# Projections of particular simplex yielding boundary of a regular polygon?

What is the maximum $$m$$ such that there is a simplex with $$n$$ vertex points in $$n-1$$ dimensions whose projection yields boundary of a regular $$m$$-gon on $$2D$$ plane?

I think you should be able to do this for arbitrary $$m$$ by taking the points $$0$$, $$e_1$$,

$$\cos\left(\frac{2\pi}{m}\right) e_1 + \sin\left(\frac{2\pi}{m}\right) e_2,$$

and

$$\cos\left(\frac{2\pi}{m} k\right) e_1 + \sin\left(\frac{2\pi}{m} k\right) e_2 + e_{k+1}$$

for $$2 \leq k \leq m -1$$.

Subtracting 0 from all the other points, we get a bunch of vectors that are linearly independent -- note $$\sin(2\pi/m)\neq 0$$ for $$m \geq 3$$ -- so these points span an $$m$$-simplex in $$\mathbb{R}^m$$.

The projection of this simplex onto the first two components gives all affine combinations of the vertices of a regular $$m$$-gon and zero, which is a solid regular $$m$$-gon.

• Does this simplex admit a non-singular linear transformation to the simplex with corner points of shape $[11\dots1100\dots00]\in\{0,1\}^{n-1}$ (all $0$s and all $1$s also considered and so we have $n$ points of which $n-1$ are linearly independent)? If the simplex vertices in your polygon also form $n-1$ independent vectors then the linear transformation should exist. – VS. Apr 16 at 10:21