Origami?

Nice problem. :) There are 56 triangles total.

Ashish posted a clear breakdown of where they are in 4 categories. However, here is an exhaustive proof that no additional triangles exist. Ashish, your method of counting is certainly faster. However, the technique I used is cool in that it will work for any triangle counting problem.

Step 1 : Identify each vertex and, for convenience, group them into equivalent classes by symmetry.

Step 2 : For each vertex, consider every possible pair of lines extending out from it. Again, group for symmetry.

Step 3 : For each pair of lines extending from a vertex, a triangle exists if and only if a point on one line is connected by a line in the diagram, directly to a point on the other line. Counting these identifies every triangle that the selected vertex is part of.

Step 4 : Add up all of the triangles counted by this method and, finally, divide this sum by three because each has been triple counted (once for each vertex.)

There are 56 triangles add up to in the Snowflake plan.

There are 5 in the big equilateral triangle part into littler equilateral triangles.

There are 3 inside the focal little equilateral triangle utilizing the core vertex.

There are 18 (3 * 6) with the additional hexagonal sides around the little equilateral triangle.

There are 30 (3 * 10) with the additional hexagonal sides around the huge equilateral triangle