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I find the following question embarrassing, but I have not been able to either resolve it, or to find a reference.

What is the vertex angle of a regular $n$-simplex?

Background: For a vertex $v$ in a convex polyhedron $P$, the vertex angle at $v$ is the proportion of the volume that $P$ occupies in a small ball around $v$. In symbols, $$\angle v=\lim_{\varepsilon\to 0} \frac{|B(v,\varepsilon)\cap P|}{|B(v,\varepsilon)|}.$$ Up to normalization, this definition agrees with the familiar definition of the angles in the plane, or the solid angle in $3$-space.

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You don't have to be embarrassed, since the question is to compute the volume of an $n-1$ dimensional spherical simplex (the one with interior dihedral angles $\alpha$, where $\cos\alpha = 1/n$ if I'm right), which could be hard if $n$ is large. At least, there is Schlafli differential formula, but I don't know if there is a "simple expression" only in terms, say, of $\Gamma$ function evaluation at rational numbers. I can only direct you to this paper and its references (notably Aomoto). – BS. Jan 31 '11 at 12:15
Your "vertex angle" is in fact $1/n$ times the usual solid angle (volume of the the spherical simplex in my comment), divided by the volume of $S^{n-1}$. This is of course due to the term $r^{n-1}dr$ under the volume integral in spherical coordinates. – BS. Jan 31 '11 at 17:36
See this meta note for citation of a recent paper on this topic: "Bounds for Pach's selection theorem and for the minimum solid angle in a simplex." – Joseph O'Rourke Nov 21 '15 at 18:02
up vote 16 down vote accepted

In the paper by John Leech, "Sphere packings in Higher Space" Canadian Journal of Mathematics, 1964, which you can find at the Google book links here, the following formulas are given for the "solid angular content at each vertex of a regular simplex":

$$2^{-n} n! f_n(n) H_n$$


$$H_n = 2 \pi^{\frac{1}{2}n} / \Gamma(n/2)$$

is the total $(n{-}1)$-dimensional surface content, $f_n(\sec 2 \alpha)=F_n(\alpha)$, and (finally!) $F_n(\alpha)$ is Schläfli's function mentioned by BS in his comment. This function is discussed in Section 7.2 (p.107ff) of Chuanming Zong's book, Sphere Packings (Google books link here) and in the paper, "Analytic structure of Schläfli function," Kazuhiko Aomoto, Nagoya Math. J. Volume 68 (1977), 1-16, also mentioned by BS. Rogers computed an asymptotic formula for $F_n(\alpha)$:

$$ \frac{ \sqrt{ 1+cn}} {\sqrt{2} e^{1/c} n!} \left( \frac{2 e}{\pi c n} \right) ^{n/2} \;,$$

where $c = (\sec 2 \alpha - (n-1))^{-1}$.

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Thanks a lot for the reference! – Boris Bukh Jan 31 '11 at 20:08
so, after pluggin in $\sec 2\alpha = n$, we get that the vertex angle is $\frac{\sqrt{n+1}}{\sqrt{2}e} \cdot \left(\frac{2e}{\pi n}\right)^{n/2} \cdot \frac{H_n}{2^n}$, which is, asymptotically, $2^{-\Theta(n\log n)}\cdot H_n$. Right? – Jan Kyncl May 13 '14 at 0:57
@JanKyncl: You're ahead of me on this. That does seem correct. – Joseph O'Rourke May 13 '14 at 10:52

In Theorem 1 in this paper (also citation 3 in that paper), a different definition of the vertex angle is given, with the value for the regular simplex computed. If you combine that with the description in Mathworld, you should be able to get an expression for your expression up there.

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I saw this paper, but I could not figure why the definition in this paper is consistent with the definition of the angle I gave. Also, from the formula in the paper, in dimension 3 I do not get the answer I expect. – Boris Bukh Jan 31 '11 at 12:05
Right, which is why I said it is a different definition (I also checked the case in 3D). I was being hopeful that with a bit of massaging around there may be a simple closed-form expression connecting that definition to your definition. But maybe I was being too optimistic. – Willie Wong Jan 31 '11 at 14:40
The determinant in that definition "approximates" the volume of the polar spherical simplex. If the given spherical simplex is considered as a cone bounded by $n$ hyperplanes, then any of the $2^n$ cones gives the same angle by this definition, but their real angles may be very different. (The formula was proved also in citation [2], which is in Czech). – Jan Kyncl May 13 '14 at 0:25

Here is a link to the original paper by Rogers mentioned by Joseph O'Rourke in his answer, with a short proof of the asymptotic formula for $F_n(\alpha)$, which is reproduced in the book Sphere Packings by Chuanming Zong.

It is also shown there that the multiplicative error in the formula is at most $1+ \frac{31}{12n} + O\left(\frac{1}{n^2}\right)$ in the case of the regular simplex, where $\sec 2\alpha=n$ (and thus $c=1$, or, $b=1$ in the paper).

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