Tetrahedral-dodecahedral honeycomb

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Tetrahedral-dodecahedral honeycomb
Type Compact uniform honeycomb
Schläfli symbol {(5,3,3,3)} or {(3,3,3,5)}
Coxeter diagram CDel label5.pngCDel branch 10r.pngCDel 3ab.pngCDel branch.png or CDel label5.pngCDel branch 01r.pngCDel 3ab.pngCDel branch.png or CDel node 1.pngCDel split1-53.pngCDel nodes.pngCDel split2.pngCDel node.png
Cells {3,3} Uniform polyhedron-33-t0.png
{5,3} Uniform polyhedron-53-t0.png
r{5,3} Uniform polyhedron-53-t1.png
Faces triangular {3}
pentagon {5}
Vertex figure Uniform t0 5333 honeycomb verf.png
rhombitetratetrahedron
Coxeter group [(5,3,3,3)]
PropertiesVertex-transitive, edge-transitive

In the geometry of hyperbolic 3-space, the tetrahedral-dodecahedral honeycomb is a compact uniform honeycomb, constructed from dodecahedron, tetrahedron, and icosidodecahedron cells, in a rhombitetratetrahedron vertex figure. It has a single-ring Coxeter diagram, CDel node 1.pngCDel split1-53.pngCDel nodes.pngCDel split2.pngCDel node.png, and is named by its two regular cells.

Contents

A geometric honeycomb is a space-filling of polyhedral or higher-dimensional cells, so that there are no gaps. It is an example of the more general mathematical tiling or tessellation in any number of dimensions.

Honeycombs are usually constructed in ordinary Euclidean ("flat") space, like the convex uniform honeycombs. They may also be constructed in non-Euclidean spaces, such as hyperbolic uniform honeycombs. Any finite uniform polytope can be projected to its circumsphere to form a uniform honeycomb in spherical space.

Images

Wide-angle perspective views
H3 5333-1000 center ultrawide.png
Centered on dodecahedron
H3 5333-0010 center ultrawide.png
Centered on icosidodecahedron

See also

Related Research Articles

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In hyperbolic geometry, the order-4 dodecahedral honeycomb is one of four compact regular space-filling tessellations of hyperbolic 3-space. With Schläfli symbol {5,3,4}, it has four dodecahedra around each edge, and 8 dodecahedra around each vertex in an octahedral arrangement. Its vertices are constructed from 3 orthogonal axes. Its dual is the order-5 cubic honeycomb.

<span class="mw-page-title-main">Order-5 dodecahedral honeycomb</span> Regular tiling of hyperbolic 3-space

In hyperbolic geometry, the order-5 dodecahedral honeycomb is one of four compact regular space-filling tessellations in hyperbolic 3-space. With Schläfli symbol {5,3,5}, it has five dodecahedral cells around each edge, and each vertex is surrounded by twenty dodecahedra. Its vertex figure is an icosahedron.

<span class="mw-page-title-main">Icosahedral honeycomb</span> Regular tiling of hyperbolic 3-space

In geometry, the icosahedral honeycomb is one of four compact, regular, space-filling tessellations in hyperbolic 3-space. With Schläfli symbol {3,5,3}, there are three icosahedra around each edge, and 12 icosahedra around each vertex, in a regular dodecahedral vertex figure.

<span class="mw-page-title-main">Order-6 tetrahedral honeycomb</span>

In hyperbolic 3-space, the order-6 tetrahedral honeycomb is a paracompact regular space-filling tessellation. It is paracompact because it has vertex figures composed of an infinite number of faces, and has all vertices as ideal points at infinity. With Schläfli symbol {3,3,6}, the order-6 tetrahedral honeycomb has six ideal tetrahedra around each edge. All vertices are ideal, with infinitely many tetrahedra existing around each vertex in a triangular tiling vertex figure.

<span class="mw-page-title-main">Order-6 dodecahedral honeycomb</span> Regular geometrical object in hyperbolic space

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<span class="mw-page-title-main">Triangular tiling honeycomb</span>

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<span class="mw-page-title-main">Order-4 octahedral honeycomb</span>

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<span class="mw-page-title-main">Dodecahedral-icosahedral honeycomb</span>

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In the geometry of hyperbolic 3-space, the tetrahedron-octahedron honeycomb is a compact uniform honeycomb, constructed from octahedron and tetrahedron cells, in a rhombicuboctahedron vertex figure.

In the geometry of hyperbolic 3-space, the tetrahedral-icosahedral honeycomb is a compact uniform honeycomb, constructed from icosahedron, tetrahedron, and octahedron cells, in an icosidodecahedron vertex figure. It has a single-ring Coxeter diagram , and is named by its two regular cells.

In the geometry of hyperbolic 3-space, the octahedron-dodecahedron honeycomb is a compact uniform honeycomb, constructed from dodecahedron, octahedron, and icosidodecahedron cells, in a rhombicuboctahedron vertex figure. It has a single-ring Coxeter diagram, , and is named by its two regular cells.

In the geometry of hyperbolic 3-space, the cubic-icosahedral honeycomb is a compact uniform honeycomb, constructed from icosahedron, cube, and cuboctahedron cells, in an icosidodecahedron vertex figure. It has a single-ring Coxeter diagram, , and is named by its two regular cells.

In the geometry of hyperbolic 3-space, the octahedron-hexagonal tiling honeycomb is a paracompact uniform honeycomb, constructed from octahedron, hexagonal tiling, and trihexagonal tiling cells, in a rhombicuboctahedron vertex figure. It has a single-ring Coxeter diagram, , and is named by its two regular cells.

In the geometry of hyperbolic 3-space, the hexagonal tiling-triangular tiling honeycomb is a paracompact uniform honeycomb, constructed from triangular tiling, hexagonal tiling, and trihexagonal tiling cells, in a rhombitrihexagonal tiling vertex figure. It has a single-ring Coxeter diagram, , and is named by its two regular cells.

In the geometry of hyperbolic 3-space, the cubic-triangular tiling honeycomb is a paracompact uniform honeycomb, constructed from cube, triangular tiling, and cuboctahedron cells, in a rhombitrihexagonal tiling vertex figure. It has a single-ring Coxeter diagram, , and is named by its two regular cells.

In the geometry of hyperbolic 3-space, the tetrahedral-triangular tiling honeycomb is a paracompact uniform honeycomb, constructed from triangular tiling, tetrahedron, and octahedron cells, in an icosidodecahedron vertex figure. It has a single-ring Coxeter diagram, , and is named by its two regular cells.

In the geometry of hyperbolic 3-space, the cubic-square tiling honeycomb is a paracompact uniform honeycomb, constructed from cube and square tiling cells, in a rhombicuboctahedron vertex figure. It has a single-ring Coxeter diagram, , and is named by its two regular cells.

In the geometry of hyperbolic 3-space, the tetrahedral-square tiling honeycomb is a paracompact uniform honeycomb, constructed from tetrahedron, cuboctahedron and square tiling cells, in a rhombicuboctahedron vertex figure. It has a single-ring Coxeter diagram, , and is named by its two regular cells.

References