P(8,3)

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P (8,3)
Rank3
TypeRegular
Elements
Faces6 octagons
Edges24
Vertices16
Vertex figureTriangle
Petrie polygons4 dodecagons
Related polytopes
Petrie dualP (8,3)π 
Abstract & topological properties
Flag count96
Euler characteristic-2
Schläfli type{8,3}
OrientableYes
Genus2
SkeletonMöbius-Kantor graph
Properties
SymmetryTucker's group, order 96

P (8,3) is an orientable abstract regular polyhedron of genus 2. It is a maximally symmetric regular tiling of the Bolza surface, with every automorphism of the Bolza surface being an automorphism of P (8,3).

Gallery[edit | edit source]

Realizations[edit | edit source]

P (8,3) has two pure symmetric realizations, both on the vertices of the tesseract; one with edges that are a subset of those of the tesseract, and one with edges that are a subset of those of the alternative tesseract. These two realizations are kappas of each other.

Related polytopes[edit | edit source]

The realization of P (8,3) on the tesseract appears as the cells of Roli's cube.[1]

It is a double cover of the cube.[2][3]

External links[edit | edit source]

References[edit | edit source]

Bibliography[edit | edit source]

  • Brahana, H. (1927), "Regular Maps and Their Groups", American Journal of Mathematics, 49 (2): 283
  • Coxeter, Donald (1977), "The pappus configuration and the self-inscribed octagon. III", Indagationes Mathematicae (Proceedings), 80 (4), doi:10.1016/1385-7258(77)90024-5
  • Tucker, Thomas (1984), "There is one group of genus 2", Journal of Combinatorial Theory, 36 (3), doi:10.1016/0095-8956(84)90032-7
  • McMullen, Peter (1992), "The regular polyhedra of type {p,3} with 2p  vertices", Geometricae Dedicata, 43 (3), doi:10.1007/BF00151518, ISSN 0046-5755
  • Marušič, Dragan; Pisanski, Tomaž (2000), "The remarkable generalized Petersen graph G (8, 3)", Mathematica Slovaca, 50: 117–121
  • Conder, Marston; Dobcsányi, Peter (2001). "Determination of all regular maps of small genus". Journal of Combinatorial Theory, Series B. 81: 224–242. doi:10.1006/jctb.2000.2008.
  • Monson, Barry (2021), On Roli's Cube, arXiv:2102.08796