Course:Harris, Fall 08: Diary Week 7

Mon:

• We explored in-depth details as to why there are only three regular tessellations:
  • Notation: $a|b$ means the number $a$ divides the number $b$ evenly.
  • Suppose we have a tessellation by regular $n$-gons; we know each of the angles is the same amount, $A=(n-2)180/n$.
  • Claim: The angle $A$ must evenly divide 360, i.e, $A|360$. Why?
    • Consider a single vertex in the tessellation: There are $m$ polygons that fit around it.
    • That means $m$ angles, all of the same size $A$.
    • And since they fit all around the vertex, they add up to 360, i.e., $mA=360$. In other words, $A|360$.
  • What are the possibilities for $m$, the number of polygons around a vertex?
    • We know of three regular tessellations:
      • By triangles (3-gons), with $m$ = 6.
      • By squares (4-gons), with $m$ = 4.
      • By hexagons (6-gons), with $m$ = 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, $m$ = 5 would mean an $n$-gon with $n$ between 4 and 3--not possible!
      • What about 7 or more polygons?
        • Looking again at the known list, $m$ = 7 or higher would mean an $n$-gon with $n$ < 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.

Wed:

• 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.

Fri:

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