Course:Harris, Fall 08: Diary Week 7

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  • We explored in-depth details as to why there are only three regular tessellations:
    • Notation: <math>a|b</math> means the number <math>a</math> divides the number <math>b</math> evenly.
    • Suppose we have a tessellation by regular <math>n</math>-gons; we know each of the angles is the same amount, <math>A = (n-2)180/n</math>.
    • Claim: The angle <math>A</math> must evenly divide 360, i.e, <math>A|360</math>. Why?
      • Consider a single vertex in the tessellation: There are <math>m</math> polygons that fit around it.
      • That means <math>m</math> angles, all of the same size <math>A</math>.
      • And since they fit all around the vertex, they add up to 360, i.e., <math>mA = 360</math>. In other words, <math>A|360</math>.
    • What are the possibilities for <math>m</math>, the number of polygons around a vertex?
      • We know of three regular tessellations:
        • By triangles (3-gons), with <math>m</math> = 6.
        • By squares (4-gons), with <math>m</math> = 4.
        • By hexagons (6-gons), with <math>m</math> = 3.
      • Any other possibilities?
        • Clearly, we cannot have 1 polygon at a vertex.
        • Nor can we have 2 polygons at a vertex, as that would mean 180-degree angles.
        • What about 5 polygons?
          • Looking at the known existing ones, <math>m</math> = 5 would mean an <math>n</math>-gon with <math>n</math> between 4 and 3--not possible!
        • What about 7 or more polygons?
          • Looking again at the known list, <math>m</math> = 7 or higher would mean an <math>n</math>-gon with <math>n</math> < 3. Again, not possible!
      • So, no, no other possibilities.
    • Thus, the only regular tessellations of the plane are the three listed.
  • We spent the rest of the time playing around with the Geometer's Sketchpad.
    • Goal was to show how an irregular quadrilateral can be made to tessellate the plane, by rotating it first around each of its four edge midpoints.
    • To be collected next time: Sketches of how this is done.
  • I handed out copies of the written classification scheme for Wallpaper Groups (available to be used on Exam 1, Friday).
    • Still to come: copies of the flow-chart, same purpose.


  • I handed out copies of the flow-chart for Wallpaper Group classification.
  • We looked at how to make Escher-like tessellations by various modifications of polygon tessellations, including
    • Transforming one side of a tessellating polygon and then translating it.
    • Transforming one side of a tessellating polygon and then translating it with a flip.
    • Transforming half of a side and then rotating it about the midpoint of that side.
    • Transforming a side and then rotating it around one endpoint.
  • We played with Geometer's Sketchpad, trying out these methods for the Escher-like Tessellation Exploration.


  • Exam 1 (I was out of town; Dr. Marks subbed.)