1. 4.4 Soothing Symmetry and Spinning Pinwheels
Friday, February 20, 2009

2. Symmetry Mirror Images Beauty?
Rigid Symmetry – motion of the plane that preserves the pattern and does not shrink, stretch, or otherwise distort the plane
Shift, rotation, flip or combination of these

3. Types of Symmetry
Line Symmetry
Rotational Symmetry

4. Symmetry of Scale Also known as scalable
If the tiles that make up the pattern can be grouped into super-tiles that still cover the plane and, if scaled down, can be rigidly moved to coincide with the original pattern
Checkerboard

5. Tessellations Tiling the plane
Regular tessellation - means a tessellation made up of congruent regular polygons

6. Semi-regular Tessellations

7. Name Some More

8. Demi-regular Tessellations

9. And Another

10. Patterns in Nature

11. Chaotic Patterns
Penrose patterns – no rigid symmetries that use only two tile shapes, kites and darts

12. Penrose Patterns

13. More about Penrose Patterns
Every tile occurs in one of 10 possible orientations in the plane

14. Penrose Tiling
December 2003: Sir Roger Penrose, the eminent British mathematician, came face to face with his own copyrighted polygon pattern in Kleenex quilted toilet paper. When his wife returned from the market with the embossed rolls, Penrose expressed "astonishment and dismay" upon seeing the use to which his discovery had been put.
Penrose devised the nonrepeating five-fold symmetrical pattern in the 1970s by using two kinds of diamond shapes—fat and thin—to create what is now called Penrose tiling. The pattern, which was thought not to exist in nature before Penrose's discovery, has subsequently been found in many physical and biological phenomena.

15. Pinwheel Pattern
1994 – John Conway of Princeton and Charles Radin of the University of Texas-Austin
Uses one single triangular tile
Symmetry of scale, but no rigid symmetry
Tiles occur in infinitely many orientations
Group by 5 to form super-tiles

18. Pinwheel Properties
Uniqueness of Scaling – there is only one way to group the Pinwheel Triangles into super-tiles to create a Pinwheel super-pattern in the plane
No rigid symmetries

19. MC Escher
M.C. Escher was a Dutch graphic artist, most recognized for spatial illusions, impossible buildings, repeating geometric patterns (tessellations), and his incredible techniques in woodcutting and lithography.
M.C. Escher was born June 1898 and died March 1972.

20. Escher’s Works

21. More Escher

22. Problem of the Day
You're a cook in a restaurant in a quaint country where clocks are outlawed. You have a four minute hourglass, a seven minute hourglass, and a pot of boiling water. A regular customer orders a nine-minute egg, and you know this person to be extremely picky and will not like it if you overcook or undercook the egg, even by a few seconds. What is the least amount of time it will take to prepare the egg, and how will you do it?