hypersolid angles and hypersine

Higher-dimensional geometry (previously "Polyshapes").

hypersolid angles and hypersine

Postby Klitzing » Sat Jan 11, 2020 12:12 pm

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 Pk, then the value of the n-dimensional sine of that vertex is

  sin(v0) = Pn-1 / ∏k=1n Pk

[…] 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, …, Sn, 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)! Sk

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 αjk between
Sj and Sk.

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 2n-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.

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Re: hypersolid angles and hypersine

Postby wendy » Tue Jan 14, 2020 9:42 am

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.
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