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The number of cubes in an octahedron formed by stacking centered squares is a centered octahedral number, the sum of two consecutive octahedral numbers. These numbers are These numbers are 1, 7, 25, 63, 129, 231, 377, 575, 833, 1159, 1561, 2047, 2625, ...
An octahedron can be any polyhedron with eight faces. In a previous example, the regular octahedron has 6 vertices and 12 edges, the minimum for an octahedron; irregular octahedra may have as many as 12 vertices and 18 edges. [24] There are 257 topologically distinct convex octahedra, excluding mirror images. More specifically there are 2, 11 ...
In mathematics, a centered octahedral number or Haüy octahedral number is a figurate number that counts the points of a three-dimensional integer lattice that lie inside an octahedron centered at the origin. [1] The same numbers are special cases of the Delannoy numbers, which count certain two-dimensional lattice paths. [2]
Patterns in Nature. Little, Brown & Co. Stewart, Ian (2001). What Shape is a Snowflake? Magical Numbers in Nature. Weidenfeld & Nicolson. Patterns from nature (as art) Edmaier, Bernard. Patterns of the Earth. Phaidon Press, 2007. Macnab, Maggie. Design by Nature: Using Universal Forms and Principles in Design. New Riders, 2012. Nakamura, Shigeki.
The compound figure comprising two such dual tetrahedra form a stellated octahedron or stella octangula. Its interior is an octahedron, and correspondingly, a regular octahedron is the result of cutting off, from a regular tetrahedron, four regular tetrahedra of half the linear size (i.e., rectifying the tetrahedron).
The non-diagonal numbers say how many of the column's element occur in or at the row's element. For example, there are 2 vertices in each edge (each edge has 2 vertices), and 2 cells meet at each face (each face belongs to 2 cells), in any regular 4-polytope. The configuration for the dual polytope can be obtained by rotating the matrix by 180 ...
The deltoidal icositetrahedron is a member of a family of duals to the uniform polyhedra related to the cube and regular octahedron. When projected onto a sphere (see right), it can be seen that the edges make up the edges of a cube and regular octahedron arranged in their dual positions. It can also be seen that the 3- and 4-fold corners can ...
The convex shapes in this family range from the octahedron itself through the regular icosahedron to the cuboctahedron, with its square faces subdivided into two right triangles in a flat plane. Extending the range of the parameter past the proportion that gives the cuboctahedron produces non-convex shapes, including Jessen's icosahedron.
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