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, A 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 Al . It is seen that 4 shot having their centres at the vertices of the tetrahedron Al 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 Al, and I at A, making Io 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 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, 10, 20; the general formula for the nth tetrahedral number is bn(n+1)(n+2) . Cubic numbers are 1, 8, 27, 64, 125, &c.; or generally n3 . Octahedral numbers are r, 6, 19, 44, &c., or generally **n(2n2+I) . Dodecahedral numbers are 1, 20, 84, 220, &c.; or generally 2n(9n'—9n+2) . Icosahedral numbers are t, 12, 48, 124, &c., or generally an(5n2—5n+2) .