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? Convex (roughly “no holes or sticking out bits”) Faces are regular polygons Interchangeable faces Interchangeable edges Interchangeable corners

3. Question: What does this give us? Answer: the Platonic solids again, plus the cuboctohedron and the icosidodecahedron. Convex Faces are regular polygons  Interchangeable faces Interchangeable edges Interchangeable corners E.g.

4. This gives us the 13 Archimedean solids… …and infinitely many prisms and antiprisms. Convex Faces are regular polygons  Interchangeable faces  Interchangeable edges Interchangeable corners

5. This additionally gives us the 92 Johnson solids. Convex Faces are regular polygons  Interchangeable faces  Interchangeable edges  Interchangeable corners

6. This gives us… …nothing new! Convex  Faces are regular polygons Interchangeable faces Interchangeable edges Interchangeable corners

7. The rhombic dodecahedron and rhombic triacontahedron. Convex  Faces are regular polygons Interchangeable faces Interchangeable edges  Interchangeable corners

8. This gives us the 13 Catalan solids, plus infinitely many bipyramids and trapezohedra. Convex  Faces are regular polygons Interchangeable faces  Interchangeable edges  Interchangeable corners

9. Just one infinite family of disphenoid tetrahedra. Convex  Faces are regular polygons Interchangeable faces  Interchangeable edges Interchangeable corners

10. Infinitely many isogonal polyhedra. Convex  Faces are regular polygons  Interchangeable faces  Interchangeable edges Interchangeable corners

11. This gives us…. …nothing new! Convex  Faces are regular polygons  Interchangeable faces Interchangeable edges  Interchangeable corners

12. The 4 Kepler-Poinsot polyhedra.  Convex Faces are (self-intersecting) regular polygons Interchangeable faces Interchangeable edges Interchangeable corners

13. The great icosidodecahedron, together with the dodecadodecahedron and its three ditrigonal variants  Convex Faces are (self-intersecting) regular polygons  Interchangeable faces Interchangeable edges Interchangeable corners

14. The 75 Uniform polyhedra…. …plus Skilling’s figure (maybe). …plus infinitely many (crossed) prisms & antiprisms  Convex Faces are (self-intersecting) regular polygons  Interchangeable faces  Interchangeable edges Interchangeable corners

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! Faces are regular polygons Interchangeable faces Interchangeable edges Interchangeable corners 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!