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Assume i got a lot of triangles where all are equilateral and of the same size. Everytime 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 trangle, the position of the new triangle is just the previous triangle times a matrice. 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?

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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

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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.

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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.

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