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I understand, that there are some combinatorial problems which are not yet solved regarding gluing triangulations in 3D. At least last time I checked, it was not yet known exactly how many triangulations can one make with $N$ tetrahedra. I have some questions, that might have an answer, just I am not aware of it.

Assume, that we are working in 3dimensional Euclidean space, with equilateral simplices. I would like to glue together tetrahedra in a combinatorial way, so do not allow for degenerate triangulations. I define the area (A) as the triangles on the boundary of the triangulation. The geometry that maximises the boundary will be a tree of tetrahedra, so the maximal edge degree (number of tetrahedra around an edge) will be 3.

Is it known how many triangulations are there inside a triangulated sphere with boundary $A$?

I understand, that there are some combinatorial problems which are not yet solved regarding gluing triangulations in 3D. At least last time I checked, it was not yet known exactly how many triangulations can one make with $N$ tetrahedra. I have some questions, that might have an answer, just I am not aware of it.

Assume, that we are working in 3dimensional Euclidean space, with equilateral simplices. I would like to glue together tetrahedra in a combinatorial way, so do not allow for degenerate triangulations. I define the area (A) as the triangles on the boundary of the triangulation. The geometry that maximises the boundary will be a tree of tetrahedra, so the maximal edge degree (number of tetrahedra around an edge) will be 3, but in other cases, edge degree can be arbitrary large.

Is it known how many triangulations are there inside a triangulated sphere with boundary $A$?

I understand,that there are some combinatorial problems which are not yet solved regarding gluing triangulations in 3D. At least last time I checked, it was not yet known exactly how many triangulations can one make with $N$ tetrahedra.

Is it known how many equilateral tetrahedra we can fit inside a triangulated sphere with boundary $A$ (in Euclidean space) ?

Is it known how many triangulations are there inside a sphere with boundary $A$?

I understand, that there are some combinatorial problems which are not yet solved regarding gluing triangulations in 3D. At least last time I checked, it was not yet known exactly how many triangulations can one make with $N$ tetrahedra. I have some questions, that might have an answer, just I am not aware of it.

Assume, that we are working in 3dimensional Euclidean space, with equilateral simplices. I would like to glue together tetrahedra in a combinatorial way, so do not allow for degenerate triangulations. I define the area (A) as the triangles on the boundary of the triangulation. The geometry that maximises the boundary will be a tree of tetrahedra, so the maximal edge degree (number of tetrahedra around an edge) will be 3.

Is it known how many triangulations are there inside a triangulated sphere with boundary $A$?

I understand,that there are some combinatorial problems which are not yet solved regarding gluing triangulations in 3D. At least last time I checked, it was not yet known exactly how many triangulations can one make with N tetrahedra.

Is it known how many equilateral tetrahedra we can fit inside a triangulated sphere with boundary A (in Euclidean space) ?

Is it known how many triangulations are there inside a sphere with boundary A?

I understand,that there are some combinatorial problems which are not yet solved regarding gluing triangulations in 3D. At least last time I checked, it was not yet known exactly how many triangulations can one make with $N$ tetrahedra.

Is it known how many equilateral tetrahedra we can fit inside a triangulated sphere with boundary $A$ (in Euclidean space) ?

Is it known how many triangulations are there inside a sphere with boundary $A$?