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In the [[geometry]] of [[Hyperbolic space|hyperbolic 3-space]], the '''Orderorder-infinite-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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In the [[geometry]] of [[Hyperbolic space|hyperbolic 3-space]], the '''Orderorder-infinite-6 triangular honeycomb''' (or '''3,∞,6 honeycomb''') is a regular space-filling [[tessellation]] (or [[honeycomb (geometry)|honeycomb]]) with [[Schläfli symbol]] {3,∞,6}. 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-6 heptagonal tiling'', {∞,6}, [[vertex figure]].
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In the [[geometry]] of [[Hyperbolic space|hyperbolic 3-space]], the '''Orderorder-infinite-infinite triangular honeycomb''' (or '''3,∞,∞ honeycomb''') is a regular space-filling [[tessellation]] (or [[honeycomb (geometry)|honeycomb]]) with [[Schläfli symbol]] {3,∞,∞}. 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 ''infinite-order heptagonal tiling'', {∞,∞}, [[vertex figure]].
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In the [[geometry]] of [[Hyperbolic space|hyperbolic 3-space]], the '''Orderorder-infinite-3 pentagonal honeycomb''' (or '''5,∞,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,∞,3}, with three ''infinite-order pentagonal tilings'' meeting at each edge. The [[vertex figure]] of this honeycomb is a heptagonal tiling, {∞,3}.
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In the [[geometry]] of [[Hyperbolic space|hyperbolic 3-space]], the '''Orderorder-infinite-3 apeirogonal honeycomb''' (or '''∞,∞,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 {∞,∞,3}, with three ''infinite-order apeirogonal tilings'' meeting at each edge. The [[vertex figure]] of this honeycomb is a heptagonal tiling, {∞,3}.
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In the [[geometry]] of [[Hyperbolic space|hyperbolic 3-space]], the '''Orderorder-infinite-4 square honeycomb''' (or '''4,∞,4 honeycomb''') a regular space-filling [[tessellation]] (or [[honeycomb (geometry)|honeycomb]]) with [[Schläfli symbol]] {4,∞,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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In the [[geometry]] of [[Hyperbolic space|hyperbolic 3-space]], the '''Orderorder-infinite-6 hexagonal honeycomb''' (or '''6,∞,6 honeycomb''') is a regular space-filling [[tessellation]] (or [[honeycomb (geometry)|honeycomb]]) with [[Schläfli symbol]] {6,∞,6}. It has six [[infinite-order hexagonal tiling]]s, {6,∞}, 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]].
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