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In three dimensions, a parallelepiped is a prism whose faces are all parallelograms. Let A, B, and C be the basis vectors defining a three-dimensional parallelepiped. Then ...

A circle packing is called rigid (or "stable") if every circle is fixed by its neighbors, i.e., no circle can be translated without disturbing other circles of the packing ...

Sprague (1963) considered the problem of "rolling" five cubes, each which an upright letter "A" on its top, on a chessboard. Here "rolling" means the cubes are moved from ...

A spheroid is an ellipsoid having two axes of equal length, making it a surface of revolution. By convention, the two distinct axis lengths are denoted a and c, and the ...

Given a circular table of diameter 9 feet, which is the minimal number of planks (each 1 foot wide and length greater than 9 feet) needed in order to completely cover the ...

A number of attractive tetrahedron 6-compounds can be constructed. The first compound (left figures) is obtained by combining three stella octangula. A second can be obtained ...

Oriented spheres in complex Euclidean three-space can be represented as lines in complex projective three-space ("Lie correspondence"), and the spheres may be thought of as ...

A graph for which the relations between pairs of vertices are symmetric, so that each edge has no directional character (as opposed to a directed graph). Unless otherwise ...

An ordered vector basisv_1,...,v_n for a finite-dimensional vector space V defines an orientation. Another basis w_i=Av_i gives the same orientation if the matrix A has a ...

The equation of motion for a membrane shaped as a right isosceles triangle of length c on a side and with the sides oriented along the positive x and y axes is given by where ...