1 / 23

The Beauty of Polyhedra

The Beauty of 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 polyhedron?. A surface consisting of polygons. What is a polygon?. Sides and corners.

oshin
Download Presentation

The Beauty of Polyhedra

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. The Beauty of 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 polyhedron? • A surface consisting of polygons.

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

  4. How many sides? • Where in Singapore is this? • How many aisles?

  5. A quick course in Greek

  6. Polyhedra • Vertices, edges and faces.

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

  8. Properties of Platonic solids Notice that V – E + F = 2 (Euler’s formula)

  9. Duality • Tetrahedron is self-dual • Cube and octahedron • Dodecahedron and icosahedron

  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? It can be inscribed in a cube, an octahedron and a rhombic dodecahedron (dual of the cuboctahedron)

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

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

More Related