— Mathematics in Art and Science — A few examples of tessellated triangles

Any triangle will tessellate (in 2-D Euclidean space) because its angles add up to 180°, but not every triangle can be used to create a variety of interesting tessellations.

One way to experiment with more appealing tessellations is to select an isosceles triangles in which one angle is an integer divisors of 360°. For example:

3	4	5	6	7	8	9	10	11	12
120°	90°	72°	60°	51.43°	45°	40°	36°	32.73°	30°

Any of the resulting isoscelese triangles, except for the equilateral triangle, will allow a good variety of non-uniform tessellations that could be used in gardens or expansive interior spaces.

Using the equilateral triangle, with angles of 60°, only allows a uniform tessellation, which is related to the tessellation of regular hexagons. Making a different pattern with equilateral triangles would involve varying their sizes.

Using an isoscelese triangle with angles of 30°, 30° and 120° allows a variety of attractive tessellations.

Using an isoscelese triangle with angles of 45°, 45° and 90° allows a surprising variety of tessellations related to the tessellation of squares.

Using an isoscelese triangle with angles of 67.5°, 67.5° and 45° allows a variety of tessellations, some of which suggest garden paths.

Using an isoscelese triangle with angles of 72°, 72° and 36° allows a variety of tessellations.