Overview

1
Overview

• 0 dimensions points
• 1 dimension lines and curves
• between 1 and 2 dimensions fractals
• 2 dimensions tilings and symmetry
• between 2 and 3 dimensions perspective
• 3 dimensions sculpture and polyhedra

2
Polyhedra

• A polyhedron is a three-dimensional figure whose
  faces (or sides) are polygons.
• Examples
• pyramids
• prisms
• boxes

3
Polyhedral Art

• Another feature of Renaissance art
• Leonardo da Vincis illustrations of Paciolis
  The Divine Proportion
• Albrecht Durers nets and Melancholia I
• Jamnitzers plates
• Modern Artists
• The works of M. C. Escher
• The domes of architect Buckminster Fuller

4
Platonic solids

• The regular polyhedra (or Platonic solids) are
  those
• whose faces are identical regular polygons, and
• whose vertex configurations are the same are each
  vertex
• Example the cube
• We denote the vertex configuration by 4.4.4 (the
  Schläfli symbol)

5
Finding the Platonic solids

• We need at least three polygons at each vertex.
  (Why?)
• The sum of the vertex angles must be less than
  360 degrees. (Why?)
• What polygons can we use (and how many of each)?

6
Constructing Platonic Solids

• Zometool pieces
• First, use blue struts only to build equilateral
  triangles, squares, and regular pentagons.
• Second, try putting these together to form a
  Platonic solid. Fill in Exercise 7 in Section
  7.2 as you do this. You should get three
  Platonic solids.

7
The other two

• Use the green struts to build equilateral
  triangles.
• Piece these green triangles together.
• This will be trickier call me over when you
  have questions.
• You can see a computer representation of each
  solid at this site.

8
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
• 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

9
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.

10
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?
• 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
• The face is connected to two or more edges
• The face is connected to one edge
• The face is connected to no edges, just a vertex
• 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