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The Menger sponge is a fractal which is the three-dimensional analog of the Sierpiński carpet. The nth iteration of the Menger sponge is implemented in the Wolfram Language ...

A minimal cover is a cover for which removal of any single member destroys the covering property. For example, of the five covers of {1,2}, namely {{1},{2}}, {{1,2}}, ...

A multiway system is a kind of substitution system in which multiple states are permitted at any stage. This accommodates rule systems in which there is more than one ...

A natural equation is an equation which specifies a curve independent of any choice of coordinates or parameterization. The study of natural equations began with the ...

O_h is the point group of symmetries of the octahedron having order 48 that includes inversion. It is also the symmetry group of the cube, cuboctahedron, and truncated ...

Every totally ordered set (A,<=) is associated with a so-called order type. Two sets A and B are said to have the same order type iff they are order isomorphic (Ciesielski ...

A set of ascending sequences in a permutation is called a run (Graham et al. 1994) or sometimes a rise (Comtet 1974, p. 241). A sorted permutation consists of a single run, ...

Every nonconstant entire function attains every complex value with at most one exception (Henrici 1988, p. 216; Apostol 1997). Furthermore, every analytic function assumes ...

The positive integers 216 and 12960000 appear in an obscure passage in Plato's The Republic. In this passage, Plato alludes to the fact that 216 is equal to 6^3, where 6 is ...

Presburger arithmetic is the first-order theory of the natural numbers containing addition but no multiplication. It is therefore not as powerful as Peano arithmetic. ...