Polygonal Numbers
POLYGONAL NUMBERS, in mathematics. Suppose we have a number of equal circular counters, then the number of counters which can be placed on a regular polygon so that the tangents to the outer rows form the regular polygon and all the 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, 10, 15 , . . and generally \n (n + i); if a square, 4, 9, 16, . . . and generally 2 ; if a pentagon, 5, 12, 22. .. and generally wfon-i); if a hexagon, 6, 15, 28, ... and generally n(m i) ; and similarly for a polygon of r sides, the general expression for the corresponding polygonal number is \n\(n i) (r 2) + 2].
Algebraically, polygonal numbers may be regarded as the sums of consecutive terms oi the arithmetical progressions having I for the first term and i, 2, 3, ... for the common differences. Taking unit common difference we have the series I; 1+2=3; 1+2+3 = 6; i + 2 + 3 + 4 = 10; or generally 1+2+3 .+ n = j n ( n -|_i) ; these are triangular numbers. With a common difference 2 we have i; 1+3 = 4; 1+3+5=9: i+3 + 5+7 = i6; or generally 1+3+5+ ... -f- (2n-i)=n 2 ; and generally for the polygonal number of the rth order we take the sums of consecutive terms of the series I, I + (r-2), 1+2 (r-2), . . . l+n-l.r-2; and hence the nth polygonal number of the rth order is the sum of n terms of this series, i.e., linear = n + Jn.n i.r 2.
The series i, 2, 3, 4, ... or generally n, are the so-called numbers " (cf. FIGURATE NUMBERS).
Note - this article incorporates content from Encyclopaedia Britannica, Eleventh Edition, (1910-1911)
