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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==
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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]].
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In the [[geometry]] of [[Hyperbolic space|hyperbolic 3-space]], the '''order-7-3 triangular honeycomb''' (or '''3,7,5 honeycomb''') is a regular space-filling [[tessellation]] (or [[honeycomb (geometry)|honeycomb]]) with [[Schläfli symbol]] {3,7,5}. It has five [[order-7 triangular tiling]], {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-5 heptagonal tiling'' [[vertex figure]].
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In the [[geometry]] of [[Hyperbolic space|hyperbolic 3-space]], the '''order-7-6 triangular honeycomb''' (or '''3,7,6 honeycomb''') is a regular space-filling [[tessellation]] (or [[honeycomb (geometry)|honeycomb]]) with [[Schläfli symbol]] {3,7,6}. It has infinitely many [[order-7 triangular tiling]], {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-6 heptagonal tiling'', {7,6}, [[vertex figure]].
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In the [[geometry]] of [[Hyperbolic space|hyperbolic 3-space]], the '''order-7-infinite triangular honeycomb''' (or '''3,7,∞ honeycomb''') is a regular space-filling [[tessellation]] (or [[honeycomb (geometry)|honeycomb]]) with [[Schläfli symbol]] {3,7,∞}. It has infinitely many [[order-7 triangular tiling]], {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 ''infinite-order heptagonal tiling'', {7,∞}, [[vertex figure]].
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