hypersolid angles and hypersine

[Klitzing]

I learned from Wolfram Alpha about hypersine, as being a dimensional analog trig function for hypersolid angles. There it is being defined by

The hypersine (n-dimensional sine function) is a function of a vertex angle of an n-dimensional parallelotope or simplex. If the content of the parallelotope is P and the contents of the n facets of the parallelotope that meet at vertex v₀ are P_k, then the value of the n-dimensional sine of that vertex is

  sin(v₀) = P^n-1 / ∏_k=1ⁿ P_k

[…] The vertex simplex of a vertex of an n-dimensional parallelotope is the simplex that has as its vertices that vertex and the n adjacent vertices of the parallelotope. Its content (and the content of all the other vertex simplices) is the content of the parallelotope divided by
n!. If the content of an n-dimensional simplex is S, and the contents of the n facets that meet at vertex v₀ are S₁, S₂, …, S_n, the simplex can be considered a vertex simplex of a parallelotope, and the facets also vertex simplices of the facets of the parallelotope with respect to the same vertex. Substituting in [the above] equation gives

  sin(v₀) = (n! S)^n-1 / ∏_k=1ⁿ (n−1)! S_k

Confering the first defining equation to the area formula of a parallelogram, it becomes obvious that this definition is nothing but the usual sine function for n=2.

This article moreover continues on how to calculate the latter formula when knowing the respective dihedral angles α_jk between
S_j and S_k.

Applying this makes it easy to get the hypersine of the corner angle of any hypercube generally:

  sin(v₀) = 1

Even the hypersine of the corner angle of any regular simplex could be calculated to:

  sin²(v₀) = (n+1)^n-1 / nⁿ = 1/(n+1) (1 + 1/n)ⁿ

But in order to calculate correspondingly the hypersine of the corner angle of the orthoplex one needs to extrapolate this function to non-simplicial corners as well. One clearly could easily subdivide the orthoplex corner symmetrically into 2^n-1 equal (mostly right-angled) simplices, but then again one needs to know about addition theorems for this hypersine function.

Is there anything being known in this direction or could anything be derived for that purpose? - I would be very grateful to learn about it.

--- rk

[wendy]

From using inside and outside polytopes, I have established this:

simplex = S(n) T^n radians
cross polytope = 2^(n-1)T(n) T^n radian

1 < S(n-1) < S(n) < sqrt(n/4)
T(n) converges on S(n)/sqrt(n).

S(8) > 4/3 ; S(24) > 1.6 from the E_8 and Leech lattice.

T^n radian means the volume of a cross polytope, whose long diagonal is a radius.

Taken together, it means the simplex-angle in 120 dimensions lies somewhere between 1.6 < S(120) < 5.5. There's some interesting stuff in 124 dimensions worth the look at, but i can't recall the result at this moment.

People have been poking these kind of formulae for a long time. Have a look at the "Schläfli function" in Coxeter's regular polytopes pp 141-144.