vertex figure

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vertex figure

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vertex figure (plural vertex figures)

1. (geometry) The figure exposed when a corner of a polyhedron or polytope is sliced off.
   • 1970, Harold Scott Macdonald Coxeter, Twisted Honeycombs, Conference Board of the Mathematical Sciences, Regional Conference Series in Mathematics: Number 4, American Mathematical Society, page 14,

     Thus the vertex figure of a polygon is the line-segment whose end-points are the vertex figures of the two edges at one vertex, the vertex figure of a polyhedron or tessellation {p, q} is the polygon {q} whose edges are the vertex figures of the q faces at one vertex, and the analogous "vertex figure" of the honeycomb of cubes (§1) is the polyhedron whose faces are the vertex figures of the eight cubic cells that come together at one vertex.

   • 1987, A. Gardiner, Mathematical Puzzling, Dover, published 1999, page 137:

     Now try to see which of these vertex figures cannot be continued to give a semiregular tiling of the whole plane.

   • 2004, Michael F. Worboys, Matt Duckham, GIS: A Computing Perspective, 2nd edition, page 188:

     At each vertex of a tessellation, the vertex figure is the polygon formed by joining in order the mid-points of all edges incident with the vertex. A tessellation of a surface for which all the participating polygons and vertex figures are regular and equal is termed a regular tessellation.

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