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Exploring Polyhedra: From Platonic Solids to Stellations

Discover the fascinating world of polyhedra, including Platonic solids, Deltahedra, and Archimedean solids. Learn about vertices, edges, and faces, and delve into stellations and coloring techniques. Unravel the mysteries of nested Platonic solids and find out how to create models using different tools. Engage in a quick course on Greek geometry and explore the beauty of polyhedral structures.

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Exploring Polyhedra: From Platonic Solids to Stellations

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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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