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Is it possible to tessellate a planar surface from triangles but with the following constraints:

  • density (average number of triangles) can be varied.
  • a finite set of unique triangles are used for the tessellation. Say 5-15 unique triangles.

Optionally also with the ability to increase density for sub areas.

If it is possible, can you point me to some relevant material.

Thanks

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    Well, then, it seems that you have your answer. How do you propose to tessellate a disk?2012-02-25

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I'm going to make an interpretation of the question, and then answer it. If my interpretation is wrong, OP can let us know.

We are given triangles $T_1,T_2,\dots,T_n$, and we want to know whether it is possible to tessellate an arbitrary polygonal region $P$ with a finite number of triangles, each triangle similar to one of those given.

I claim it's not possible. Let $P$ have an angle that is smaller than any of the angles in the triangles. Then there is no way to get to that angle.

Now, what if we are allowed to pick the triangles $T_1,T_2,\dots,T_n$ after we have seen the region $P$? If $n$ is fixed, we're still out of luck. The angles we can get lie in an extension field of transcendence degree at most $2n+1$ over the rationals, so if we are faced with a region with more than $2n+1$ algebraically independent angles, we can't tessellate it.

In short, under the sort of assumptions I've been making, the class of tessellatable polygonal regions is very restricted.

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    The Penrose picture shows that one particular polygon can be tessellated with a small number of basic shapes of triangle. Notice that as you go around the perimeter of that polygon, you never come to any really, really sharp angles. If you choose your triangles in advance, and then someone confronts you with a polygon with an angle sharper than any of the ones in your triangles, you are out of luck.2012-02-26