Order-infinite-3 triangular honeycomb: Difference between revisions

Revision as of 03:16, 28 September 2017

Revision as of 03:18, 28 September 2017

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

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|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]]

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|bgcolor=#efdcc3|Dual||[[Order-infinite-3 hexagonal honeycomb|{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 '''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

|[[File:Hyperbolic honeycomb 3-i-6 poincare.png|240px]]<BR>[[Poincaré disk model]]

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

|}

{{-}}

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

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

!bgcolor=#efdcc3 colspan=2|Order-infinite-7 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|[[Schläfli symbol]]s||{3,&infin;,7}

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

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

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

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

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

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

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

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|bgcolor=#efdcc3|Properties||Regular

|}

In the [[geometry]] of [[Hyperbolic space|hyperbolic 3-space]], the '''order-infinite-7 triangular honeycomb''' (or '''3,&infin;,6 honeycomb''') is a regular space-filling [[tessellation]] (or [[honeycomb (geometry)|honeycomb]]) with [[Schläfli symbol]] {3,&infin;,7}. 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-7 apeirogonal tiling'', {&infin;,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

|}

{{-}}