bgcolor=#e7dcc3 colspan=2 | Order-5 hexagonal tiling honeycomb | |
---|---|---|
Perspective projection view from center of Poincaré disk model | ||
Type | Hyperbolic regular honeycomb Paracompact uniform honeycomb | |
Schläfli symbol | ||
Coxeter-Dynkin diagrams | ↔ | |
Cells | ||
Faces | ||
Edge figure | ||
Vertex figure | icosahedron | |
Dual | Order-6 dodecahedral honeycomb | |
Coxeter group | \overline{HV}3 | |
Properties | Regular |
The Schläfli symbol of the order-5 hexagonal tiling honeycomb is . Since that of the hexagonal tiling is, this honeycomb has five such hexagonal tilings meeting at each edge. Since the Schläfli symbol of the icosahedron is, the vertex figure of this honeycomb is an icosahedron. Thus, 20 hexagonal tilings meet at each vertex of this honeycomb.[1]
A lower-symmetry construction of index 120, [6,(3,5)<sup>*</sup>], exists with regular dodecahedral fundamental domains, and an icosahedral Coxeter-Dynkin diagram with 6 axial infinite-order (ultraparallel) branches.
The order-5 hexagonal tiling honeycomb is similar to the 2D hyperbolic regular paracompact order-5 apeirogonal tiling,, with five apeirogonal faces meeting around every vertex.
The order-5 hexagonal tiling honeycomb is a regular hyperbolic honeycomb in 3-space, and one of 11 which are paracompact.
There are 15 uniform honeycombs in the [6,3,5] Coxeter group family, including this regular form, and its regular dual, the order-6 dodecahedral honeycomb.
The order-5 hexagonal tiling honeycomb has a related alternation honeycomb, represented by ↔, with icosahedron and triangular tiling cells.
It is a part of sequence of regular hyperbolic honeycombs of the form, with hexagonal tiling facets:
It is also part of a sequence of regular polychora and honeycombs with icosahedral vertex figures:
bgcolor=#e7dcc3 colspan=2 | Rectified order-5 hexagonal tiling honeycomb | |
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Type | Paracompact uniform honeycomb | |
Schläfli symbols | r or t1 | |
Coxeter diagrams | ↔ | |
Cells | r or h2 | |
Faces | ||
Vertex figure | pentagonal prism | |
Coxeter groups | {\overline{HV}3} {\overline{HP}3} | |
Properties | Vertex-transitive, edge-transitive |
It is similar to the 2D hyperbolic infinite-order square tiling, r with pentagon and apeirogonal faces. All vertices are on the ideal surface.
bgcolor=#e7dcc3 colspan=2 | Truncated order-5 hexagonal tiling honeycomb | |
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Type | Paracompact uniform honeycomb | |
Schläfli symbol | t or t0,1 | |
Coxeter diagram | ||
Cells | ||
Faces | ||
Vertex figure | pentagonal pyramid | |
Coxeter groups | {\overline{HV}}3 | |
Properties | Vertex-transitive |
bgcolor=#e7dcc3 colspan=2 | Bitruncated order-5 hexagonal tiling honeycomb | |
---|---|---|
Type | Paracompact uniform honeycomb | |
Schläfli symbol | 2t or t1,2 | |
Coxeter diagram | ↔ | |
Cells | ||
Faces | ||
Vertex figure | digonal disphenoid | |
Coxeter groups | {\overline{HV}}3 {\overline{HP}}3 | |
Properties | Vertex-transitive |
bgcolor=#e7dcc3 colspan=2 | Cantellated order-5 hexagonal tiling honeycomb | |
---|---|---|
Type | Paracompact uniform honeycomb | |
Schläfli symbol | rr or t0,2 | |
Coxeter diagram | ||
Cells | ||
Faces | ||
Vertex figure | wedge | |
Coxeter groups | {\overline{HV}}3 | |
Properties | Vertex-transitive |
bgcolor=#e7dcc3 colspan=2 | Cantitruncated order-5 hexagonal tiling honeycomb | |
---|---|---|
Type | Paracompact uniform honeycomb | |
Schläfli symbol | tr or t0,1,2 | |
Coxeter diagram | ||
Cells | ||
Faces | ||
Vertex figure | mirrored sphenoid | |
Coxeter groups | {\overline{HV}}3 | |
Properties | Vertex-transitive |
The cantitruncated order-5 hexagonal tiling honeycomb, t0,1,2, has truncated icosahedron, truncated trihexagonal tiling, and pentagonal prism facets, with a mirrored sphenoid vertex figure.
bgcolor=#e7dcc3 colspan=2 | Runcinated order-5 hexagonal tiling honeycomb | |
---|---|---|
Type | Paracompact uniform honeycomb | |
Schläfli symbol | t0,3 | |
Coxeter diagram | ||
Cells | ||
Faces | ||
Vertex figure | irregular triangular antiprism | |
Coxeter groups | {\overline{HV}}3 | |
Properties | Vertex-transitive |
bgcolor=#e7dcc3 colspan=2 | Runcitruncated order-5 hexagonal tiling honeycomb | |
---|---|---|
Type | Paracompact uniform honeycomb | |
Schläfli symbol | t0,1,3 | |
Coxeter diagram | ||
Cells | ||
Faces | ||
Vertex figure | isosceles-trapezoidal pyramid | |
Coxeter groups | {\overline{HV}}3 | |
Properties | Vertex-transitive |
The runcicantellated order-5 hexagonal tiling honeycomb is the same as the runcitruncated order-6 dodecahedral honeycomb.
bgcolor=#e7dcc3 colspan=2 | Omnitruncated order-5 hexagonal tiling honeycomb | |
---|---|---|
Type | Paracompact uniform honeycomb | |
Schläfli symbol | t0,1,2,3 | |
Coxeter diagram | ||
Cells | ||
Faces | ||
Vertex figure | irregular tetrahedron | |
Coxeter groups | {\overline{HV}}3 | |
Properties | Vertex-transitive |
bgcolor=#e7dcc3 colspan=2 | Alternated order-5 hexagonal tiling honeycomb | |
---|---|---|
Type | Paracompact uniform honeycomb Semiregular honeycomb | |
Schläfli symbol | h | |
Coxeter diagram | ↔ | |
Cells | ||
Faces | ||
Vertex figure | truncated icosahedron | |
Coxeter groups | {\overline{HP}}3 | |
Properties | Vertex-transitive, edge-transitive, quasiregular |
bgcolor=#e7dcc3 colspan=2 | Cantic order-5 hexagonal tiling honeycomb | |
---|---|---|
Type | Paracompact uniform honeycomb | |
Schläfli symbol | h2 | |
Coxeter diagram | ↔ | |
Cells | ||
Faces | ||
Vertex figure | triangular prism | |
Coxeter groups | {\overline{HP}}3 | |
Properties | Vertex-transitive |
bgcolor=#e7dcc3 colspan=2 | Runcic order-5 hexagonal tiling honeycomb | |
---|---|---|
Type | Paracompact uniform honeycomb | |
Schläfli symbol | h3 | |
Coxeter diagram | ↔ | |
Cells | ||
Faces | ||
Vertex figure | triangular cupola | |
Coxeter groups | {\overline{HP}}3 | |
Properties | Vertex-transitive |
bgcolor=#e7dcc3 colspan=2 | Runcicantic order-5 hexagonal tiling honeycomb | |
---|---|---|
Type | Paracompact uniform honeycomb | |
Schläfli symbol | h2,3 | |
Coxeter diagram | ↔ | |
Cells | ||
Faces | ||
Vertex figure | rectangular pyramid | |
Coxeter groups | {\overline{HP}}3 | |
Properties | Vertex-transitive |