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GRED 505 (Topics in mathematics for elementary teachers)
Assignment 6: Application of fractions to tessellations
A tessellation is a tiling of the plane using one or more shapes in a repeated pattern with
no holes (gaps) or overlaps.
Figure 1. Tessellations with (arbitrary) triangle and quadrilateral.
Figure 1 shows that triangles and quadrilaterals can tessellate the plane. From here it
follows that the sum of internal angles of any triangle equals to 180 and the sum of
internal angles of any quadrilateral equals to 360.
Consider edge-to-edge tessellation with three regular polygons with the number of sides
equals to n1, n2, and n3 respectively. For example, when n1=4, n2=6, and n3=12 we have
the case of a square (four sides), regular hexagon (six sides), and regular dodecagon (12
sides). Incidentally, 1/4+1/6+1/12=1/2. It can be shown that, in general, the equality
1/ n1+1/ n2+1/ n3=1/2 implies that regular polygons with the number of sides equals to n1,
n2, and n3, respectively, enable edge-to-edge tessellation. (Apparently, not any three
regular polygons can tessellate. Indeed, equilateral triangle, square, and regular pentagon
do not tessellate. Why?)
Instructor Sergei Abramovich
