Hypersine

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

(1)

Changing the length of an edge of the parallelotope by a factor changes the content by the same factor and the contents of all but one of the facets by the same factor. Thus, a change in edge length does not affect the value of the right-hand side, and the sine function is dependent solely on the angles between the edges of the parallelotope, not their lengths. In addition, the sines of all of the vertex angles of the parallelotope are the same, since the opposite facets have the same content, and one of each pair of opposite facets meets at each vertex. If we extend the facets at a vertex, all of the vertex angles thus formed have the same sine, as they are simply translations of the vertex angles of the parallelotope.

The maximum value of the -dimensional sine is one, when the parallelotope is an orthotope and all of its vertex angles are right angles. The minimum value of the -dimensional sine is zero, for a parallelotope or simplex that lies in a space of lower dimension and is therefore degenerate. If a simplex has a right angle (one at which all of the edges are mutually orthogonal), it is a right simplex, and the sums of the squares of the sines of the vertex angles other than the right angle is one. The -dimensional sine of any vertex angle of an -dimensional right simplex is the ratio of the content of the opposite facet to the content of the facet opposite the right angle.

The vertex simplex of a vertex of an -dimensional parallelotope is the simplex that has as its vertices that vertex and the adjacent vertices of the parallelotope. Its content (and the content of all the other vertex simplices) is the content of the parallelotope divided by . If the content of an -dimensional simplex is , and the contents of the facets that meet at vertex are , 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 equation (1) gives

(2)

If we label the remaining facet of the simplex , we may derive

(3)

Since the right-hand side is independent of which vertex of the simplex is chosen, the left-hand side is the same for all vertices, so that there is a law of sines for the -dimensional simplex such that the ratio of the sine of a vertex to the content of the opposite facet is the same for all vertices.

For a simplex in elliptic or spherical space, the sine of a vertex of the simplex is equal to the polar sine of the facet of its polar simplex for which that vertex is a pole. The edges of that facet have an arc length in radians that is supplementary to the corresponding dihedral angle between facets of the vertex angle. Using this, we can apply a formula for the polar sine of the facet of the polar simplex in terms of its edges to calculate the -dimensional sine from the dihedral angles between the facets that meet at that angle. If the dihedral angle between facets and is , then

(4)

For a tetrahedron with dihedral angles , , and between the facets meeting at vertex , we have

(5)

The content of a parallelotope is equal to the product of the polar sine of a vertex angle and the edges meeting at that angle. If we identify as the vertex angle of facet of a parallelotope, we can substitute in Equation (1) and cancel out the edges to obtain

(6)

The polar sine is the same for all vertices of a parallelotope, and it is easily calculated from the plane angles between edges of the -dimensional angle.

In two-dimensional space, the -dimensional sine of the vertex of a triangle is the same as the sine of the vertex angle. In three-dimensional space, the -dimensional sine of the vertex angle of a tetrahedron with face angles , , and at that vertex is given by

(7)