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 v_{0}are P_{k}, then the value of the n-dimensional sine of that vertex is

sin(v_{0}) = P^{n-1}/ ∏_{k=1}^{n}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 v_{0}are S_{1}, S_{2}, …, 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(v_{0}) = (n! S)^{n-1}/ ∏_{k=1}^{n}(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(v

_{0}) = 1

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

sin

^{2}(v

_{0}) = (n+1)

^{n-1}/ n

^{n}= 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