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Polyhedra

Polyhedra. Helmer ASLAKSEN Department of Mathematics National University of Singapore aslaksen@math.nus.edu.sg www.math.nus.edu.sg/aslaksen/polyhedra/. What is a polygon?. Sides and corners. Regular polygon: Equal sides and equal angles. For n greater than 3, we need both.

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Polyhedra

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

  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

  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

  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 • http://www.math.nus.edu.sg/aslaksen/

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