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In hexagonal close packing, layers of spheres are packed so that spheres in alternating layers overlie one another. As in cubic close packing, each sphere is surrounded by 12 ...

In general, an icosidodecahedron is a 32-faced polyhedron. A number of such solids are illustrated above. "The" (quasiregular) icosidodecahedron is the 32-faced Archimedean ...

Let (xi_1,xi_2) be a locally Euclidean coordinate system. Then ds^2=dxi_1^2+dxi_2^2. (1) Now plug in dxi_1=(partialxi_1)/(partialx_1)dx_1+(partialxi_1)/(partialx_2)dx_2 (2) ...

The Connell sequence is the sequence obtained by starting with the first positive odd number (1), taking the next two even numbers (2, 4), the next three odd numbers (5, 7, ...

The bound for the number of colors which are sufficient for map coloring on a surface of genus g, gamma(g)=|_1/2(7+sqrt(48g+1))_| is the best possible, where |_x_| is the ...

A polygonal number of the form n(3n-1)/2. The first few are 1, 5, 12, 22, 35, 51, 70, ... (OEIS A000326). The generating function for the pentagonal numbers is ...

Tietze's graph is the cubic graph on 12 nodes and 18 edges illustrated above in a number of embeddings. It is implemented in the Wolfram Language as GraphData["TietzeGraph"]. ...

Call a number of the form n^2-k a "near-square number." Numbers of the form n^2-5 for n=1, 2, ... are -4, -1, 4, 11, 20, 31, 44, 59, 76, 95, ... (OEIS A028875). These are ...

Convergents of the pi continued fractions are the simplest approximants to pi. The first few are given by 3, 22/7, 333/106, 355/113, 103993/33102, 104348/33215, ... (OEIS ...

Given two starting numbers (a_1,a_2), the following table gives the unique sequences {a_i} that contain no three-term arithmetic progressions. Sloane sequence A003278 1, 2, ...