The relations connecting the angle-sums and volume of a polytope in space of n dimensions

Abstract

1.1. In two-dimensional spherical or elliptic geometry we have the familiar relation between the area of a triangle and its angle-sum, ∆ = k 2 (∑ α ‒ π ), (1.11) 2 π being the measure of the whole angle at a point, and k the space-constant, or, in spherical geometry, the radius of the sphere. It is well known that in three-dimensional spherical or elliptic geometry there is no corresponding relation involving the volume of a tetrahedron. For elliptic or hyperbolic space of four dimensions it was proved by Dehn that the volume of a simplex can be expressed linearly in terms of the sums of the dihedral angles (angles at a face), angles at an edge, and angles at a vertex, but for space of five dimensions the linear relations do not involve the volume. He indicates also, in a general way, the extensions of these results for spaces of any odd or even dimensions. He shows further that these results are connected with the form of the Euler polyhedral theorem, which is expressed by a linear relation connecting the numbers of boundaries of different dimensions, and which for space of odd dimensions is not homogeneous, e. g ., N 2 — N 1 + N 0 = 2 in three dimen­sions, but for space of even dimensions is homogeneous, e. g ., N 1 — N 0 = 0 in two dimensions, N 3 — N 2 + N 1 — N 0 = 0 in four. The connection, as Dehn points out, was made use of by Legendre in a proof which he gave for the Euler formula in three dimensions. 1.2. Dehn extends this connection in detail for four and five dimensions, and states the following general results in space of dimensions R n for simplexes and for polytopes bounded entirely by simplexes : (1) In R n there are ½ n + 1 or ½( n + 1) relations (according as n is even or odd) between the numbers of boundaries of a polytope bounded by simplexes.