0
$\begingroup$

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 $A$?

Improve this question
$\endgroup$
3
  • 1
    $\begingroup$ How can the "how many" questions make sense without any reference to size or distance? And by "inside" are you referring to the bounded portion of the complement of the sphere in euclidean space, or something else? (I also notice that the dimensions of the sphere and of the euclidean space are left unmentioned.) $\endgroup$
    – Daniel Asimov
    Commented Oct 2 at 18:15
  • 1
    $\begingroup$ @DanielAsimov : Might the phrase "with boundary $A$" be a "reference to size of distance"? Maybe it means $A$ is a polyhedron whose vertices are on the sphere, to the question is about the number of triangulations by tetrhedra of a polyhedron inscribed in a sphere. $\endgroup$
    – Michael Hardy
    Commented Oct 3 at 17:19
  • $\begingroup$ I wrote, that "gluing triangulations in 3D", but if im not precise enough, then I will reformulate my question. $\endgroup$
    – Kregnach
    Commented Oct 3 at 18:46

0

Reset to default

You must log in to answer this question.

Browse other questions tagged .