1. 4.5 More Platonic Solids
Wednesday, March 3, 2004

2. 4 6 3 8 12 20 30 5 Vertices V Edges E Faces F Faces at each vertex
Sides of each face
Tetrahedron 4 6 3
Cube 8 12
Octahedron
Dodecahedron 20 30 5
Icosahedron

3. 4 6 3 8 12 20 30 5 Vertices V Edges E Faces F Faces at each vertex
Sides of each face
Tetrahedron 4 6 3
Cube 8 12
Octahedron
Dodecahedron 20 30 5
Icosahedron

4. Some Relationships Faces of cube = Vertices of Octahedron
Vertices of cube = Faces of Octahedron

5. Duality Process of creating one solid from another
Faces Vertices

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

7. 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

8. Truncated Cube

9. Solid
(pretruncating)
Truncated Vertices
Edges
Faces
Tetrahedron
Cube
Octahedron
Dodecahedron
Icosahedron

10. Solid
(pretruncating)
Truncated Vertices
Edges
Faces
Tetrahedron 12 18 8
Cube 14 36 24
Octahedron
Dodecahedron 32 90 60
Icosahedron

11. Solid
(pretruncating)
Truncated Vertices
Edges
Faces
Tetrahedron 8 18 12
Cube 24 36 14
Octahedron
Dodecahedron 60 90 32
Icosahedron

12. 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

13. 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

14. Two Dimensions: The Pentagon

15. Octagon

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

17. What About 3-Dimensions?
Assignment:
Page 286 III.2 Stellated Solids