Lesson 1
Ritual Geometry
Mandala
Lesson 2
Group Elements
Color Theory
Lesson 3
Groups and Groups Acting on Sets
Block printing
Lesson 4
Klimt and the Computer
Color and symmetry in modern art I
Lesson 5
Islam
Islamic art
Lesson 6
Penrose and Rice
Color and symmetry in modern art II
Lesson 7
Escher 1
Escher 2
Lesson 8
Hundertwasser & Griffeath
Pattern and Modern Painting
Brian P. Hoke:
Cellular Automata and Art
Student's Work


Goals: To make "artistic" decisions in a mathematical context.
To develop the metacognitive skills that apply in both fields.
To investigate some questions going beyond usual symmetry, in
particular Penrose tiles.
 1. What questions of a mathematical nature have we asked so far about
symmetries? (Relationship among, composition of symmetries, classification of
pictures according to them, how many, divisibility properties, how they relate to
the "symmetric object".) Make a list of these. What makes a good question in
math (or art)? Put criteria on board.
SOME THINGS STUDENTS SAY
A question whose answer isn't obvious and neither is how to get it.
A question that captures the essence of the phenomenon you are looking at.
A question that is clearly stated so that someone else would know if they had the answer.
 2. Most of these questions were initiated by me. But asking the question is
the "artistic" part of mathematics, analogous to an artistic question such as,
"How do I express three dimensional space on a flat piece of paper?" So, now it
is your turn. Divide into pairs and look at your examples of wallpaper patterns.
Think of three questions about the nature of their symmetries. You have given
criteria for what makes a good question, so try to make sure your questions are
good ones.
(We then make list on the board. Look for questions leading toward the idea of
wierd and different tilings.)
 3. Here are some examples of such questions.
How would one arrange patterns according to increased complexity? Can one tile
the plane with regular pentagons? If one can't tile the plane with a regular
pentagon, how about irregular ones? How about irregular tilings? Could we get a
fivefold symmetry with them?
 4. Pentagons: Marjorie Rice.
Does everybody know about Martin Gardner's mathematical games column in
Scientific American? It ran for years. One year he published a claim by R.B.
Kershner to have found all types of convex polygons that tile the plane
(nonregular). (Does everyone know what convex is?) A reader soon wrote back
with a pentagonal counterexample, which Gardner published. This was around 1975.
One Gardner fan was Marjorie Rice, a San Diego housewife and mother of five, who
read her son's S.A. regularly. She thought it would be fun to look for other
pentagonal examples. It took her two years and she found three additional new
tilings. Her most advanced diploma was a high school degree for which she had
taken only one general math course. The degree was from 1939. The moral to the
story? Never underestimate your mother!
"For All Practical Purposes: Introduction to Contemporary Mathematics"
W. H. Freeman and Company, New York
copyright 1994 by COMAP, Inc., Third Edition
p. 704
Is this periodic or not? What is the fundamental region?
Rice liked the work of M.C. Escher and used her tilings as a basis for art.
"For All Practical Purposes: Introduction to Contemporary Mathematics"
W. H. Freeman and Company, New York
Color Plate 2, p.558
A modification of a Penrose tiling by refashioning the kites and darts into
bird shapes.
[Tiling by Roger Penrose.]
 5. The issue of fivefold symmetry is tied up with irregular tilings. A
nonperiodic tiling is a filling of the plane with shape so that there
is no translational symmetry. There could, however, be a rotational symmetry
somewhere.
 6. Usually if a collection of tiles has a nonperiodic tiling then
the same tiles could be rearranged to form a periodic one. Here is an example of
such a tiling with only one shape tile.
"An Explorer's Guide"
by Marjorie Senechal and the students of "Tilings" (1995 and 1996)
Figure 15 A tiling produced by a rule. P.13
(We demonstrate this by actually doing a blockprint with three sizes of L shape
tile. This is easy
to do with Safetycut and has the advantage of making the process clear.) What
is the proof that you can fill the plane with these?
 7. In 1964 a set of tiles was discovered that would only tile
nonperiodically. How many different shapes do you think were needed? (20,000!)
In 1975 Roger Penrose came up with an example with only two tiles.
Made from a parallelogram whose small angle 72 degrees. What's special about 72?
Cut into two shapes.
"For All Practical Purposes: Introduction to Contemporary Mathematics"
W. H. Freeman and Company, New York
p.710
 8. Diagonal of the parallelogram is divided in the ratio of
1 : [ 1 + sqr (5) ] / 2.
What is so special about this ratio? There is a rule that says you can't ever
put them together to make the parallelogram again. This rule is equivalent to
making small cuts on the appropriate edge.
These are called the Kite and Dart.
Each pair of people has a kite and a dart made from safetycut.
Tile a region with them. Is the result symmetric? How so? Is it more or less
complex than the ones we have studied so far? What makes it complex?
Can you make a mandala with fivefold symmetry out of them?
 9. Kite and Dart are examples of Penrose tiles. How can you prove that
they tile the plane?
Slide of Kites and Darts, parsed to larger ones. This suggests a proof similar to
the one for the L shaped block.
Do these ever occur in nature? What do you think? Would nature do such a thing?
It does indeed, and they are called quasicrystals.
"For All Practical Purposes: Introduction to Contemporary Mathematics"
W. H. Freeman and Company, New York
p.714
(a) Penrose tile,
(b) 3d structure based on this tile give a theoretical diffraction pattern.
(c) Diffraction pattern of Manganesealuminum alloy.
(d) Scanning tunnelling microscope image of Aluminumcoppercobalt alloy.
(e) Corresponding tiling. Aluminum is open centers, copper and cobalt
are closed centers. ref. For All Practical Purposes.
Where can I see more? U. Minn. Geometry
Center Website gives more, plus explains how you construct them from a
three dimensional object. And there are a finite number of ways to tile for a five
fold mandala. (2)
