Dense polytope

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A dense polytope is a polytope whose vertices are arbitrarily close to each other. A polytope which is not dense is called discrete.[1] Dense polytopes can have discrete elements and figures but still be dense, for example {5/2,5/2}. Dense polytopes are almost always considered degenerate, and are usually found when a uniform polytope with a finite circumradius does not close.

Dense polygons[edit | edit source]

Dense regular polygons are polygons constructed with segments that have an irrational internal angle. For example, repeating a chord of one radian over and over does not ever close at the initial point, as is irrational. This means that there are an infinite number of points on the finite circumcircle, so its vertices must become arbitrarily close.

Dense polyhedra[edit | edit source]

For example, {4,5/2}. This tiles the sphere such that squares are arranged in a pentagrammic order around a vertex. It is possible to construct such an arrangement of squares around a vertex, and there is a proper angle deficit, however it does not correspond to any spherical symmetry group and therefore does not ever close.

Dense tilings[edit | edit source]

There are two classes of dense regular apeirohedra in 𝔼2 .[2] The first are the planar apeirohedra, these are polytopes of the form {p,q} where 1/p +1/q  = 1/2, i.e. the angular defect is a multiple of 2π  other than 0 (when the angular defect is zero a discrete tiling occurs). For example, {5,10/3} and {10/3,5} have an angular defect of -4π  and thus are dense regular apeirohedra.

The other class are regular skew apeirohedra, the consist of the Petrials of the above,[2] having zigzag faces. They can also be derived as the apeir of any polygon which has no integral coordinates in any dimension.

See also[edit | edit source]

References[edit | edit source]

  1. McMullen (2004)
  2. 2.0 2.1 McMullen (2004:29)

Bibliography[edit | edit source]

  • McMullen, Peter (2004). "Regular Polytopes of Full Rank" (PDF). Discrete Computational Geometry.