Order-infinite-3 triangular honeycomb: Difference between revisions

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

|-

|bgcolor=#efdcc3|[[Schläfli symbol]]s||{3,∞∞,3}

|-

|bgcolor=#efdcc3|[[Coxeter diagram]]s||{{CDD|node_1|3|node|infin|node|3|node}}

|-

|bgcolor=#efdcc3|Cells||[[Infinite-order triangular tiling|{3,∞∞}]] [[File:H2 tiling 23i-4.png|40px]]

|-

|bgcolor=#efdcc3|Faces||[[Triangle|{3}]]

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|bgcolor=#efdcc3|Edge figure||[[Triangle|{3}]]

|-

|bgcolor=#efdcc3|Vertex figure||[[heptagonalOrder-3 apeirogonal tiling|{∞∞,3}]] [[File:H2 tiling 23i-1I-3-dual.pngsvg|50px]]

|-

|bgcolor=#efdcc3|Dual||Self-dual

|-

|bgcolor=#efdcc3|[[Coxeter-Dynkin diagram#Lorentzian groups|Coxeter group]]||[3,∞∞,3]

|-

|bgcolor=#efdcc3|Properties||Regular

|}

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

== Geometry==

It has three [[Infinite-order triangular tiling]] {3,∞∞} around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many triangular tilings existing around each vertex in an [[heptagonalorder-3 apeirogonal tiling]] [[vertex figure]].

{| class=wikitable width=640

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== Related polytopes and honeycombs ==

It a part of a sequence of self-dual regular honeycombs: {''p'',∞,''p''}.

It is a part of a sequence of regular honeycombs with [[Infinite-order triangular tiling]] [[cell (geometry)|cells]]: {3,∞∞,''p''}.

It isais a part of a sequence of regular honeycombs with [[heptagonalorder-3 apeirogonal tiling]] [[vertex figures]]: {''p'',∞∞,3}.

It is a part of a sequence of self-dual regular honeycombs: {''p'',∞,''p''}.

{{Clear}}

{{-}}

=== Order-infinite-4 triangular honeycomb===

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

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

|-

|bgcolor=#efdcc3|[[Schläfli symbol]]s||{3,&infin;∞,4}<BR>{3,∞^1,1}

|-

|bgcolor=#efdcc3|[[Coxeter diagram]]s||{{CDD|node_1|3|node|infin|node|4|node}}<BR>{{CDD|node_1|3|node|infin|node|4|node_h0}} = {{CDD|node_1|3|node|split1-ii|nodes}}

|-

|bgcolor=#efdcc3|Cells||[[Infinite-order triangular tiling|{3,&infin;∞}]] [[File:H2 tiling 23i-4.png|40px]]

|-

|bgcolor=#efdcc3|Faces||[[Triangle|{3}]]

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|bgcolor=#efdcc3|Edge figure||[[square|{4}]]

|-

|bgcolor=#efdcc3|Vertex figure||[[Order-4 hexagonalapeirogonal 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-infinite-3 square honeycomb|{4,&infin;∞,3}]]

|-

|bgcolor=#efdcc3|[[Coxeter-Dynkin diagram#Lorentzian groups|Coxeter group]]||[3,&infin;∞,4]<BR>[3,∞^1,1]

|-

|bgcolor=#efdcc3|Properties||Regular

|}

In the [[geometry]] of [[Hyperbolic space|hyperbolic 3-space]], the '''order-infinite-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 [[Infiniteinfinite-order triangular tiling]]s, {3,&infin;∞}, around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many Infiniteinfinite-order triangular tilings existing around each vertex in an [[order-4 hexagonalapeirogonal tiling]] [[vertex arrangementfigure]].

{| class=wikitable width=480

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|}

It has a second construction as a uniform honeycomb, [[Schläfli symbol]] {3,&infin;∞^1,1}, Coxeter diagram, {{CDD|node_1|3|node|split1-ii|nodes}}, with alternating types or colors of Infiniteinfinite-order triangular tiling cells. In [[Coxeter notation]] the half symmetry is [3,&infin;∞,4,1⁺] = [3,&infin;∞^1,1].

{{-Clear}}

=== Order-infinite-5 triangular honeycomb===

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

|-

|bgcolor=#efdcc3|[[Schläfli symbol]]s||{3,&infin;∞,5}

|-

|bgcolor=#efdcc3|[[Coxeter diagram#Lorentzian groups|Coxeter diagrams]]||{{CDD|node_1|3|node|infin|node|5|node}}

|-

|bgcolor=#efdcc3|Cells||[[Infinite-order triangular tiling|{3,&infin;∞}]] [[File:H2 tiling 23i-4.png|40px]]

|-

|bgcolor=#efdcc3|Faces||[[Triangle|{3}]]

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|bgcolor=#efdcc3|Edge figure||[[pentagon|{5}]]

|-

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

|-

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

|-

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

|-

|bgcolor=#efdcc3|Properties||Regular

|}

In the [[geometry]] of [[Hyperbolic space|hyperbolic 3-space]], the '''order-infinite-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 heptagonalapeirogonal tiling'' [[vertex figure]].

{| class=wikitable width=480

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|}

{{-Clear}}

===Order-infinite-6 triangular honeycomb===

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

|-

|bgcolor=#efdcc3|[[Schläfli symbol]]s||{3,&infin;∞,6}<BR>{3,(&infin;∞,3,&infin;∞)}

|-

|bgcolor=#efdcc3|[[Coxeter diagram#Lorentzian groups|Coxeter diagrams]]||{{CDD|node_1|3|node|infin|node|6|node}}<BR>{{CDD|node_1|3|node|infin|node|6|node_h0}} = {{CDD|node_1|3|node|split1-ii|branch}}

|-

|bgcolor=#efdcc3|Cells||[[Infinite-order triangular tiling|{3,&infin;∞}]] [[File:H2 tiling 23i-4.png|40px]]

|-

|bgcolor=#efdcc3|Faces||[[Triangle|{3}]]

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|bgcolor=#efdcc3|Edge figure||[[Hexagon|{6}]]

|-

|bgcolor=#efdcc3|Vertex figure||[[Order-6 heptagonalapeirogonal 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-infinite-3 hexagonal honeycomb|{6,&infin;∞,3}]]

|-

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

|-

|bgcolor=#efdcc3|Properties||Regular

|}

In the [[geometry]] of [[Hyperbolic space|hyperbolic 3-space]], the '''order-infinite-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 heptagonalapeirogonal tiling'', {&infin;∞,6}, [[vertex figure]].

{| class=wikitable width=480

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|[[File:H3_3i6_UHS_plane_at_infinity.png|240px]]<BR>Ideal surface

|}

{{-Clear}}

===Order-infinite-7 triangular honeycomb===

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

!bgcolor=#efdcc3 colspan=2|Order-infinite-7 triangular honeycomb

|-

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

|-

|bgcolor=#efdcc3|[[Schläfli symbol]]s||{3,∞,7}

|-

|bgcolor=#efdcc3|[[Coxeter diagram#Lorentzian groups|Coxeter diagrams]]||{{CDD|node_1|3|node|infin|node|7|node}}

|-

|bgcolor=#efdcc3|Cells||[[Infinite-order triangular tiling|{3,∞}]] [[File:H2 tiling 23i-4.png|40px]]

|-

|bgcolor=#efdcc3|Faces||[[Triangle|{3}]]

|-

|bgcolor=#efdcc3|Edge figure||[[Heptagon|{7}]]

|-

|bgcolor=#efdcc3|Vertex figure||[[Order-7 apeirogonal tiling|{∞,7}]] [[File:H2 tiling 27i-4.png|40px]]

|-

|bgcolor=#efdcc3|Dual||[[Order-infinite-3 heptagonal honeycomb|{7,∞,3}]]

|-

|bgcolor=#efdcc3|[[Coxeter-Dynkin diagram#Lorentzian groups|Coxeter group]]||[3,∞,7]

|-

|bgcolor=#efdcc3|Properties||Regular

|}

In the [[geometry]] of [[Hyperbolic space|hyperbolic 3-space]], the '''order-infinite-7 triangular honeycomb''' (or '''3,∞,6 honeycomb''') is a regular space-filling [[tessellation]] (or [[honeycomb (geometry)|honeycomb]]) with [[Schläfli symbol]] {3,∞,7}. It has infinitely many [[infinite-order triangular tiling]], {3,∞}, 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-7 apeirogonal tiling'', {∞,7}, [[vertex figure]].

{| class=wikitable

<!--|[[File:Hyperbolic honeycomb 3-i-7 poincare.png|240px]]<BR>[[Poincaré disk model]]-->

|[[File:H3_3i7_UHS_plane_at_infinity.png|240px]]<BR>Ideal surface

|}

{{Clear}}

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

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

|-

|bgcolor=#efdcc3|[[Schläfli symbol]]s||{3,&infin;∞,∞}<BR>{3,(&infin;∞,∞,&infin;∞)}

|-

|bgcolor=#efdcc3|[[Coxeter diagram#Lorentzian groups|Coxeter diagrams]]||{{CDD|node_1|3|node|infin|node|infin|node}}<BR>{{CDD|node_1|3|node|infin|node|infin|node_h0}} = {{CDD|node_1|3|node|split1-ii|branch|labelinfin}}

|-

|bgcolor=#efdcc3|Cells||[[Infinite-order triangular tiling|{3,&infin;∞}]] [[File:H2 tiling 23i-4.png|40px]]

|-

|bgcolor=#efdcc3|Faces||[[Triangle|{3}]]

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|bgcolor=#efdcc3|Edge figure||[[Apeirogon|{∞}]]

|-

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

|-

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

|-

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

|-

|bgcolor=#efdcc3|Properties||Regular

|}

In the [[geometry]] of [[Hyperbolic space|hyperbolic 3-space]], the '''order-infinite-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 heptagonalapeirogonal tiling'', {&infin;∞,∞}, [[vertex figure]].

{| class=wikitable width=480

Line 173 ⟶ 207:

|}

It has a second construction as a uniform honeycomb, [[Schläfli symbol]] {3,(&infin;∞,∞,&infin;∞)}, Coxeter diagram, {{CDD|node_1|3|node|infin|node|infin|node_h0}} = {{CDD|node_1|3|node|split1-ii|branch|labelinfin}}, with alternating types or colors of infinite-order triangular tiling cells. In Coxeter notation the half symmetry is [3,&infin;∞,∞,1⁺] = [3,((&infin;∞,∞,&infin;∞))].

{{Clear}}

{{-}}

=== Order-infinite-3 square honeycomb===

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

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

|-

|bgcolor=#efdcc3|[[Schläfli symbol]]||{4,&infin;∞,3}

|-

|bgcolor=#efdcc3|[[Coxeter diagram]]||{{CDD|node_1|4|node|infin|node|3|node}}

|-

|bgcolor=#efdcc3|Cells||[[Infinite-order square tiling|{4,&infin;∞}]] [[File:H2_tiling_24i-4.png|80px]]

|-

|bgcolor=#efdcc3|Faces||[[Square|{4}]]

|-

|bgcolor=#efdcc3|[[Vertex figure]]||[[heptagonalOrder-3 apeirogonal tiling|{&infin;∞,3}]]

|-

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

|-

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

|-

|bgcolor=#efdcc3|Properties||Regular

|}

In the [[geometry]] of [[Hyperbolic space|hyperbolic 3-space]], the '''order-infinite-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 ''order-infinite-3 square honeycomb'' is {4,&infin;∞,3}, with three orderinfinite-4order heptagonalsquare tilings meeting at each edge. The [[vertex figure]] of this honeycomb is aan order-3 heptagonalapeirogonal tiling, {&infin;∞,3}.

{| class=wikitable

Line 214 ⟶ 249:

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

|-

|bgcolor=#efdcc3|[[Schläfli symbol]]||{5,&infin;∞,3}

|-

|bgcolor=#efdcc3|[[Coxeter diagram]]||{{CDD|node_1|5|node|infin|node|3|node}}

|-

|bgcolor=#efdcc3|Cells||[[Infinite-order pentagonal tiling|{5,&infin;∞}]] [[File:H2_tiling_25i-4.png|80px]]

|-

|bgcolor=#efdcc3|Faces||[[Pentagon|{5}]]

|-

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

|-

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

|-

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

|-

|bgcolor=#efdcc3|Properties||Regular

|}

In the [[geometry]] of [[Hyperbolic space|hyperbolic 3-space]], the '''order-infinite-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}.

{| class=wikitable

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

|-

|bgcolor=#efdcc3|[[Schläfli symbol]]||{6,&infin;∞,3}

|-

|bgcolor=#efdcc3|[[Coxeter diagram]]||{{CDD|node_1|6|node|infin|node|3|node}}

|-

|bgcolor=#efdcc3|Cells||[[infinite-order hexagonal tiling|{6,&infin;∞}]] [[File:H2_tiling_26i-4.png|80px]]

|-

|bgcolor=#efdcc3|Faces||[[Hexagon|{6}]]

|-

|bgcolor=#efdcc3|[[Vertex figure]]||[[heptagonalorder-3 apeirogonal tiling|{&infin;∞,3}]]

|-

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

|-

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

|-

|bgcolor=#efdcc3|Properties||Regular

|}

In the [[geometry]] of [[Hyperbolic space|hyperbolic 3-space]], the '''order-infinite-3 hexagonal honeycomb''' (or '''6,&infin;∞,3 honeycomb''') a regular space-filling [[tessellation]] (or [[honeycomb (geometry)|honeycomb]]). Each infinite cell consists of aan [[order-63 hexagonalapeirogonal 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-infinite-3 hexagonal honeycomb'' is {6,&infin;∞,3}, with three orderinfinite-5order hexagonal tilings meeting at each edge. The [[vertex figure]] of this honeycomb is aan order-3 heptagonalapeirogonal tiling, {&infin;∞,3}.

{| class=wikitable

Line 270 ⟶ 305:

|}

{{-Clear}}

=== Order-infinite-3 heptagonal honeycomb===

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

!bgcolor=#efdcc3 colspan=2|Order-infinite-3 heptagonal honeycomb

|-

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

|-

|bgcolor=#efdcc3|[[Schläfli symbol]]||{7,∞,3}

|-

|bgcolor=#efdcc3|[[Coxeter diagram]]||{{CDD|node_1|7|node|infin|node|3|node}}

|-

|bgcolor=#efdcc3|Cells||[[infinite-order heptagonal tiling|{7,∞}]] [[File:H2_tiling_27i-4.png|80px]]

|-

|bgcolor=#efdcc3|Faces||[[Heptagon|{7}]]

|-

|bgcolor=#efdcc3|[[Vertex figure]]||[[order-3 apeirogonal tiling|{∞,3}]]

|-

|bgcolor=#efdcc3|Dual||[[Order-infinite-7 triangular honeycomb|{3,∞,7}]]

|-

|bgcolor=#efdcc3|[[Coxeter–Dynkin diagram#Ranks 4.E2.80.9310|Coxeter group]]||[7,∞,3]

|-

|bgcolor=#efdcc3|Properties||Regular

|}

In the [[geometry]] of [[Hyperbolic space|hyperbolic 3-space]], the '''order-infinite-3 heptagonal honeycomb''' (or '''7,∞,3 honeycomb''') a regular space-filling [[tessellation]] (or [[honeycomb (geometry)|honeycomb]]). Each infinite cell consists of an [[infinite-order 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 ''order-infinite-3 heptagonal honeycomb'' is {7,∞,3}, with three infinite-order heptagonal tilings meeting at each edge. The [[vertex figure]] of this honeycomb is an order-3 apeirogonal tiling, {∞,3}.

{| class=wikitable

<!--|[[File:Hyperbolic honeycomb 7-i-3 poincare.png|240px]]<BR>[[Poincaré disk model]]-->

|[[File:H3_7i3_UHS_plane_at_infinity.png|240px]]<BR>Ideal surface

|}

{{Clear}}

=== Order-infinite-3 apeirogonal honeycomb===

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

|-

|bgcolor=#efdcc3|[[Schläfli symbol]]||{&infin;∞,&infin;∞,3}

|-

|bgcolor=#efdcc3|[[Coxeter diagram]]||{{CDD|node_1|infin|node|infin|node|3|node}}

|-

|bgcolor=#efdcc3|Cells||[[infinite-order apeirogonal tiling|{&infin;∞,&infin;∞}]] [[File:H2_tiling_2ii-1.png|80px]]

|-

|bgcolor=#efdcc3|Faces||[[Apeirogon]] {&infin;∞}

|-

|bgcolor=#efdcc3|[[Vertex figure]]||[[heptagonalinfinite-order apeirogonal tiling|{&infin;∞,3}]]

|-

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

|-

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

|-

|bgcolor=#efdcc3|Properties||Regular

|}

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

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

{| class=wikitable

Line 311 ⟶ 379:

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

|-

|bgcolor=#efdcc3|[[Schläfli symbol]]||{4,&infin;∞,4}

|-

|bgcolor=#efdcc3|[[Coxeter diagram#Lorentzian groups|Coxeter diagrams]]||{{CDD|node_1|4|node|infin|node|4|node}}<BR>{{CDD|node_1|4|node|infin|node|4|node_h0}} = {{CDD|node_1|4|node|split1-ii|nodes}}

|-

|bgcolor=#efdcc3|Cells||[[infinite-order square tiling|{4,&infin;∞}]] [[File:H2 tiling 24i-4.png|60px]]

|-

|bgcolor=#efdcc3|Faces||[[Square|{4}]]

Line 321 ⟶ 389:

|bgcolor=#efdcc3|Edge figure||[[Square|{4}]]

|-

|bgcolor=#efdcc3|Vertex figure||[[Order-4 heptagonalapeirogonal tiling|{&infin;∞,4}]]<BR>{∞,∞}

|-

|bgcolor=#efdcc3|Dual||self-dual

|-

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

|-

|bgcolor=#efdcc3|Properties||Regular

|}

In the [[geometry]] of [[Hyperbolic space|hyperbolic 3-space]], the '''order-infinite-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 [[orderinfinite-5order square tiling]]s existing around each edge and with an [[order-4 heptagonalapeirogonal tiling]] [[vertex figure]].

{| class=wikitable

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|}

It has a second construction as a uniform honeycomb, [[Schläfli symbol]] {4,∞^1,1}, Coxeter diagram, {{CDD|node_1|4|node|split1-ii|nodes}}, with alternating types or colors of cells. In Coxeter notation the half symmetry is [4,∞,4,1⁺] = [4,∞^1,1].

{{-}}

{{Clear}}

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

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

|-

|bgcolor=#efdcc3|[[Schläfli symbol]]||{5,&infin;∞,5}

|-

|bgcolor=#efdcc3|[[Coxeter diagram#Lorentzian groups|Coxeter diagrams]]||{{CDD|node_1|5|node|infin|node|5|node}}

|-

|bgcolor=#efdcc3|Cells||[[heptagonalinfinite-order pentagonal tiling|{5,&infin;∞}]] [[File:H2 tiling 25i-1.png|60px]]

|-

|bgcolor=#efdcc3|Faces||[[pentagon|{5}]]

Line 356 ⟶ 426:

|bgcolor=#efdcc3|Edge figure||[[pentagon|{5}]]

|-

|bgcolor=#efdcc3|Vertex figure||[[Order-5 heptagonalapeirogonal tiling|{&infin;∞,5}]]

|-

|bgcolor=#efdcc3|Dual||self-dual

|-

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

|-

|bgcolor=#efdcc3|Properties||Regular

|}

In the [[geometry]] of [[Hyperbolic space|hyperbolic 3-space]], the '''order-infinite-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 pentagonalapeirogonal tiling]] [[vertex figure]].

{| class=wikitable

|[[File:Hyperbolic honeycomb 5-i-5 poincare.png|240px]]<BR>[[Poincaré disk model]]

|[[File:H3_555_UHS_plane_at_infinityH3_5i5_UHS_plane_at_infinity.png|240px]]<BR>Ideal surface

|}

{{-Clear}}

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

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

|-

|bgcolor=#efdcc3|[[Schläfli symbol]]s||{6,&infin;∞,6}<BR>{6,(&infin;∞,3,&infin;∞)}

|-

|bgcolor=#efdcc3|[[Coxeter diagram#Lorentzian groups|Coxeter diagrams]]||{{CDD|node_1|6|node|infin|node|6|node}}<BR>{{CDD|node_1|6|node|infin|node|6|node_h0}} = {{CDD|node_1|6|node|split1-ii|branch}}

|-

|bgcolor=#efdcc3|Cells||[[orderinfinite-5order heptagonalhexagonal tiling|{6,&infin;∞}]] [[File:H2 tiling 25i-4.png|60px]]

|-

|bgcolor=#efdcc3|Faces||[[hexagon|{6}]]

Line 390 ⟶ 460:

|bgcolor=#efdcc3|Edge figure||[[hexagon|{6}]]

|-

|bgcolor=#efdcc3|Vertex figure||[[Order-6 heptagonalhexagonal tiling|{&infin;∞,6}]] [[File:H2 tiling 25i-4.png|40px]]<BR>{(5,3,5)} [[File:H2 tiling 35i-1.png|40px]]

|-

|bgcolor=#efdcc3|Dual||self-dual

|-

|bgcolor=#efdcc3|[[Coxeter-Dynkin diagram#Lorentzian groups|Coxeter group]]||[6,&infin;∞,6]<BR>[6,((&infin;∞,3,&infin;∞))]

|-

|bgcolor=#efdcc3|Properties||Regular

|}

In the [[geometry]] of [[Hyperbolic space|hyperbolic 3-space]], the '''order-infinite-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 heptagonalapeirogonal tiling]] [[vertex arrangementfigure]].

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It has a second construction as a uniform honeycomb, [[Schläfli symbol]] {6,(&infin;∞,3,&infin;∞)}, Coxeter diagram, {{CDD|node_1|6|node|split1-ii|branch}}, with alternating types or colors of cells. In Coxeter notation the half symmetry is [6,&infin;∞,6,1⁺] = [6,((&infin;∞,3,&infin;∞))].

{{Clear}}

=== Order-infinite-7 heptagonal honeycomb===

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

!bgcolor=#efdcc3 colspan=2|Order-infinite-7 heptagonal honeycomb

|-

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

|-

|bgcolor=#efdcc3|[[Schläfli symbol]]s||{7,∞,7}

|-

|bgcolor=#efdcc3|[[Coxeter diagram#Lorentzian groups|Coxeter diagrams]]||{{CDD|node_1|7|node|infin|node|7|node}}

|-

|bgcolor=#efdcc3|Cells||[[order-5 heptagonal tiling|{7,∞}]] [[File:H2 tiling 27i-4.png|60px]]

|-

|bgcolor=#efdcc3|Faces||[[heptagon|{7}]]

|-

|bgcolor=#efdcc3|Edge figure||[[heptagon|{7}]]

|-

|bgcolor=#efdcc3|Vertex figure||[[Order-7 apeirogonal tiling|{∞,7}]] [[File:H2 tiling 27i-4.png|40px]]

|-

|bgcolor=#efdcc3|Dual||self-dual

|-

|bgcolor=#efdcc3|[[Coxeter-Dynkin diagram#Lorentzian groups|Coxeter group]]||[7,∞,7]

|-

|bgcolor=#efdcc3|Properties||Regular

|}

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

{| class=wikitable

<!--|[[File:Hyperbolic honeycomb 7-i-7 poincare.png|240px]]<BR>[[Poincaré disk model]]-->

|[[File:H3_7i7_UHS_plane_at_infinity.png|240px]]<BR>Ideal surface

|}

{{-Clear}}

=== Order-infinite-infinite apeirogonal honeycomb ===

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|bgcolor=#efdcc3|Properties||Regular

|}

In the [[geometry]] of [[Hyperbolic space|hyperbolic 3-space]], the '''order-infinite-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 heptagonalapeirogonal tiling]] [[vertex figure]].

{| class=wikitable

|[[File:Hyperbolic honeycomb i-5i-i poincare.png|240px]]<BR>[[Poincaré disk model]]

|[[File:H3_i5i_UHS_plane_at_infinityH3_iii_UHS_plane_at_infinity.png|240px]]<BR>Ideal surface

|}

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{{reflist}}

*[[H. S. M. Coxeter|Coxeter]], ''[[Regular Polytopes (book)|Regular Polytopes]]'', 3rd. ed., Dover Publications, 1973. {{ISBN|0-486-61480-8}}. (Tables I and II: Regular polytopes and honeycombs, pp.&nbsp;294–296)

* ''The Beauty of Geometry: Twelve Essays'' (1999), Dover Publications, {{LCCN|99035678}}, {{ISBN|0-486-40919-8}} (Chapter 10, [http://www.mathunion.org/ICM/ICM1954.3/Main/icm1954.3.0155.0169.ocr.pdf Regular Honeycombs in Hyperbolic Space] {{Webarchive|url=https://web.archive.org/web/20160610043106/http://www.mathunion.org/ICM/ICM1954.3/Main/icm1954.3.0155.0169.ocr.pdf |date=2016-06-10 }}) Table III

* [[Jeffrey Weeks (mathematician)|Jeffrey R. Weeks]] ''The Shape of Space, 2nd edition'' {{ISBN|0-8247-0709-5}} (Chapters 16–17: Geometries on Three-manifolds I, II)

* George Maxwell, ''Sphere Packings and Hyperbolic Reflection Groups'', JOURNAL OF ALGEBRA 79,&infin;878-97 (1982) [http://www.sciencedirect.com/science/article/pii/0021869382903180]

* Hao Chen, Jean-Philippe Labbé, ''Lorentzian Coxeter groups and Boyd-Maxwell ball packings'', (2013)[httphttps://arxiv.org/abs/1310.8608]

* [https://arxiv.org/abs/1511.02851 Visualizing Hyperbolic Honeycombs arXiv:1511.02851] Roice Nelson, Henry Segerman (2015)

==External links==

* [https://www.youtube.com/watch?v=GRo_FQm2KRc Hyperbolic Catacombs Carousel: {3,&infin;∞,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)

* [[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. [https://web.archive.org/web/20161109004910/http://math.uchicago.edu/~dannyc/papers/kleinian_mtf_Feb_2014.pdf]

[[Category:HoneycombsRegular (geometry)3-honeycombs]]