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Tilings and Polyhedra

Tilings and Polyhedra. Helmer ASLAKSEN Department of Mathematics National University of Singapore aslaksen@math.nus.edu.sg www.math.nus.edu.sg/aslaksen/polyhedra/. Why are we interested in this?. They look nice! They teach us mathematics. Mathematics is the abstract study of patterns.

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Tilings and Polyhedra

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

  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

  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

  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? • It’s the icosahedron!

  30. The rest of the rest • Prism and antiprism.

  31. Are there any more? • Miller’s solid or Sommerville’s 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/

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