Lesson 6:   Penrose and Rice

Lesson 1
Ritual Geometry


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

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

To make "artistic" decisions in a mathematical context. To develop the meta-cognitive 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.


    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 (non-regular). (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

    Marjorie Rice's Fish

    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.]

    Color Plate

  • 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 Safety-cut 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

    Penrose tiles

  • 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.

    group work

    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

    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

    Penrose tile

    (a) Penrose tile,
    (b) 3-d structure based on this tile give a theoretical diffraction pattern.
    (c) Diffraction pattern of Manganese-aluminum alloy.
    (d) Scanning tunnelling microscope image of Aluminum-copper-cobalt 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)

    students work

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© Copyright 1996, Pippa Drew and Dorothy Wallace, Dartmouth College