An icosahedral number is a

figurate number The term figurate number is used by different writers for members of different sets of numbers, generalizing from triangular numbers to different shapes (polygonal numbers) and different dimensions (polyhedral numbers). The term can mean * polyg ...

that represents an icosahedron. The ''n''th icosahedral number is given by the formula :

The first such numbers are 1, 12, 48, 124, 255, 456, 742, 1128, 1629, 2260, 3036, 3972, 5083, … .

History

The first study of icosahedral numbers appears to have been by

René Descartes René Descartes ( or ; ; Latinized: Renatus Cartesius; 31 March 1596 – 11 February 1650) was a French philosopher, scientist, and mathematician, widely considered a seminal figure in the emergence of modern philosophy and science. Ma ...

, around 1630, in his ''De solidorum elementis''. Prior to Descartes, figurate numbers had been studied by the ancient Greeks and by

Johann Faulhaber Johann Faulhaber (5 May 1580 – 10 September 1635) was a German mathematician. Born in Ulm, Faulhaber was a trained weaver who later took the role of a surveyor of the city of Ulm. He collaborated with Johannes Kepler and Ludolph van Ceulen. Bes ...

, but only for

polygonal number In mathematics, a polygonal number is a number represented as dots or pebbles arranged in the shape of a regular polygon. The dots are thought of as alphas (units). These are one type of 2-dimensional figurate numbers. Definition and examples T ...

pyramidal number A pyramidal number is a figurate number that represents a pyramid with a polygonal base and a given number of triangular sides. A pyramidal number is the number of points in a pyramid where each layer of the pyramid is an -sided polygon of points. ...

s, and

cubes In geometry, a cube is a three-dimensional space, three-dimensional solid object bounded by six square (geometry), square faces, Facet (geometry), facets or sides, with three meeting at each vertex (geometry), vertex. Viewed from a corner it i ...

. Descartes introduced the study of figurate numbers based on the

Platonic solid In geometry, a Platonic solid is a convex, regular polyhedron in three-dimensional Euclidean space. Being a regular polyhedron means that the faces are congruent (identical in shape and size) regular polygons (all angles congruent and all e ...

s and some semiregular polyhedra; his work included the icosahedral numbers. However, ''De solidorum elementis'' was lost, and not rediscovered until 1860. In the meantime, icosahedral numbers had been studied again by other mathematicians, including

Friedrich Wilhelm Marpurg Friedrich Wilhelm Marpurg (21 November 1718 – 22 May 1795) was a German music critic, music theorist and composer. He was friendly and active with many figures of the Enlightenment of the 18th century. Life Little is known of Marpurg's ear ...

in 1774, Georg Simon Klügel in 1808, and Sir Frederick Pollock in 1850.

References

Figurate numbers {{Num-stub