This pattern in quite amazing when you think about it.
It is made up of four different kinds of triangle: one irregular triangle which can be of any shape we like, and three different sizes of equilateral triangle.
Every equilateral triangle is surrounded by three copies of the irregular triangle (top left). The centres of these triangles form an equilateral triangle.
Every copy of the irregular triangle is surrounded by three equilateral triangles, one of each size (bottom left). The centres of these triangles form an equilateral triangle (this is Napoleon's theorem.)
We can extend the patter as far as we like, so it tiles the whole plane.
If we join up all the triangle centres we get a hexagonal lattice, like a honeycomb.
If we join up the centres of all the copies of a particular triangle (all the equilateral triangles of a particular size, or all the copies of the irregular triangle), we get a triangular lattice.
It also has a rich set of symmetries. There are only 17 basic 'wallpaper patterns' in terms of symmetries but I haven't figured out which one this is yet.