The centered icosahedral numbers and cuboctahedral numbers are two different names for the same sequence of numbers, describing two different representations for these numbers as three-dimensional
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 ...
s. As centered icosahedral numbers, they are
centered number
The centered polygonal numbers are a class of series of figurate numbers, each formed by a central dot, surrounded by polygonal layers of dots with a constant number of sides. Each side of a polygonal layer contains one more dot than each side i ...
s representing points arranged in the shape of a
regular icosahedron
In geometry, a regular icosahedron ( or ) is a convex polyhedron with 20 faces, 30 edges and 12 vertices. It is one of the five Platonic solids, and the one with the most faces.
It has five equilateral triangular faces meeting at each vertex. It ...
. As cuboctahedral numbers, they represent points arranged in the shape of a
cuboctahedron
A cuboctahedron is a polyhedron with 8 triangular faces and 6 square faces. A cuboctahedron has 12 identical vertices, with 2 triangles and 2 squares meeting at each, and 24 identical edges, each separating a triangle from a square. As such, it ...
, and are a
magic number for the
face-centered cubic lattice. The centered icosahedral number for a specific
is given by
The first such numbers are
References
* .
{{Num-stub
Figurate numbers