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The solid cut from a horizontal cylinder of length L and radius R by a single plane oriented parallel to the cylinder's axis of symmetry (i.e., a portion of a horizontal ...

Let a set of vertices A in a connected graph G be called convex if for every two vertices x,y in A, the vertex set of every (x,y) graph geodesic lies completely in A. Also ...

A link invariant defined for a two-component oriented link as the sum of +1 crossings and -1 crossing over all crossings between the two links divided by 2. For components ...

Let E be the largest and e the smallest power of l in the HOMFLY polynomial of an oriented link, and i be the braid index. Then the Morton-Franks-Williams inequality holds, ...

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 ...