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Hyperbolic 2-space, H 2, which was the first instance studied, is also called the hyperbolic plane. It is also sometimes referred to as Lobachevsky space or Bolyai–Lobachevsky space after the names of the author who first published on the topic of hyperbolic geometry .
John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, The Symmetries of Things 2008, ISBN 978-1-56881-220-5 (Chapter 19, The Hyperbolic Archimedean Tessellations) "Chapter 10: Regular honeycombs in hyperbolic space".
In dimension 3, the fractional linear action of PGL(2, C) on the Riemann sphere is identified with the action on the conformal boundary of hyperbolic 3-space induced by the isomorphism O + (1, 3) ≅ PGL(2, C). This allows one to study isometries of hyperbolic 3-space by considering spectral properties of representative complex matrices.
The runcicantic square tiling honeycomb, h 2,3 {4,4,3}, ↔ , is a paracompact uniform honeycomb in hyperbolic 3-space. It has truncated square tiling, truncated cuboctahedron, and truncated octahedron facets in a mirrored sphenoid vertex figure.
The group SO + (1,n) is the full group of orientation-preserving isometries of the n-dimensional hyperbolic space. In more concrete terms, SO + (1,n) can be split into n(n-1)/2 rotations (formed with a regular Euclidean rotation matrix in the lower-right block) and n hyperbolic translations, which take the form
The distance between two points in the half-plane model can be computed in terms of Euclidean distances in an isosceles trapezoid formed by the points and their reflection across the x-axis: a "side length" s, a "diagonal" d, and two "heights" h 1 and h 2. It is the logarithm dist(p 1, p 2) = log ((s + d) 2 /h 1 h 2) Distance between two points ...
It is part of a sequence of regular polychora, which include the 5-cell {3,3,3}, tesseract {4,3,3}, and 120-cell {5,3,3} of Euclidean 4-space, along with other hyperbolic honeycombs containing tetrahedral vertex figures.
John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, The Symmetries of Things 2008, ISBN 978-1-56881-220-5 (Chapter 19, The Hyperbolic Archimedean Tessellations) "Chapter 10: Regular honeycombs in hyperbolic space". The Beauty of Geometry: Twelve Essays. Dover Publications. 1999. ISBN 0-486-40919-8. LCCN 99035678.