Aperiodic Tessellation Exercises

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  1. Look at Escher's prints Mosaic and Mosaic II.
    1. Are these periodic tessellations or non-periodic tessellations?
    2. What is the only tile in these prints that is not a living creature?
    3. What is the focal point of Mosaic II?
    4. Describe how Escher has carefully chosen which tiles are white and which are black in these prints.
    5. Draw a tessellation where it is impossible to color tiles black and white so that no two of the same color share an edge.
  2. Use the random number generator http://www.random.org/bytes to generate 8 random binary bytes, and then create an 8x8 random tessellation using 45°-45°-90° triangles as described in the text.
  3. The Voderberg tiling
    In the Voderberg tiling, shown at right, how many different shapes of tile are there? How many tiles touch the innermost blue tile? (Only count tiles which touch along an edge, not corner touches)
  4. The L substitution tiling

    Using a sheet of graph paper, create the substitution tiling using the L rep-tile. At each step, the previous picture is expanded by a factor of two and then each of the L-triomino-level-0.png shapes are subdivided into four smaller L-triomino-level-0.png. Draw the next step in the process shown at right. You'll want to start with an L whose long sides are 16 graph paper squares.

  5. The P pentomino is also a rep-tile. Draw the next substitution step for this tiling: P-pentomino-rep-tile.svg
  6. Draw the curved colored markings on the Penrose tiles shown below. (The markings are shown here: Image:Penrose-kite-dart-marked.svg)
  7. Which of these configurations satisfy the matching rules? That is, which of these are legal arrangements of Penrose tiles?
    Penrose-good-1.svg Penrose-bad-1.svg Penrose-bad-2.svg Penrose-bad-3.svg Penrose-good-2.svg
    A B C D E
  8. This configuration of Penrose tiles is known as the 'King'. What tile or tiles must fit into the gap marked A? What tile or tiles must fit into the gap marked B?
    Penrose Tile 'King'

Instructor:Aperiodic Tessellation Exercises Solutions (restricted access)