1. Tilings and Polyhedra Helmer ASLAKSEN Department of Mathematics National University of Singapore aslaksen@math.nus.edu.sg www.math.nus.edu.sg/aslaksen/polyhedra/

2. They look nice! They teach us mathematics. Mathematics is the abstract study of patterns. Be conscious of shapes, structure and symmetry around you! Why are we interested in this?

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

4. A quick course in Greek 34567 TriTetraPentaHexaHepta 89101220 OctaEnneaDecaDodecaIcosa

5. 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

6. What is a tiling? Tilings or tessellations are coverings of the plane with tiles.

7. Assumptions about tilings 1 The tiles are regular polygons. The tiling is edge-to-edge. This means that two tiles intersect along a common edge, only at a common vertex or not at all.

8. Assumptions about tilings 2 All the vertices are of the same type. This means that the same types of polygons meet in the same order (ignoring orientation) at each vertex.

9. Regular or Platonic tilings A tiling is called Platonic if it uses only one type of polygons. Only three types of Platonic tilings. There must be at least three polygons at each vertex. There cannot be more than six. There cannot be five.

10. Archimedean or semiregular tilings There are eight tilings that use more than one type of tiles. They are called Archimedean or semiregular tilings.

11. Picture of tilings

12. More pictures 1

13. More pictures 2

14. More pictures 3

15. A trick picture

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

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

18. Polyhedra Vertices, edges and faces.

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

20. Properties of Platonic solids FacesEdgesVerticesSides of face Faces at vertex Tet46433 Cub612843 Oct812634 Dod12302053 Ico20301235

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

22. 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.

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

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

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

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

27. Why rhombicuboctahedron?

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

29. Why no snub tetrahedron? Its the icosahedron!

30. The rest of the rest Prism and antiprism.

31. Are there any more? Millers solid or Sommervilles solid. The vertices are congruent, but not equivalent!

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

33. Nested Platonic Solids

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

35. Web http://www.math.nus.edu.sg/aslaksen/