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Polyhedral Numbers

1911 Encyclopedia Britannica

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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 A 1, 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 Ai. It is seen that 4 shot having their centres at the vertices of the tetrahedron Ai 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 Ar, and i at A, making ro in all. To add a third layer, we will require 4 along each base, i.e. 9, and r in the centre. Hence in the tetrahedron A3 we have 20 shot. The numbers I, 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, Io, 20; the general formula for the nth tetrahedral number is bn(n+I)(n+2). Cubic numbers are 1, 8, 27, 64, 125, &c.; or generally n 3. Octahedral numbers are 1, 6, 1 9, 44, &c., or generally 3n(2n 2 +1). Dodecahedral numbers are I, 20, 84, 220, &c.; or generally n(9n 2 -9n+2). Icosahedral numbers are I, 12, 48, 124, &c., or generally zn(5n2-5n+2).