1911 Encyclopædia Britannica/Polyhedral Numbers
POLYHEDRAL NUMBERS, in mathematics. These numbers
are related to the polyhedra (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 covertical edges of
a regular tetrahedron. Divide AB,
. . . into parts each equal to A1, so
that tetrahedra having the common
vertex A are obtained, whose linear
dimensions increase arithmetically.
Imagine that we have a number of
spheres (or shot) of a diameter equal
to the distance A1. It is seen that
4 shot having their centres at the
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 A1, and 1 at A, making 10 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 1, 4, 10, 20; the general formula for the 𝑛th tetrahedral number is 1/6𝑛(𝑛+1)(𝑛+2). Cubic numbers are 1, 8, 27, 64, 125, &c; or generally 𝑛3. Octahedral numbers are 1, 6, 19, 44, &c., or generally 1/3𝑛(2𝑛2+1). Dodecahedral numbers are 1, 20, 84, 220, &c.; or generally 1/2𝑛(9𝑛2−9𝑛+2). icosahedral numbers are 1, 12, 48, 124, &c., or generally 1/2𝑛(5𝑛2−5𝑛+2).