In my previous post, I showed you some adventurous ideas on how
to tile floors. In this post, we examine other types of tilings.
In the first figure, the tiling is composed of squares and
equilateral triangles. In the second figure, the tiling is
composed of hexagons (6-sided polygons) and equilateral
triangles. In the third figure, the tiling is made up of
dodegacons (12-sided polygons), squares, and equilateral
triangles. The tilings above and in the previous post do not
only display beautiful patterns. Mathematical considerations are
needed to create such tilings. For instance, in the third
figure, there are points where the vertices ofa a dodecahedron,
a square, and two equilateral triangles meet. The interior
angles of these polygons measure 150, 90, 120 degrees
respectively. Observe that the angle measures add up to 360
degrees. Only combinations of polygons whose interior angles add
up to 360 degrees will tile without gaps or overlaps (Can you
see why?).