Forming a ring with equilateral triangle of the same size Assume I got a lot of triangles where all are equilateral and of the same size. Every time I would like to stick one edge of a triangle to another edge of triangle. My question is: Is it possible to stick triangle edge to edge such that it forms a ring like structure? What a ring like structure is mean that the we can get the same position of the first triangle after many sticking processes. 
What I have thought is that each time we stick a triangle, the position of the new triangle is just the previous triangle times a matrix. But i don't know how to show it is possible or not possible to form a ring. I guess the ans should be no, but is there any proof to this case?
 A: Yes, in many ways. We can for example tile the plane using equilateral triangles, and find many ring-like patterns. The most primitive is obtained by joining the centre of a regular hexagon to its vertices.  We get a ring with no "hole." But by continuing the pattern of equilateral triangles, we can produce infinitely many different-shaped rings with holes.
One of the simplest comes from splitting each equilateral triangle that made up our regular hexagon into four equilateral triangles, and picking up just the $18$ small equilateral triangles that are on the outside periphery of our hexagon. 
A: If you have $n$ triangles, $n \ge 3$, one approach is to take a regular $n$-gon and erect an equilateral triangle on each side.  You can go inward or outward, but it seems from your question that you should always go the same direction.  For $n \lt 6$ the inward case will have the triangles overlapping, but maybe that is OK.  For $n=6$ the sides overlap and you get a piece of the tiling described by André Nicolas.  For $n \gt 6$ they all fit inside just fine.
