Figurate Number

A figurate number, also (but mostly in texts from the 1500 and 1600s) known as a figural number (Simpson and Weiner 1992, p. 587), is a number that can be represented by a regular geometrical arrangement of equally spaced points. If the arrangement forms a regular polygon, the number is called a polygonal number. The polygonal numbers illustrated above are called triangular, square, pentagonal, and hexagonal numbers, respectively. Figurate numbers can also form other shapes such as centered polygons, L-shapes, three-dimensional solids, etc.

The th regular -polytopic number is given by

			(1)
			(2)
			(3)

where  is the multichoose function,  is a binomial coefficient, and  is a rising factorial. Special cases therefore include the triangular numbers

(4)

tetrahedral numbers

(5)

pentatope numbers

(6)

and so on (Dickson 2005, p. 7).

The following table lists the most common types of figurate numbers.

figurate number	formula
biquadratic number
centered cube number
centered pentagonal number
centered square number
centered triangular number
cubic number
decagonal number
gnomonic number
Haűy octahedral number
Haűy rhombic dodecahedral number
heptagonal number
hex number
heptagonal pyramidal number
hexagonal number
hexagonal pyramidal number
octagonal number
octahedral number
pentagonal number
pentagonal pyramidal number
pentatope number
pronic number
rhombic dodecahedral number
square number
square pyramidal number
stella octangula number
tetrahedral number
triangular number
truncated octahedral number
truncated tetrahedral number

REFERENCES:

Conway, J. H. and Guy, R. K. The Book of Numbers. New York: Springer-Verlag, pp. 30-62, 1996.

Dickson, L. E. "Polygonal, Pyramidal, and Figurate Numbers." Ch. 1 in History of the Theory of Numbers, Vol. 2: Diophantine Analysis. New York: Chelsea, pp. 1-39, 2005.

Goodwin, P. "A Polyhedral Sequence of Two." Math. Gaz. 69, 191-197, 1985.

Guy, R. K. "Figurate Numbers." §D3 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 147-150, 1994.

Kraitchik, M. "Figurate Numbers." §3.4 in Mathematical Recreations. New York: W. W. Norton, pp. 66-69, 1942.

Savin, A. "Shape Numbers." Quantum 11, 14-18, 2000.

Simpson, J. A. and Weiner, E. S. C. (Preparers). The Compact Oxford English Dictionary, 2nd ed. Oxford, England: Clarendon Press, 1992.