Order-infinite-3 triangular honeycomb: Difference between revisions

Revision as of 01:28, 28 September 2017

Revision as of 01:31, 28 September 2017

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!bgcolor=#efdcc3 colspan=2|Order-infinityinfinite-3 triangular honeycomb

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|bgcolor=#efdcc3|Type||[[List of regular polytopes#Tessellations of hyperbolic 3-space|Regular honeycomb]]

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|bgcolor=#efdcc3|Properties||Regular

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

== Geometry==

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=== Order-infinityinfinite-4 triangular honeycomb===

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

!bgcolor=#efdcc3 colspan=2|Order-infinityinfinite-4 triangular honeycomb

|-

|bgcolor=#efdcc3|Type||[[List of regular polytopes#Tessellations of hyperbolic 3-space|Regular honeycomb]]

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|bgcolor=#efdcc3|Vertex figure||[[Order-4 hexagonal tiling|{&infin;,4}]] [[File:H2 tiling 24i-1.png|50px]]<BR>r{&infin;,&infin;} [[File:H2_tiling_2ii-2.png|50px]]

|-

|bgcolor=#efdcc3|Dual||[[Order-infinityinfinite-3 square honeycomb|{4,&infin;,3}]]

|-

|bgcolor=#efdcc3|[[Coxeter-Dynkin diagram#Lorentzian groups|Coxeter group]]||[3,&infin;,4]

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|bgcolor=#efdcc3|Properties||Regular

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

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

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=== Order-infinityinfinite-5 triangular honeycomb===

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

!bgcolor=#efdcc3 colspan=2|Order-infinityinfinite-5 triangular honeycomb

|-

|bgcolor=#efdcc3|Type||[[List of regular polytopes#Tessellations of hyperbolic 3-space|Regular honeycomb]]

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|bgcolor=#efdcc3|Vertex figure||[[Order-5 hexagonal tiling|{&infin;,5}]] [[File:H2 tiling 25i-1.png|40px]]

|-

|bgcolor=#efdcc3|Dual||[[Order-infinityinfinite-3 pentagonal honeycomb|{5,&infin;,3}]]

|-

|bgcolor=#efdcc3|[[Coxeter-Dynkin diagram#Lorentzian groups|Coxeter group]]||[3,&infin;,5]

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|bgcolor=#efdcc3|Properties||Regular

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In the [[geometry]] of [[Hyperbolic space|hyperbolic 3-space]], the '''Orderorder-infinityinfinite-3 triangular honeycomb''' (or '''3,&infin;,5 honeycomb''') is a regular space-filling [[tessellation]] (or [[honeycomb (geometry)|honeycomb]]) with [[Schläfli symbol]] {3,&infin;,5}. It has five [[infinite-order triangular tiling]], {3,&infin;}, around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many infinite-order triangular tilings existing around each vertex in an ''order-5 heptagonal tiling'' [[vertex figure]].

{| class=wikitable width=480

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===Order-infinityinfinite-6 triangular honeycomb===

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

!bgcolor=#efdcc3 colspan=2|Order-infinityinfinite-6 triangular honeycomb

|-

|bgcolor=#efdcc3|Type||[[List of regular polytopes#Tessellations of hyperbolic 3-space|Regular honeycomb]]

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|bgcolor=#efdcc3|Vertex figure||[[Order-6 heptagonal tiling|{&infin;,6}]] [[File:H2 tiling 26i-4.png|40px]]<BR>{(&infin;,3,&infin;)} [[File:H2 tiling 3ii-2.png|40px]]

|-

|bgcolor=#efdcc3|Dual||[[Order-infinityinfinite-3 hexagonal honeycomb|{6,&infin;,3}]]

|-

|bgcolor=#efdcc3|[[Coxeter-Dynkin diagram#Lorentzian groups|Coxeter group]]||[3,&infin;,6]

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|bgcolor=#efdcc3|Properties||Regular

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In the [[geometry]] of [[Hyperbolic space|hyperbolic 3-space]], the '''Order-infinityinfinite-6 triangular honeycomb''' (or '''3,&infin;,6 honeycomb''') is a regular space-filling [[tessellation]] (or [[honeycomb (geometry)|honeycomb]]) with [[Schläfli symbol]] {3,&infin;,6}. It has infinitely many [[infinite-order triangular tiling]], {3,&infin;}, around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many infinite-order triangular tilings existing around each vertex in an ''order-6 heptagonal tiling'', {&infin;,6}, [[vertex figure]].

{| class=wikitable width=480

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===Order-infinityinfinite-infinite triangular honeycomb===

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

!bgcolor=#efdcc3 colspan=2|Order-infinityinfinite-infinite triangular honeycomb

|-

|bgcolor=#efdcc3|Type||[[List of regular polytopes#Tessellations of hyperbolic 3-space|Regular honeycomb]]

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|bgcolor=#efdcc3|Vertex figure||[[Infinite-order heptagonal tiling|{&infin;,∞}]] [[File:H2 tiling 2ii-4.png|40px]]<BR>{(&infin;,∞,&infin;)} [[File:H2 tiling iii-4.png|40px]]

|-

|bgcolor=#efdcc3|Dual||[[Order-infinityinfinite-3 apeirogonal honeycomb|{∞,&infin;,3}]]

|-

|bgcolor=#efdcc3|[[Coxeter-Dynkin diagram#Lorentzian groups|Coxeter group]]||[∞,&infin;,3]<BR>[3,((&infin;,∞,&infin;))]

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|bgcolor=#efdcc3|Properties||Regular

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

{| class=wikitable width=480

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=== Order-infinityinfinite-3 square honeycomb===

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

!bgcolor=#efdcc3 colspan=2|Order-infinityinfinite-3 square honeycomb

|-

|bgcolor=#efdcc3|Type||[[List of regular polytopes#Tessellations of hyperbolic 3-space|Regular honeycomb]]

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|bgcolor=#efdcc3|[[Vertex figure]]||[[heptagonal tiling|{&infin;,3}]]

|-

|bgcolor=#efdcc3|Dual||[[Order-infinityinfinite-4 triangular honeycomb|{3,&infin;,4}]]

|-

|bgcolor=#efdcc3|[[Coxeter–Dynkin diagram#Ranks 4.E2.80.9310|Coxeter group]]||[4,&infin;,3]

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|bgcolor=#efdcc3|Properties||Regular

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In the [[geometry]] of [[Hyperbolic space|hyperbolic 3-space]], the '''Orderorder-infinityinfinite-3 square honeycomb''' (or '''4,&infin;,3 honeycomb''') a regular space-filling [[tessellation]] (or [[honeycomb (geometry)|honeycomb]]). Each infinite cell consists of a [[heptagonal 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 ''Orderorder-infinityinfinite-3 square honeycomb'' is {4,&infin;,3}, with three order-4 heptagonal tilings meeting at each edge. The [[vertex figure]] of this honeycomb is a heptagonal tiling, {&infin;,3}.

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=== Order-infinityinfinite-3 pentagonal honeycomb===

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

!bgcolor=#efdcc3 colspan=2|Order-infinityinfinite-3 pentagonal honeycomb

|-

|bgcolor=#efdcc3|Type||[[List of regular polytopes#Tessellations of hyperbolic 3-space|Regular honeycomb]]

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|bgcolor=#efdcc3|[[Vertex figure]]||[[heptagonal tiling|{&infin;,3}]]

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|bgcolor=#efdcc3|Dual||[[Order-infinityinfinite-5 triangular honeycomb|{3,&infin;,5}]]

|-

|bgcolor=#efdcc3|[[Coxeter–Dynkin diagram#Ranks 4.E2.80.9310|Coxeter group]]||[5,&infin;,3]

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|bgcolor=#efdcc3|Properties||Regular

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In the [[geometry]] of [[Hyperbolic space|hyperbolic 3-space]], the '''Order-infinityinfinite-3 pentagonal honeycomb''' (or '''5,&infin;,3 honeycomb''') a regular space-filling [[tessellation]] (or [[honeycomb (geometry)|honeycomb]]). Each infinite cell consists of an [[infinite-order pentagonal 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 ''order-6-3 pentagonal honeycomb'' is {5,&infin;,3}, with three ''infinite-order pentagonal tilings'' meeting at each edge. The [[vertex figure]] of this honeycomb is a heptagonal tiling, {&infin;,3}.

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=== Order-infinityinfinite-3 hexagonal honeycomb===

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

!bgcolor=#efdcc3 colspan=2|Order-infinityinfinite-3 hexagonal honeycomb

|-

|bgcolor=#efdcc3|Type||[[List of regular polytopes#Tessellations of hyperbolic 3-space|Regular honeycomb]]

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|bgcolor=#efdcc3|[[Vertex figure]]||[[heptagonal tiling|{&infin;,3}]]

|-

|bgcolor=#efdcc3|Dual||[[Order-infinityinfinite-6 triangular honeycomb|{3,&infin;,6}]]

|-

|bgcolor=#efdcc3|[[Coxeter–Dynkin diagram#Ranks 4.E2.80.9310|Coxeter group]]||[6,&infin;,3]

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|bgcolor=#efdcc3|Properties||Regular

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In the [[geometry]] of [[Hyperbolic space|hyperbolic 3-space]], the '''Orderorder-infinityinfinite-3 hexagonal honeycomb''' (or '''6,&infin;,3 honeycomb''') a regular space-filling [[tessellation]] (or [[honeycomb (geometry)|honeycomb]]). Each infinite cell consists of a [[order-6 hexagonal 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 ''Orderorder-infinityinfinite-3 hexagonal honeycomb'' is {6,&infin;,3}, with three order-5 hexagonal tilings meeting at each edge. The [[vertex figure]] of this honeycomb is a heptagonal tiling, {&infin;,3}.

{| class=wikitable

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|bgcolor=#efdcc3|[[Vertex figure]]||[[heptagonal tiling|{&infin;,3}]]

|-

|bgcolor=#efdcc3|Dual||[[Order-infinityinfinite-infinite triangular honeycomb|{3,&infin;,&infin;}]]

|-

|bgcolor=#efdcc3|[[Coxeter–Dynkin diagram#Ranks 4.E2.80.9310|Coxeter group]]||[&infin;,&infin;,3]

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|bgcolor=#efdcc3|Properties||Regular

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In the [[geometry]] of [[Hyperbolic space|hyperbolic 3-space]], the '''Order-infinityinfinite-4 square honeycomb''' (or '''4,&infin;,4 honeycomb''') a regular space-filling [[tessellation]] (or [[honeycomb (geometry)|honeycomb]]) with [[Schläfli symbol]] {4,&infin;,4}.

All vertices are ultra-ideal (existing beyond the ideal boundary) with four [[order-5 square tiling]]s existing around each edge and with an [[order-4 heptagonal tiling]] [[vertex figure]].

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=== Order-infinite-5 pentagonal honeycomb===

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

!bgcolor=#efdcc3 colspan=2|Order-infinityinfinite-5 pentagonal honeycomb

|-

|bgcolor=#efdcc3|Type||[[List of regular polytopes#Tessellations of hyperbolic 3-space|Regular honeycomb]]

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|bgcolor=#efdcc3|Properties||Regular

|}

In the [[geometry]] of [[Hyperbolic space|hyperbolic 3-space]], the '''order-infinityinfinite-5 pentagonal honeycomb''' (or '''5,&infin;,5 honeycomb''') a regular space-filling [[tessellation]] (or [[honeycomb (geometry)|honeycomb]]) with [[Schläfli symbol]] {5,&infin;,5}.

All vertices are ultra-ideal (existing beyond the ideal boundary) with five infinite-order pentagonal tilings existing around each edge and with an [[order-5 pentagonal tiling]] [[vertex figure]].

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=== Order-infinite-6 hexagonal honeycomb===

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

!bgcolor=#efdcc3 colspan=2|Order-infinityinfinite-6 hexagonal honeycomb

|-

|bgcolor=#efdcc3|Type||[[List of regular polytopes#Tessellations of hyperbolic 3-space|Regular honeycomb]]

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|bgcolor=#efdcc3|Properties||Regular

|}

In the [[geometry]] of [[Hyperbolic space|hyperbolic 3-space]], the '''Order-infinityinfinite-6 hexagonal honeycomb''' (or '''6,&infin;,6 honeycomb''') is a regular space-filling [[tessellation]] (or [[honeycomb (geometry)|honeycomb]]) with [[Schläfli symbol]] {6,&infin;,6}. It has six [[infinite-order hexagonal tiling]]s, {6,&infin;}, around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many hexagonal tilings existing around each vertex in an [[order-6 heptagonal tiling]] [[vertex arrangement]].

{| class=wikitable

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|bgcolor=#efdcc3|Properties||Regular

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

{| class=wikitable