Overview

1
Overview

• 0 dimensions points
• 1 dimension lines and curves
• between 1 and 2 dimensions fractals
• 2 dimensions
• polygon
• tilings/tesselations
• symmetry

2
A M. C. Escher symmetry drawing from
www.mcescher.com
A tiling from Cairo (found at Tilings from
Historical Sources
3
Polygons

• A polygon is a many-angled figure (often called
  an n-gon for specific values of n).
• A polygon is called regular if all sides and
  vertex angles are congruent.

from mathworld.wolfram.com
4
Angles in Polygons

• Theorem The angles in any triangle add to 180
  degrees.
• Use this fact to find the vertex angles for
  regular triangles, squares, and regular
  pentagons.
• Fill in the list of vertex angles on page 86.

5
Tilings

• A tiling (or tessellation) is a filling-up of the
  plane with a shape or shapes that meet edge to
  edge and vertex to vertex.
• A tiling is regular if the only shape used is a
  regular polygon and the basic pattern is
  repeated.
• Basic example tilings with squares.
• Notation we call this tiling 4.4.4.4 (indicating
  that 4 squares meet at a vertex)

6
Regular Tilings

• To find all regular tilings, note that one shape
  is used to surround a point.
• The vertex angle must evenly divide 360 degrees.
• Find and notate all other regular tilings.

7
Semiregular tilings

• A tiling is semiregular if it can be formed in a
  regular way (i.e., each vertex has the same
  configuration) with two or more regular polygons.
• Example 4.8.8
• Why does this work?

8
Our first goal

• We need to find the different configurations of
  regular polygons that can fit around a point.
• Build the justifications for Rules 1, 2, 3, 4,
  and 5 on pages 86-88.
• Build the list of possible configurations on page
  88.