Order-6 pentagonal tiling | |
---|---|
![]() Poincaré disk model of the hyperbolic plane | |
Type | Hyperbolic regular tiling |
Vertex configuration | 56 |
Schläfli symbol | {5,6} |
Wythoff symbol | 6 | 5 2 |
Coxeter diagram | ![]() ![]() ![]() ![]() ![]() |
Symmetry group | [6,5], (*652) |
Dual | Order-5 hexagonal tiling |
Properties | Vertex-transitive, edge-transitive, face-transitive |
In geometry, the order-6 pentagonal tiling is a regular tiling of the hyperbolic plane. It has Schläfli symbol of {5,6}.
Uniform coloring
editThis regular tiling can also be constructed from [(5,5,3)] symmetry alternating two colors of pentagons, represented by t1(5,5,3).
Symmetry
editThis tiling represents a hyperbolic kaleidoscope of 6 mirrors defining a regular hexagon fundamental domain, and 5 mirrors meeting at a point. This symmetry by orbifold notation is called *33333 with 5 order-3 mirror intersections.
Related polyhedra and tiling
editThis tiling is topologically related as a part of sequence of regular tilings with order-6 vertices with Schläfli symbol {n,6}, and Coxeter diagram
, progressing to infinity.
Regular tilings {n,6} | ||||||||
---|---|---|---|---|---|---|---|---|
Spherical | Euclidean | Hyperbolic tilings | ||||||
![]() {2,6} ![]() ![]() ![]() ![]() ![]() | ![]() {3,6} ![]() ![]() ![]() ![]() ![]() | ![]() {4,6} ![]() ![]() ![]() ![]() ![]() | ![]() {5,6} ![]() ![]() ![]() ![]() ![]() | ![]() {6,6} ![]() ![]() ![]() ![]() ![]() | ![]() {7,6} ![]() ![]() ![]() ![]() ![]() | ![]() {8,6} ![]() ![]() ![]() ![]() ![]() | ... | ![]() {∞,6} ![]() ![]() ![]() ![]() ![]() |
Uniform hexagonal/pentagonal tilings | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|
Symmetry: [6,5], (*652) | [6,5]+, (652) | [6,5+], (5*3) | [1+,6,5], (*553) | ||||||||
![]() ![]() ![]() ![]() ![]() | ![]() ![]() ![]() ![]() ![]() | ![]() ![]() ![]() ![]() ![]() | ![]() ![]() ![]() ![]() ![]() | ![]() ![]() ![]() ![]() ![]() | ![]() ![]() ![]() ![]() ![]() | ![]() ![]() ![]() ![]() ![]() | ![]() ![]() ![]() ![]() ![]() | ![]() ![]() ![]() ![]() ![]() | ![]() ![]() ![]() ![]() ![]() | ||
![]() | ![]() | ![]() | ![]() | ![]() | ![]() | ![]() | ![]() | ![]() | |||
{6,5} | t{6,5} | r{6,5} | 2t{6,5}=t{5,6} | 2r{6,5}={5,6} | rr{6,5} | tr{6,5} | sr{6,5} | s{5,6} | h{6,5} | ||
Uniform duals | |||||||||||
![]() ![]() ![]() ![]() ![]() | ![]() ![]() ![]() ![]() ![]() | ![]() ![]() ![]() ![]() ![]() | ![]() ![]() ![]() ![]() ![]() | ![]() ![]() ![]() ![]() ![]() | ![]() ![]() ![]() ![]() ![]() | ![]() ![]() ![]() ![]() ![]() | ![]() ![]() ![]() ![]() ![]() | ![]() ![]() ![]() ![]() ![]() | ![]() ![]() ![]() ![]() ![]() | ||
![]() | ![]() | ![]() | ![]() | ![]() | ![]() | ![]() | |||||
V65 | V5.12.12 | V5.6.5.6 | V6.10.10 | V56 | V4.5.4.6 | V4.10.12 | V3.3.5.3.6 | V3.3.3.5.3.5 | V(3.5)5 |
[(5,5,3)] reflective symmetry uniform tilings | ||||||
---|---|---|---|---|---|---|
![]() | ![]() | ![]() | ![]() | ![]() | ![]() | ![]() |
References
edit- 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.
See also
edit![](http://upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/30px-Commons-logo.svg.png)
Wikimedia Commons has media related to Order-6 pentagonal tiling.
External links
edit🔥 Top keywords: Akademia e Shkencave e RPS te ShqiperiseAlexandria Ocasio-CortezBilderberg GroupCristiano RonaldoDong XiaowanMinecraftOperation GladioPrimal cutRiot FestStrictly Come Dancing (series 7)Main PageSpecial:SearchUEFA Euro 2024Wikipedia:Featured picturesJulian AssangeShifty ShellshockFlorida PanthersList of Stanley Cup championsUEFA European ChampionshipPaul MauriceConnor McDavidMatthew TkachukCleopatraDeaths in 2024House of the Dragon2024 ICC Men's T20 World CupKeith Tkachuk2024 Copa AméricaCrazy TownStella AssangeChelsea ManningYouTubeElon MuskRoberto LuongoAnna Hall (athlete)ICC Men's T20 World CupMckenna GraceStanley CupJJ RedickConn Smythe Trophy.xxxInside Out 2Danny MakkelieZaheer IqbalAfghanistan national cricket teamMoney in the Bank (2024)Gordon Todd SkinnerProject 2025Dr Disrespect