POLYHEDRAL NUMBERS

This article appears in Volume V22, Page 27 of the Encyclopedia Britannica.

Encyclopedia Britannica - Main :: PIG-POL

POLYHEDRAL NUMBERS , in

mathematics

MATHEMATICS (Gr. /saBnµar1Kil, Sc. vOcvn or E7rlQTt'µn; from p iN.La, "learning" or "science ")

. These numbers are related to the polyhedra (see

POLYHEDRON

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

) 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

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

SPHERES, MUSIC OF THE

(or shot) of a

diameter

DIAMETER (from the Cr. &a, through, µErpov, measure)

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

ILLUSTRATION

may serve for a definition of polyhedral numbers: a polyhedral number represents the number of equal

spheres

SPHERES, MUSIC OF THE

which can be placed within a

polyhedron

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

so that the spheres

touch

TOUCH (derived through Fr. toucher from a common Teutonic and Indo-Germanic root, cf. " tug," " tuck," O. H. Ger. zucchen, to twitch or draw)

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

FORMULA
FORMULA (Lat. diminutive of forma, shape,
pattern
- PATTERN
, &c., especially used of rules of judicial
procedure
- PROCEDURE (Fr. procedure, from Lat. procedere, to go for-ward)
)

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(5n25n+2).

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