1. Polyhedra Helmer ASLAKSEN Department of Mathematics
National University of Singapore

2. What is a polygon? Sides and corners.
Regular polygon: Equal sides and equal angles.
For n greater than 3, we need both.

3. A quick course in Greek 3 4 5 6 7 Tri Tetra Penta Hexa Hepta 8 9 10 1220
Octa
Ennea
Deca
Dodeca
Icosa

4. More about polygons
The vertex angle in a regular n-gon is 180 (n-2)/n. To see this, divide the polygon into n triangles.
3: 60
4: 90
5: 108
6: 120

5. Polyhedra What is a polyhedron? Platonic solids Deltahedra
Archimedean solids
Colouring Platonic solids
Stellation

6. What is a polyhedron? Solid or surface?
A surface consisting of polygons.

7. Polyhedra
Vertices, edges and faces.

8. Platonic solids
Euclid: Convex polyhedron with congruent, regular faces.

9. Properties of Platonic solids
Faces
Edges
Vertices
Sides
of face
Faces at
vertex
Tet 4 6 3
Cub 12 8
Oct
Dod 30 20 5
Ico

10. Colouring the Platonic solids
Octahedron: 2 colours
Cube and icosahedron: 3
Tetrahedron and dodecahedron: 4

11. Euclid was wrong!
Platonic solids: Convex polyhedra with congruent, regular faces and the same number of faces at each vertex.
Freudenthal and Van der Waerden, 1947.

12. Deltahedra Polyhedra with congruent, regular, triangular faces.
Cube and dodecahedron only with squares and regular pentagons.

13. Archimedean solids
Regular faces of more than one type and congruent vertices.

14. Truncation Cuboctahedron and icosidodecahedron.
A football is a truncated icosahedron!

15. The rest Rhombicuboctahedron and great rhombicuboctahedron
Rhombicosidodecahedron and great rhombicosidodecahedron
Snub cube and snub dodecahedron

16. Why rhombicuboctahedron?

17. Why snub? Left snub cube equals right snub octahedron.
Left snub dodecahedron equals right snub icosahedron.

18. Why no snub tetrahedron?
It’s the icosahedron!

19. The rest of the rest
Prism and antiprism.

20. Are there any more? Miller’s solid or Sommerville’s solid.
The vertices are congruent, but not equivalent!

21. Stellations of the dodecahedron
The edge stellation of the icosahedron is a face stellation of the dodecahedron!

22. Nested Platonic Solids

23. How to make models
Paper
Zome
Polydron/Frameworks
Jovo

24. Web