POLYGONAL NUMBERS

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

Encyclopedia Britannica - Main :: PIG-POL

POLYGONAL NUMBERS , in

mathematics

MATHEMATICS (Gr. /saBnľar1Kil, Sc. vOcvn or E7rlQTt'ľn; from p iN.La, "learning" or "science ")

. Suppose we have a number of equal circular counters, then the number of counters which can be placed on a

regular

REGULAR

polygon

POLYGON (Gr. rroXus, many, and ywvia, an
angle
- ANGLE (from the Lat. angulus, a corner, a diminutive, of which the primitive form,
  angus
  - ANGUS, EARLS OF
  , does not occur in Latin; cognate are the Lat. angere, to compress into a bend or to strangle, and the Gr. ayKOS, a bend; both connected with the Aryan root ank-, to
)

so that the tangents to the outer rows form the

regular

REGULAR

polygon

POLYGON (Gr. rroXus, many, and ywvia, an
angle
- ANGLE (from the Lat. angulus, a corner, a diminutive, of which the primitive form,
  angus
  - ANGUS, EARLS OF
  , does not occur in Latin; cognate are the Lat. angere, to compress into a bend or to strangle, and the Gr. ayKOS, a bend; both connected with the Aryan root ank-, to
)

and all the

internal

INTERNAL

counters are in contact with its neighbours, is a " polygonal number " of the order of the polygon. If the polygon be a triangle then it is readily seen that the numbers are 3, 6, ro, 15 .. and generally Zn (n + I) ; if a square, 4, 9, 16, . . . and generally n'; if a pentagon, 5, 12, 22... and generally n(3n--1); if a hexagon, 6, 15, 28, . . . and generally n(2n r) ; and similarly for a polygon of r sides, the general expression for the corresponding polygonal number is 2n[(nI) (r2)+2].
Algebraically, polygonal numbers may be regarded as the sums of consecutive terms of the arithmetical progressions having 1 for the first

term

TERM

and 1, 2, 3, . . . for the common

differences

DIFFERENCES, CALCULUS OF (Theory of Finite Differences)

. Taking unit common difference we have the

series

SERIES (a Latin word from serere, to join)

1; 1+2=3; 1+2+3 =6; I+2+3+4= 10; cr generally I+2+3 ,, + n= an(n+r); these are triangular numbers. With a common difference 2 we have 1; 1+3=4; 1+3+5=9; 1+3+5+7=16; or generally 1+3+5+ . . . +- (2n1) =n2; and generally for the polygonal number of the rth order we take the sums of consecutive terms of the

series

SERIES (a Latin word from serere, to join)

1, 1+(r-2), 1+2 (r-2), . . 1+n-I.r2;
and hence the nth polygonal number of the rth order is the sum of is terms of this series, i.e.,
1+I+(r2)+I+2(r2)+ ... +(I+nI.r2)
=n + ;n.n I.r -2.
The series 1, 2, 3, 4, . . . or generally n, are the so-called, " linear numbers " (cf. FIGURATE NUMBERS).

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