# Regular triangulation of hypercube

I have started studying regular subdivisions of the $$n$$-cube, and came across the following post: Regularity of Delaunay triangulation of a hypercube.

My question is whether the "standard triangulation" of the $$n$$-cube described in the answer of that post is always regular. I know that the answer is yes in two and three dimensions, and, intuitively, I would guess that it is also true for arbitrary $$n$$. I'd be grateful for any help.

(A subdivision is regular if it can be obtained by lifting the vertices to $$\mathbb{R}^{n+1}$$, and projecting the lower faces of the convex hull of the lifted vertices to $$\mathbb{R}^n$$.)

• More generally the canonical triangulation of the order polytope of any poset (with simplices corresponding to linear extensions) is regular. But this question seems a little too low-level for this site. Feb 25, 2020 at 21:41
• See pp. 439-440 of csun.edu/~ctoth/Handbook/chap16.pdf. Feb 25, 2020 at 21:43
• @SamHopkins Thank you! I‘m new to this topic, and wasn‘t aware that it’s that simple. Feb 26, 2020 at 7:40