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POLYHEDRAL NUMBERS

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Originally appearing in Volume V22, Page 27 of the 1911 Encyclopedia Britannica.
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POLYHEDRAL See also:

NUMBERS , in See also:mathematics. These numbers are related to the polyhedra (see See also:POLYHEDRON) in a manner similar to the relation between polygonal numbers (see above) and polygons. Take the See also:case of See also:tetrahedral numbers. Let AB, A AC; AD be three covertical edges of a See also:regular See also:tetrahedron. See also:Divide AB, . . . into parts each equal to A 1, so that tetrahedra having the See also:common vertex A are obtained, whose linear dimensions increase arithmetically. Imagine that we have a number of See also:spheres (or shot) of a See also:diameter equal to the distance Al. It is seen that 4 shot having their centres at the vertices of the tetrahedron Al will See also:form a See also:pyramid. In the case of the tetrahedron of edge See also:A2 we require 3 along each See also:side of the See also: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 See also:

illustration may serve for a See also:definition of polyhedral numbers: a polyhedral number represents the number of equal spheres which can be placed within a polyhedron so that the spheres See also: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 See also:general See also: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).

End of Article: POLYHEDRAL NUMBERS

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POLYHEDRON (Gr. rain, many, ESpa, a base)

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