# Course:Harris, Fall 08: Diary Week 7

Mon:

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