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Whilst working on a Tangram problem, I came across the need to find the total number of convex shapes that can be produced from $16$ identical (isosceles) right angle triangles (since the Tangram can be subdivided in this way).

This should be something that can be computed by hand, but I am curious if there are any references that explore producing a general solution to this problem for $n$ (isosceles) right angle triangles and the total number of shapes that they can produce?

Is this something that has been explored before? And if so is there a result that I can reference?

To be make the problem rigorous:

  • I am only allowing for rotation and translation of the triangles
  • I would specifically be interested in the number of distinct convex polygons that can be made (however, if there are references on the instances where the shapes need not be distinct, then these would also be worth mentioning)

As mentioned in the comments, I have posted a request for a specific computational solution for $n=16$ on MSE. This question is fundamentally different, as I am looking for a reference request that generalizes the problem solution. Whereas, the other post does not request a solution that generalizes.

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    $\begingroup$ Simulposted to m.se, math.stackexchange.com/questions/4671241/… thereby abusing both sites. $\endgroup$ Commented Apr 2, 2023 at 12:13
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    $\begingroup$ This is a reference request. The mse question is a specific request for a computational solution to the specific instance of $n=16$ - not a reference request $\endgroup$ Commented Apr 2, 2023 at 12:26
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    $\begingroup$ (i) I think it should be explained why this is a valid research problem (if it is such a one). (ii) Anyhow, you can try to count the number $C_n$ of the convex shapes for, say, $n=1,\dots,10$ triangles and then try to find the sequence of the $C_n$'s on the OEIS. $\endgroup$ Commented Apr 2, 2023 at 14:30
  • $\begingroup$ When you post two questions so closely related to each other, you should link each post to the other. $\endgroup$ Commented Apr 2, 2023 at 22:10

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The following facts (and more) can all be found at:

OEIS A245676: Number of convex polyaboloes (or convex polytans): number of distinct convex shapes that can be formed with n congruent isosceles right triangles. Reflections are not counted as different.

Let $a(n)$ denote this sequence. Then:

  • $$a(n) = A093709(n) + A292146(n) + A292147(n) + A292148(n) + A292149(n) + A292150(n)$$

  • As far as I can tell, the most complete list that exists gives values of $a(n)$ for $n \in \{1, \cdots , 750 \}$. This is produced by Douglas J. Durian and can be found at the following link.

  • For $n \in \{1, \cdots , 20 \}$, Douglas J. Durian has also produced a PowerPoint visualising all possible convex polygons that can be produced for each $n$.

  • $a(n) < A006074 \space \space \space \text{for} \space n>2$

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