give us? Answer: the Platonic solids again, plus the cuboctohedron and the icosidodecahedron.

  4. This gives us the 13 Archimedean solids… …and infinitely many prisms and antiprisms.

  5. This additionally gives us the 92 Johnson solids.

  6. This gives us… …nothing new!

  7. The rhombic dodecahedron and rhombic triacontahedron.

  8. This gives us the 13 Catalan solids, plus infinitely many bipyramids and trapezohedra.

  9. Just one infinite family of disphenoid tetrahedra.

  10. Infinitely many isogonal polyhedra.

  11. This gives us…. …nothing new!

  12. The 4 Kepler-Poinsot polyhedra.

  13. The great icosidodecahedron, together with the dodecadodecahedron and its three ditrigonal variants

  14. The 75 Uniform polyhedra…. …plus Skilling’s figure (maybe). …plus infinitely many (crossed) prisms & antiprisms

  15. Other refinements Convex means – very approximately – “no sticking out bits” and “no holes”. We can disassemble this. Roughly speaking, allowing holes but not sticking out bits (or self-intersection) gives… …the Stewart toroids.

  16. Hidden assumptions! Did I mention that all the different corners must be connected somehow? Dropping this gives… …5 regular compound polyhedra

  17. Who said it had to be finite? What’s the difference between a tiling and a polyhedron? Not much! The 3 regular Euclidean tilings… …plus infinitely many hyperbolic & elliptic tilings …plus 3 infinite regular skew polyhedra

  18. Into higher dimensions What are the ‘Platonic solids’ in 4 dimensions? • Pentachoron • Hypercube • Hexadecachoron • Icositetrachoron • Hecatonicosachoron • Hexacosichoron And then all the non-convex, semiregular, isogonal, uniform, infinite,….

  19. Even higher dimensions! In dimensions 5 and higher there are only ever 3 regular polytopes (Ludwig Schläfli): • Simplex (tetrahedron) • Hypercube • Orthoplex (octahedron) And there are no non-convex ones!

  20. Thank you!

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