1911 Encyclopædia Britannica/Polyhedral Numbers

POLYHEDRAL NUMBERS, in mathematics. These numbers are related to the polyhedral (see POLYHEDRON) in a manner similar to the relation between polygonal numbers (see above) and polygons. Take the case of tetrahedral numbers. Let AB, AC, AD be three co vertical edges of

a regular tetrahedron. Divide AB,

into parts each equal to AVI, so

that tetrahedral having the common

vertex A are obtained, whose linear

2 dimensions increase arithmetically.

3 Imagine that we have a number of

spheres (or shot) of a diameter equal

B D to the distance A1. It is seen that

4 shot having their centres at the

C vertices of the tetrahedron A1 will form a pyramid. In the case of the tetrahedron

of edge A2 we require 3 along each side of the base, i.e. 6, 3 along the base of AI, and I at A, making IO in all. To add a. third layer, we will require 4 along each base, i.e. 9, and 1 in the centre. Hence in the tetrahedron A3 we have 20 shot. The numbers 1, 4, 10, 20 are polyhedral numbers, and from their association with the tetrahedron are termed “ tetrahedral numbers.”

This illustration may serve for a definition of polyhedral numbers: a polyhedral number represents the number of equal spheres which can be placed within a polyhedron so that the spheres touch one another or the sides of the polyhedron. In the case of the tetrahedron we have seen the numbers to be I, 4, IO, 20; the general formula for the nth tetrahedral number is § n(n+1)(n-|-2). Cubic numbers are 1, 8, 27, 64, 125, &c; or generally na. Octahedral numbers are 1, 6, 19,44, &c., or generally § n(2n2+1). Dodecahedral numbers are 1, 20, 84, 220, &c.; or generally § n(9n2-9n-|-2). icosahedral numbers are 1, 12, 48, 124, &c., or generally § n(5n2-5n+2).