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A sphenic number is a positive integer n which is the product of exactly three distinct primes. The first few sphenic numbers are 30, 42, 66, 70, 78, 102, 105, 110, 114, ... ...

Johnson solid J_(86).

Johnson solid J_(88).

Let two spheres of radii R and r be located along the x-axis centered at (0,0,0) and (d,0,0), respectively. Not surprisingly, the analysis is very similar to the case of the ...

A sphere is defined as the set of all points in three-dimensional Euclidean space R^3 that are located at a distance r (the "radius") from a given point (the "center"). Twice ...

A 4-sphere has positive curvature, with R^2=x^2+y^2+z^2+w^2 (1) 2x(dx)/(dw)+2y(dy)/(dw)+2z(dz)/(dw)+2w=0. (2) Since r=xx^^+yy^^+zz^^ (3) dw = -(xdx+ydy+zdz)/w (4) = ...

Smale (1958) proved that it is mathematically possible to turn a sphere inside-out without introducing a sharp crease at any point. This means there is a regular homotopy ...

Sphere line picking is the selection of pairs of points corresponding to vertices of a line segment with endpoints on the surface of a sphere. n random line segments can be ...

Define the packing density eta of a packing of spheres to be the fraction of a volume filled by the spheres. In three dimensions, there are three periodic packings for ...

To pick a random point on the surface of a unit sphere, it is incorrect to select spherical coordinates theta and phi from uniform distributions theta in [0,2pi) and phi in ...