Review of Platonic solids

1
Review of Platonic solids

• tetrahedron (3.3.3)
• cube (4.4.4)
• octahedron (3.3.3.3)
• dodecahedron (5.5.5)
• icosahedron (3.3.3.3.3)

2
Edge and vertex counting

• Determine the number of vertices and edges on a
  cube by
• counting the vertices on each face
• counting the edges on each face
• Repeat with the dodecahedron

3
Dual polyhedra

• The dual of a polyhedron is created by placing a
  vertex at the center of each face and connecting
  vertices corresponding to adjacent faces.
• Examples
• dual of a right square pyramid
• dual of a rectangular box

4
Duals of Platonic solids

• Go to this applet to find the duals of the
  Platonic solids.
• Find mathematical relationships between the faces
  of a Platonic solid and the number of vertices of
  its dual.

5
Even more embedding

• Go to this applet to see how Platonic solids can
  be placed inside each other.
• Kepler thought the orbits of the then-planets
  could be explained using nested Platonic solids.
• See an image from his 1596 book Mysterium
  Cosmographicum

6
Representing polyhedra

• Nets
• A net is a planar shape that can be folded into a
  polyhedra.
• Examples and nonexamples for the cube.
• Example for the tetrahedron
• How can we compute edges and faces from a net?

7
Representing Polyhedra

• Schlegel diagrams
• The picture created by suspending a light source
  directly above the center of one of the faces
• See this picture
• Create the Schlegel diagram for a cube
• How can we compute edges, faces, and vertices
  from a Schlegel diagram?

8
Eulers formula

• For each of your solids, find v e f.
• Eulers formula
• Holds for any polyhedron that is convex that
  is, any line connecting any two points on faces
  of the polyhedron lies inside the polyhedron.
• Example show this for a right square pyramid.

9
Why does this work?

• Start with a cube and start removing faces.
• What happens to ? when you remove a face?
• What happens to ? when you remove other faces?

10
Proof of Eulers formula

• A general proof
• Remove one face
• When you remove other faces (never breaking the
  object into two pieces), you are in one of three
  cases
• You remove only one edge
• You remove all edges except one
• You remove all edges
• Show that the Euler characteristic doesnt change
  in any of these cases.
• You end up with one polygon that has ? equal to 1.

11
Exam

• Exam next Wednesday (4/16) that covers
• wallpaper classification and construction
• perspective
• polyhedra