Order-7-3 triangular honeycomb: Difference between revisions

Revision as of 23:01, 26 January 2018

Revision as of 04:13, 7 February 2018

Line 22:

|bgcolor=#e7dcc3|Properties||Regular

|}

In the [[geometry]] of [[Hyperbolic space|hyperbolic 3-space]], the '''order-7-3 triangular honeycomb''' (or '''3,7,3 honeycomb''') is a regular space-filling [[tessellation]] (or [[honeycomb (geometry)|honeycomb]]) with [[Schläfli symbol]] {3,7,3}.

== Geometry==

Line 41:

{{-}}

=== Order-7-4 triangular honeycomb===

{| class="wikitable" align="right" style="margin-left:10px" width=240

Line 65 ⟶ 66:

|bgcolor=#e7dcc3|Properties||Regular

|}

In the [[geometry]] of [[Hyperbolic space|hyperbolic 3-space]], the '''order-7-4 triangular honeycomb''' (or '''3,7,4 honeycomb''') is a regular space-filling [[tessellation]] (or [[honeycomb (geometry)|honeycomb]]) with [[Schläfli symbol]] {3,7,4}.

It has four [[order-7 triangular tiling]]s, {3,7}, around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many order-7 triangular tilings existing around each vertex in an [[order-4 hexagonal tiling]] [[vertex arrangement]].

Line 77 ⟶ 78:

{{-}}

=== Order-7-5 triangular honeycomb===

{| class="wikitable" align="right" style="margin-left:10px" width=280

Line 176 ⟶ 178:

{{-}}

=== Order-7-3 square honeycomb===

{| class="wikitable" align="right" style="margin-left:10px"

Line 278 ⟶ 281:

|bgcolor=#e7dcc3|Type||[[List of regular polytopes#Tessellations of hyperbolic 3-space|Regular honeycomb]]

|-

|bgcolor=#e7dcc3|[[Schläfli symbol]]||{∞∞,7,3}

|-

|bgcolor=#e7dcc3|[[Coxeter diagram]]||{{CDD|node_1|infin|node|7|node|3|node}}

Line 284 ⟶ 287:

|bgcolor=#e7dcc3|Cells||[[Order-7 apeirogonal tiling|{∞,7}]] [[File:H2_tiling_27i-1.png|80px]]

|-

|bgcolor=#e7dcc3|Faces||[[Apeirogon]] {∞∞}

|-

|bgcolor=#e7dcc3|[[Vertex figure]]||[[heptagonal tiling|{7,3}]]

Line 290 ⟶ 293:

|bgcolor=#e7dcc3|Dual||[[Order-7-infinite triangular honeycomb|{3,7,∞}]]

|-

|bgcolor=#e7dcc3|[[Coxeter–Dynkin diagram#Ranks 4.E2.80.9310|Coxeter group]]||[∞∞,7,3]

|-

|bgcolor=#e7dcc3|Properties||Regular

|}

In the [[geometry]] of [[Hyperbolic space|hyperbolic 3-space]], the '''order-7-3 apeirogonal honeycomb''' (or '''∞∞,7,3 honeycomb''') a regular space-filling [[tessellation]] (or [[honeycomb (geometry)|honeycomb]]). Each infinite cell consists of an [[order-7 apeirogonal tiling]] whose vertices lie on a [[Hypercycle (geometry)|2-hypercycle]], each of which has a limiting circle on the ideal sphere.

The [[Schläfli symbol]] of the apeirogonal tiling honeycomb is {∞∞,7,3}, with three ''order-7 apeirogonal tilings'' meeting at each edge. The [[vertex figure]] of this honeycomb is a heptagonal tiling, {7,3}.

The "ideal surface" projection below is a plane-at-infinity, in the Poincare half-space model of H3. It shows a [[Apollonian gasket]] pattern of circles inside a largest circle.

Line 433 ⟶ 436:

|bgcolor=#e7dcc3|Properties||Regular

|}

In the [[geometry]] of [[Hyperbolic space|hyperbolic 3-space]], the '''order-7-infinite apeirogonal honeycomb''' (or '''∞∞,7,∞∞ honeycomb''') is a regular space-filling [[tessellation]] (or [[honeycomb (geometry)|honeycomb]]) with [[Schläfli symbol]] {∞,7,∞}. It has infinitely many [[order-7 apeirogonal tiling]] {∞,7} around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many order-7 apeirogonal tilings existing around each vertex in an [[infinite-order heptagonal tiling]] [[vertex figure]].

{| class=wikitable

Line 458 ⟶ 461:

* [https://www.youtube.com/watch?v=GRo_FQm2KRc Hyperbolic Catacombs Carousel: {3,7,3} honeycomb] [[YouTube]], Roice Nelson

*[[John Baez]], ''Visual insights'': [http://blogs.ams.org/visualinsight/2014/08/01/733-honeycomb/ {7,3,3} Honeycomb] (2014/08/01) [http://blogs.ams.org/visualinsight/2014/08/14/733-honeycomb-meets-plane-at-infinity/ {7,3,3} Honeycomb Meets Plane at Infinity] (2014/08/14)

[[Category:Honeycombs (geometry)]]

* [[Danny Calegari]], [http://lamington.wordpress.com/2014/03/04/kleinian-a-tool-for-visualizing-kleinian-groups/Kleinian Kleinian, a tool for visualizing Kleinian groups, Geometry and the Imagination] 4 March 2014. [http://math.uchicago.edu/~dannyc/papers/kleinian_mtf_Feb_2014.pdf]