1. 4.5 Platonic Solids Wednesday, February 25, 2009

2. Symmetry in 3-D  Sphere – looks the same from any vantage point  Other symmetric solids? CONSIDER REGULAR POLYGONS

3. Start in The Plane  Two-dimensional symmetry  Circle is most symmetrical  Regular polygons – most symmetrical with straight sides

4. 2D to 3D  Planes to solids  Sphere – same from all directions  Platonic solids Made up of flat sides to be as symmetric as possible Faces are identical regular polygons Number of edges coming out of any vertex should be the same for all vertices

5. Five Platonic Solids  Cube Most familiar  Tetrahedron  Octahedron  Dodecahedron  Icosahedron

6. Powerful?  Named after Plato  Euclid wrote about them  Pythagoreans held them in awe

7. VerticesEdgesFacesFaces at each vertex Sides of each face Tetrahedron Cube Octahedron Dodecahedron Icosahedron

8. Vertices V Edges E Faces F Faces at each vertex Sides of each face Tetrahedron 46433 Cube 812634 Octahedron 612843 Dodecahedron 20301235 Icosahedron 12302053

9. Vertices V Edges E Faces F Faces at each vertex Sides of each face Tetrahedron 46433 Cube 812634 Octahedron 612843 Dodecahedron 20301235 Icosahedron 12302053

10. Some Relationships  Faces of cube = Vertices of Octahedron  Vertices of cube = Faces of Octahedron

11. Duality  Process of creating one solid from another  Faces - - - Vertices

12. Euler's polyhedron theorem  V + F - E = 2

13. Archimedean Solids  Allow more than one kind of regular polygon to be used for the faces  13 Archimedean Solids (semiregular solids)  Seven of the Archimedean solids are derived from the Platonic solids by the process of "truncation", literally cutting off the corners  All are roughly ball-shaped

14. Truncated Cube

15. Archimedean Solids

16. Soccer Ball – 12 pentagons, 20 hexagons

17. Solid (pretruncating) Truncated Vertices EdgesFaces Tetrahedron Cube Octahedron Dodecahedron Icosahedron

18. Solid (pretruncating) Truncated Vertices EdgesFaces Tetrahedron12188 Cube143624 Octahedron143624 Dodecahedron329060 Icosahedron329060

19. Solid (post-truncating) Truncated Vertices EdgesFaces Tetrahedron81812 Cube243614 Octahedron243614 Dodecahedron609032 Icosahedron609032

20. Some Relationships  New F = Old F + Old V  New E = Old E + Old V x number of faces that meet at a vertex  New V = Old V x number of faces that meet at a vertex

21. Stellating  Stellation is a process that allows us to derive a new polyhedron from an existing one by extending the faces until they re-intersect

22. Two Dimensions: The Pentagon

23. Octagon

24. How Many Stellations?  Triangle and Square  Pentagon and Hexagon  Heptagon and Octagon  N-gon?

25. Problem of the Day  How can a woman living in New Jersey legally marry 3 men, without ever getting a divorce, be widowed, or becoming legally separated?