The shape of the triangle in a tiling is important. If we start with just
any triangle, for example the one on the left below,
then the copies of the triangle don't fit nicely next to each
other. There are gaps or overlaps as shown on the right:
It turns out that in order to fit nicely into a tiling, each angle in
a triangle must each be 180/n degrees for some whole number n. So the
possible angles are 180/2 = 90, 180/3 = 60, 180/4 = 45, 180/5 = 36,
180/6 = 30, and so on. We'll call these angles nice angles. The three angles do not have
to be equal to each other, but each one must be nice. In
addition, remember that the angles in a triangle have to add up to 180
degrees. If you experiment around a little bit you can convince
yourself that the only combinations of 3 nice angles which add up
to 180 are (90,60,30), (90,45,45), and (60,60,60).
This means that there are only three different triangle shapes that
can be used to make a triangle tiling.
The tilings in this article so far all use a (90,60,30) triangle.
The tilings look different from each other because the triangle
is colored differently, but it's the same shape in each case.
Read on about
triangle to bend gives rise to 3D shapes!