Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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?

share|cite|improve this question
The answer will depend on the triangles you have. – Gerry Myerson Mar 27 '12 at 11:51
Edited. The triangle is equilateral and of the same size – Mathematics Mar 27 '12 at 17:52
up vote 3 down vote accepted

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.

share|cite|improve this answer
Edited. The triangle is equilateral and of the same size – Mathematics Mar 27 '12 at 16:33
Wait, how do you split an equilateral triangle into three equilateral triangles? Four, sure, but three? I count 18 triangles in your construction. It is possible to arrange 12 triangles around a triangular hole. – Gerry Myerson Mar 27 '12 at 23:54
Yeah, I counted wrong. Ran out of fingers. – André Nicolas Mar 28 '12 at 1:56

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.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.