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
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.
See also Biquadratic Number ,
Centered Cube Number ,
Centered Pentagonal Number ,
Centered Polygonal Number ,
Centered
Square Number ,
Centered Triangular Number ,
Cubic Number ,
Decagonal
Number ,
Figurate Number Triangle ,
Gnomonic Number ,
Heptagonal
Number ,
Heptagonal Pyramidal Number ,
Hex Number ,
Hex
Pyramidal Number ,
Hexagonal Number ,
Hexagonal
Pyramidal Number ,
Multichoose ,
Nexus
Number ,
Octagonal Number ,
Octahedral
Number ,
Pentagonal Number ,
Pentagonal
Pyramidal Number ,
Pentatope Number ,
Polygonal
Number ,
Pronic Number ,
Pyramidal
Number ,
Rhombic Dodecahedral Number ,
Square Number ,
Square
Pyramidal Number ,
Stella Octangula Number ,
Tetrahedral Number ,
Triangular
Number ,
Truncated Octahedral Number ,
Truncated Tetrahedral Number
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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. Referenced on Wolfram|Alpha Figurate Number
Cite this as:
Weisstein, Eric W. "Figurate Number."
From MathWorld --A Wolfram Web Resource. https://mathworld.wolfram.com/FigurateNumber.html
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