A hyperbolic polyhedron is a polyhedron in a hyperbolic
geometry .
In three-dimensional hyperbolic space , a convex hyperbolic polyhedron is the intersection of a finite number of hyperbolic half-spaces .
Its faces lie in hyperbolic
planes (Virto 2010, pp. 3-4).
Regular skew polyhedra have also been studied
in hyperbolic space (Garner 1967).
Trott (2004, §8.3.10) also uses the name "hyperbolic" for decorative deformations of Euclidean Platonic solids . He explicitly
describes his cover-art dodecahedron as a heuristic
construction rather than a genuine hyperbolic
dodecahedron .
See also Hyperbolic Cube ,
Hyperbolic Dodecahedron ,
Hyperbolic Icosahedron ,
Hyperbolic Octahedron ,
Hyperbolic
Space ,
Hyperbolic Tetrahedron ,
Regular
Skew Polyhedron
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References Garner, C. W. L. "Regular Skew Polyhedra in Hyperbolic Three-Space." Canad. J. Math. 19 , 1179-1186, 1967.
https://doi.org/10.4153/CJM-1967-106-9 . Hodgson,
C. D. and Rivin, I. "A Characterization of Compact Convex Polyhedra in
Hyperbolic 3-Space." Invent. Math. 111 , 77-111, 1993. https://doi.org/10.1007/BF01231281 . Kellerhals,
R. "Shape and Size Through Hyperbolic Eyes." Math. Intell. 17 ,
21-30, 1995. https://doi.org/10.1007/BF03024894 . Kellerhals,
R. "Nichteuklidische Geometrie und Volumina hyperbolischer Polyeder." Math.
Semesterber. 43 , 155-168, 1996. https://doi.org/10.1007/s005910050019 . Ratcliffe,
J. G. Foundations
of Hyperbolic Manifolds. https://doi.org/10.1007/978-1-4757-4013-4 . Rivin,
I. "A Characterization of Ideal Polyhedra in Hyperbolic 3-Space." Ann.
Math. 143 , 51-70, 1996. https://doi.org/10.2307/2118652 . Rivin,
I. and Hodgson, C. D. "Corrigendum." Invent. Math. 117 ,
359, 1994. https://doi.org/10.1007/BF01232247 . Thurston,
W. P. Three-Dimensional
Geometry and Topology, Vol. 1. Trott, M. "The Cover Image: Hyperbolic Platonic
Bodies." §8.3.10 in The
Mathematica GuideBook for Graphics. https://www.mathematicaguidebooks.org/ . Virto,
J. "On the Characterization of Polyhedra in Hyperbolic 3-Space." 23 Jun
2010. https://arxiv.org/abs/1006.4445 . Referenced
on Wolfram|Alpha Hyperbolic Polyhedron
Cite this as:
Weisstein, Eric W. "Hyperbolic Polyhedron."
From MathWorld https://mathworld.wolfram.com/HyperbolicPolyhedron.html
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