Recently, I concern with the volume of the outer parallel body of a hypersimplex that is defined as follows $$ \mathcal{H}_s(n,k)=\left\{ (x_1,\cdots,x_n):\sum_{i=1}^n x_i=k,x_i\in[0,1] \right\}, $$ where $k$ is some integer satisfying $k\ge 2$ and $k\le n/2$.

Thus, I want to konw how to compute the external angle of each face (of any fixed dimension) of $\mathcal{H}_s(n,k)$?


1 Answer 1


The first thing here would be to "see" what you are describing here. So, what really "is" that $\mathcal{H}_s(n,k)$?

The set of coordinates you allow within the set are taken from the unit $n$D hypercube $\left[0,1\right]^n$, or as a Schläfli symbol $\{4,3,3,...,3\} =: \{4,3^{n-2}\}$. That "sum = $k$" property of that set just selects coordinates from an affine hyperplane orthogonal to the vertex-first diagonal of that hypercube running through the various vertex layers. For sure, the thing you thus are after will be the intersection of that affine hyperplane with that hypercube.

Thus $\mathcal{H}_s(n,0)$ generally describes just the single vertex at the origin itself. Thence "external angles" (probably: ditopal angles) of that "shape" cannot be defined here. Next $\mathcal{H}_s(n,1)$ generally will be just the regular simplex, which is obtained at that first faceting level. All its edge sizes can be seen to be of length $\sqrt{2}$. Up to that scaling factor this is nothing but what gets described by the Schläfli symbol $\{3,3,...,3\} =: \{3^{n-2}\}$.

Thus the very first question would be to provide the ditopal angle between either facet pair of the general $(n-1)$D regular simplex $\mathcal{H}_s(n,1)$, given as a function of $n$. This quantity can generally be given as $\arccos\left(\frac1{n-1}\right)$.

When we now proceed towards values of $k\gt1$, our sectioning affine hyperplane would proceed vertex level by vertex level deeper into the hypercube along that diagonal. But there we "see" that the former simplex also gets more and more truncated, as our sectioning plane would intersect the coordinate axis already beyond that limiting value of $x_i\le1$. In fact, what we there would obtain then is $\mathcal{H}_s(n,k) = (k-1)\text{-rect}\left(\{3^{n-2}\}\right) = \left\{{3^{k-1}\;\;\,\atop 3^{n-k-1}}\right\}$, where "rect" means the operator of the according (multi-)rectification and the final symbol is Coxeter's generalized Schläfli symbol.

Now your quest was to provide the various ditopal angles of those $\mathcal{H}_s(n,k)$, i.e. of these multi-rectified regular simplices. As it happens only 2 different values would occure here. One of them will be the remaining value from the simplex, i.e. $\arccos\left(\frac1{n-1}\right)$, the other one would be just its complemental angle $\arccos\left(-\frac1{n-1}\right)$.

For instance the dihedral angle of the tetrahedron, which occurs as section of the 4D tesseract, is $\arccos\left(\frac13\right) = 70.528779°$, while the (here only) dihedral angle of the octahedron, i.e. the rectified tetrahedron, which also occurs as a deeper section of the 4D tesseract, is $\arccos\left(-\frac13\right) = 109.471221°$. The tetrahedron itself would be described herein by $\mathcal{H}_s(4,1)$, while the octahedron by $\mathcal{H}_s(4,2)$. The rectified pentachoron or $\mathcal{H}_s(5,2)$ would use both dichoral angles $\arccos\left(\frac14\right) = 75.522488°$ (between tetrahedronal and octahedronal facets) and $\arccos\left(-\frac14\right) = 104.477512°$ (between two octahedronal facets).

--- rk


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