Triangulating a hypercube

Definitions and mathematical statement

Let $C_d=[0,1]^d$. A triangulation of $C_d$ is a finite collection of nondegenerate closed $d$-simplices whose union is $C_d$, such that any two intersect in a common face or not at all. Its size is the number of its $d$-dimensional simplices, not the total number of faces.

In the standard vertex-triangulation problem every simplex vertex belongs to $\{0,1\}^d$. Define \[ \tau_d=\min\{\#\mathcal T:\mathcal T\text{ triangulates }C_d \text{ using only cube vertices}\}. \] Determine $\tau_d$ and its asymptotic growth as $d\to\infty$, including sharp exponential improvements over the elementary $d!$ subdivision. That subdivision orders the $d$ coordinates in every possible way and uses one simplex for each permutation.

If extra vertices are permitted, define $\tau_d^{\mathrm{St}}$ by the same minimum with arbitrary vertices in $C_d$. These are Steiner vertices, and $\tau_d^{\mathrm{St}}\le\tau_d$. The two conventions should be distinguished when reporting a minimum or an asymptotic bound. A dissection allowing nonmatching faces, or a demand that all simplices be congruent, is also a different problem. No congruence condition is imposed here.

Short English statement

What is the smallest triangulation of a high-dimensional cube, and how does its size grow with dimension? Distinguish cube-vertex triangulations from those allowed to insert extra vertices.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 3900005 (AMR-038-0005), AMR Open Problem Lists. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Eppstein - The Geometry Junkyard: Open Problems, Triangulating a hypercube. Original source indexed by AMR-038-0005. https://ics.uci.edu/~eppstein/junkyard/open.html.
• S3 David Eppstein, Geometry Junkyard, Cube triangulation. https://ics.uci.edu/~eppstein/junkyard/cube-triangulation.html.