Infinite-order apeirogonal tiling: Difference between revisions
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In geometry, the infinite-order apeirogonal tiling is a regular tiling of the hyperbolic plane. It has Schläfli symbol of {∞,∞}, which means it has countably infinitely many apeirogons around all its ideal vertices.
Symmetry
This tiling represents the fundamental domains of *∞∞ symmetry.
Uniform colorings
This tiling can also be alternately colored in the [(∞,∞,∞)] symmetry from 3 generator positions.
| Domains | 0 | 1 | 2 |
|---|---|---|---|
 symmetry: [(∞,∞,∞)] Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination
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 t0{(∞,∞,∞)} Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination
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 t1{(∞,∞,∞)} Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination
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 t2{(∞,∞,∞)} Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination Error creating thumbnail: Unable to save thumbnail to destination
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Related polyhedra and tiling
The union of this tiling and its dual can be seen as orthogonal red and blue lines here, and combined define the lines of a *2∞2∞ fundamental domain.
- a{∞,∞} or Error creating thumbnail: Unable to save thumbnail to destinationError creating thumbnail: Unable to save thumbnail to destinationError creating thumbnail: Unable to save thumbnail to destinationError creating thumbnail: Unable to save thumbnail to destinationError creating thumbnail: Unable to save thumbnail to destination=Error creating thumbnail: Unable to save thumbnail to destinationError creating thumbnail: Unable to save thumbnail to destinationError creating thumbnail: Unable to save thumbnail to destinationError creating thumbnail: Unable to save thumbnail to destination∪Error creating thumbnail: Unable to save thumbnail to destinationError creating thumbnail: Unable to save thumbnail to destinationError creating thumbnail: Unable to save thumbnail to destinationError creating thumbnail: Unable to save thumbnail to destination
See also
- Tilings of regular polygons
- List of uniform planar tilings
- List of regular polytopes
References
- 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.
External links
- Weisstein, Eric W.. "Hyperbolic tiling". http://mathworld.wolfram.com/HyperbolicTiling.html.
- Weisstein, Eric W.. "Poincaré hyperbolic disk". http://mathworld.wolfram.com/PoincareHyperbolicDisk.html.
- Hyperbolic and Spherical Tiling Gallery
- KaleidoTile 3: Educational software to create spherical, planar and hyperbolic tilings
- Hyperbolic Planar Tessellations, Don Hatch
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