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- In [[geometry]], many uniform tilings on sphere, euclidean plane, and hyperbolic plane can be made by [[Wythoff construction]] within a fundamental triangle ...any of the numbers defining a hyperbolic or Euclidean tiling makes another hyperbolic tiling.) ...54 KB (7,476 words) - 11:59, 22 January 2025
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- ...hyperbolic plane]], or some other two-dimensional space by [[apeirogon]]s. Tilings of this type include: *[[Order-3 apeirogonal tiling]], hyperbolic tiling with 3 apeirogons around a vertex ...1 KB (156 words) - 07:38, 7 November 2024
- ...dcc3|Type||[[List of regular polytopes#Tessellations of hyperbolic 3-space|Hyperbolic regular honeycomb]]<BR>[[Paracompact uniform honeycomb#.3B6.2C6.2C3.5D fami ...ellation]]s (or [[honeycomb (geometry)|honeycombs]]) in [[Hyperbolic space|hyperbolic 3-space]]. It is called ''paracompact'' because it has infinite [[Cell (geo ...18 KB (2,379 words) - 03:03, 10 January 2025
- ...dcc3|Type||[[List of regular polytopes#Tessellations of hyperbolic 3-space|Hyperbolic regular honeycomb]]<BR>[[Paracompact uniform honeycomb#.5B6.2C3.2C6.5D fami ...hexagonal tiling]] whose vertices lie on a [[horosphere]]: a flat plane in hyperbolic space that approaches a single [[ideal point]] at infinity. ...24 KB (3,253 words) - 10:02, 4 September 2024
- ...dcc3|Type||[[List of regular polytopes#Tessellations of hyperbolic 3-space|Hyperbolic regular honeycomb]]<BR>[[Paracompact uniform honeycomb#.5B4.2C4.2C4.5D fami ...4,4}, it has four [[square tiling]]s around each edge, and infinite square tilings around each vertex in a [[square tiling]] [[vertex figure]].<ref>Coxeter '' ...20 KB (2,773 words) - 16:01, 8 December 2024
- In three-dimensional hyperbolic geometry, the '''alternated hexagonal tiling honeycomb''', h{6,3,3}, {{CDD| [[File:Hyperbolic subgroup tree 336-direct.png|120px|thumb|left|[[Coxeter_diagram#Subgroup_re ...10 KB (1,293 words) - 19:45, 8 January 2025
- ...dcc3|Type||[[List of regular polytopes#Tessellations of hyperbolic 3-space|Hyperbolic regular honeycomb]]<BR>[[Paracompact uniform honeycomb]] ...[[hexagonal tiling]] whose vertices lie on a [[horosphere]], a surface in hyperbolic space that approaches a single [[ideal point]] at infinity. ...21 KB (2,783 words) - 10:49, 9 January 2025
- ...dcc3|Type||[[List of regular polytopes#Tessellations of hyperbolic 3-space|Hyperbolic regular honeycomb]]<BR>[[Paracompact uniform honeycomb#.5B6.2C3.2C4.5D fami ...hexagonal tiling]] whose vertices lie on a [[horosphere]]: a flat plane in hyperbolic space that approaches a single [[ideal point]] at infinity. ...27 KB (3,638 words) - 20:49, 16 January 2025
- ...dcc3|Type||[[List of regular polytopes#Tessellations of hyperbolic 3-space|Hyperbolic regular honeycomb]]<BR>[[Paracompact uniform honeycomb#.5B6.2C3.2C4.5D fami ...hexagonal tiling]] whose vertices lie on a [[horosphere]], a flat plane in hyperbolic space that approaches a single [[ideal point]] at infinity. ...23 KB (2,891 words) - 21:26, 9 January 2025
- ...dcc3|Type||[[List of regular polytopes#Tessellations of hyperbolic 3-space|Hyperbolic regular honeycomb]]<BR>[[Paracompact uniform honeycomb#.5B4.2C4.2C3.5D fami ..., it has three [[square tiling]]s, {4,4}, around each edge, and six square tilings around each vertex, in a [[cube|cubic]] {4,3} [[vertex figure]].<ref>Coxete ...26 KB (3,482 words) - 20:50, 16 January 2025
- {{Short description|Tiling of the hyperbolic plane}} ...edge lies on a [[horocycle]] (shown as circles interior to the disk) or a hyperbolic line (arcs perpendicular to the disk boundary). The horocycles and lines ar ...26 KB (3,830 words) - 00:35, 11 January 2025
- ...yperbolic small dodecahedral honeycomb|dodecahedral tessellation]] in '''[[Hyperbolic 3-manifold|H<sup>3</sup>]]'''. Note the recursive structure: each pentagon ...architecture, biology, and computer science, as well as in the study of [[hyperbolic manifold]]s. [[Substitution tiling]]s are a well-studied type of subdivisio ...21 KB (3,302 words) - 16:05, 5 June 2024
- In [[geometry]], many uniform tilings on sphere, euclidean plane, and hyperbolic plane can be made by [[Wythoff construction]] within a fundamental triangle ...any of the numbers defining a hyperbolic or Euclidean tiling makes another hyperbolic tiling.) ...54 KB (7,476 words) - 11:59, 22 January 2025
- ...dcc3|Type||[[List of regular polytopes#Tessellations of hyperbolic 3-space|Hyperbolic regular honeycomb]]<BR>[[Paracompact uniform honeycomb#.3B4.2C4.2C4.5D fami ...dral honeycomb''' is a regular paracompact honeycomb in [[Hyperbolic space|hyperbolic 3-space]]. It is ''paracompact'' because it has infinite [[vertex figure]]s ...14 KB (1,931 words) - 20:51, 16 January 2025
- ...dcc3|Type||[[List of regular polytopes#Tessellations of hyperbolic 3-space|Hyperbolic regular honeycomb]]<BR>[[Paracompact uniform honeycomb#.6B6.2C3.2C4.5D fami ...ssellation]] (or [[honeycomb (geometry)|honeycomb]]) in [[Hyperbolic space|hyperbolic 3-space]]. It is ''paracompact'' because it has [[vertex figure]]s composed ...21 KB (2,757 words) - 20:50, 16 January 2025
- {{Short description|Model of hyperbolic geometry}} ...re disc hyperbolic parallel lines.svg|300px|right|thumb|Poincaré disk with hyperbolic parallel lines]] ...25 KB (3,976 words) - 09:29, 16 December 2024
- ...Springer Science & Business Media|date=2010|isbn=9780387754772}}</ref> The tilings obtained from an aperiodic set of tiles are often called [[aperiodic tiling | H<sup>2</sup> || [[Hyperbolic space|hyperbolic plane]] || plane, where the [[parallel postulate]] does not hold ...37 KB (4,805 words) - 17:46, 8 March 2024
- {{short description|Tiling of hyperbolic 3-space by uniform polyhedra}} {{unsolved|mathematics|Find the complete set of hyperbolic uniform honeycombs.}} ...88 KB (11,289 words) - 10:45, 9 January 2025
- ...rünbaum |first2=Geoffrey |last2=Shepard |author-link2=G.C. Shephard |title=Tilings by Regular Polygons |date=November 1977 |url=http://www.maa.org/sites/defau ...g|190px|thumb|The fundamental domain of the [[Klein quartic]] is a regular hyperbolic 14-sided [[tetradecagon]], with an area of <math>8\pi</math>.]] ...18 KB (2,403 words) - 04:15, 5 February 2025
- ...d their classification is related to the theory of [[Riemann surface]]s, [[hyperbolic geometry]], and [[Galois theory]]. Regular maps are classified according to Regular maps exist as torohedral polyhedra as finite portions of Euclidean tilings, wrapped onto the surface of a [[duocylinder]] as a [[flat torus]]. These a ...16 KB (2,291 words) - 21:17, 6 December 2024
- ...ntly strong small cancellation conditions are [[word-hyperbolic group|word hyperbolic]] and have [[Word problem for groups|word problem]] solvable by '''Dehn's a ...drawing the [[Cayley graph]] of such a group in the [[Hyperbolic manifold|hyperbolic plane]] and performing curvature estimates via the [[Gauss–Bonnet theorem]] ...32 KB (4,718 words) - 11:34, 5 June 2024