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Regular tiling of hyperbolic 3-space
In geometry, the icosahedral honeycomb is one of four compact, regular, space-filling tessellations (or honeycombs) in hyperbolic 3-space. With Schläfli
Icosahedral_honeycomb
Honeycomb made from unique polyhedrons
In the geometry of hyperbolic 3-space, the dodecahedral-icosahedral honeycomb is a uniform honeycomb, constructed from dodecahedron, icosahedron, and
Dodecahedral-icosahedral honeycomb
Dodecahedral-icosahedral_honeycomb
Regular tiling of hyperbolic 3-space
cubes around each vertex. It is dual with the order-4 dodecahedral honeycomb. A geometric honeycomb is a space-filling of polyhedral or higher-dimensional
Order-5_cubic_honeycomb
120-cell honeycomb. It can be constructed by replacing the great dodecahedral cells of the great 120-cell honeycomb with their icosahedral convex hulls
Order-5 icosahedral 120-cell honeycomb
Order-5_icosahedral_120-cell_honeycomb
Tiling of hyperbolic 3-space by uniform polyhedra
120 with a dodecahedral fundamental domain. There are also 23 paracompact Coxeter groups of rank 4 that produce paracompact uniform honeycombs with infinite
Uniform honeycombs in hyperbolic space
Uniform_honeycombs_in_hyperbolic_space
Regular polytope whose 2D form is a pentagon
It can be named by its Schläfli symbol as {5, 3n − 2} (dodecahedral) or {3n − 2, 5} (icosahedral). The family starts as 1-polytopes and ends with n = 5
Pentagonal_polytope
regular dual, the order-6 dodecahedral honeycomb. The order-5 hexagonal tiling honeycomb has a related alternation honeycomb, represented by ↔ , with icosahedron
Order-5 hexagonal tiling honeycomb
Order-5_hexagonal_tiling_honeycomb
tiling honeycomb Order-2 hexagonal tiling honeycomb Order-4 dodecahedral honeycomb Order-5 dodecahedral honeycomb Order-5 cubic honeycomb Icosahedral honeycomb
List_of_mathematical_shapes
Polyhedron with 2 faces
A dihedron (pl. dihedra) is a type of polyhedron, made of two polygon faces which share the same set of n edges. In three-dimensional Euclidean space,
Dihedron
dodecahedral honeycomb Order-5 cubic honeycomb Order-5 dodecahedral honeycomb Icosahedral honeycomb Apeirogonal prism Apeirohedron Bicupola Bifrusta Triangular
List of polygons, polyhedra and polytopes
List_of_polygons,_polyhedra_and_polytopes
Tessellation of convex uniform polyhedron cells
In geometry, uniform honeycombs in hyperbolic space are tessellations of convex uniform polyhedron cells. In 3-dimensional hyperbolic space there are 23
Paracompact uniform honeycombs
Paracompact_uniform_honeycombs
with hemi-icosahedral cells. 120-cell – regular 4-polytope with dodecahedral cells Order-5 dodecahedral honeycomb - regular hyperbolic honeycomb with same
57-cell
Polyhedron with two kinds of faces
cases include the Euclidean cubic honeycomb {4,3,4} with cubic cells, and compact hyperbolic {5,3,4} with dodecahedral cells, and paracompact {6,3,4} with
Quasiregular_polyhedron
Ordered chemical structure with no repeating pattern
a local icosahedral order. The icosahedral order is in equilibrium in the liquid state for the stable quasicrystals, whereas the icosahedral order prevails
Quasicrystal
Operation in Euclidean geometry
regular polyhedra: t{q,p}. Two are honeycombs on the 3-sphere, one a honeycomb in Euclidean 3-space, and two are honeycombs in hyperbolic 3-space. Rectification
Bitruncation
Four-dimensional analog of the dodecahedron
Schläfli symbol {5,3,3}. It is also called a C120, dodecaplex (short for "dodecahedral complex"), hyperdodecahedron, polydodecahedron, hecatonicosachoron, dodecacontachoron
120-cell
3-space honeycomb or 4-polytope = 0 {\displaystyle =0} : Euclidean 3-space honeycomb < 0 {\displaystyle <0} : Hyperbolic 3-space honeycomb These constraints
List_of_regular_polytopes
Class of 4-dimensional polytopes
completes his Ph.D. dissertation The Theory of Uniform Polytopes and Honeycombs under advisor Coxeter, completes the basic theory of uniform polytopes
Uniform_4-polytope
3D shape made of polyhedra sharing a common center
and icosahedral dual compounds are the first stellations of the cuboctahedron and icosidodecahedron, respectively. The small stellated dodecahedral (or
Polytope_compound
Polyhedron with 12 faces
are constructed as stellations of the convex form. All of these have icosahedral symmetry, order 120. Some dodecahedra have the same combinatorial structure
Dodecahedron
Notation for polytopes and tessellations
made of dodecahedral cells {5,3}, and has 3 cells around each edge. There is one regular tessellation of Euclidean 3-space: the cubic honeycomb, with a
Schläfli_symbol
Polyhedron with regular congruent polygons as faces
Platonic solids: Tetrahedral Octahedral (or cubic) Icosahedral (or dodecahedral) Any shapes with icosahedral or octahedral symmetry will also contain tetrahedral
Regular_polyhedron
Four-dimensional analogues of the regular polyhedra in three dimensions
600-cell), pI=polyicosahedron {3,5,5/2} (an icosahedral 120-cell), and pD=polydodecahedron {5,3,3} (a dodecahedral 120-cell), with prefix modifiers: g, a,
Regular_4-polytope
Four-dimensional shape
A lower symmetry form of the as 16-cell. Cubic bipyramid Dodecahedral bipyramid Icosahedral bipyramid https://www.bendwavy.org/klitzing/incmats/tedpy
Tetrahedral_bipyramid
Four-dimensional geometric object with flat sides
cells) 27 unique convex dual uniform honeycombs, including: Rhombic dodecahedral honeycomb Disphenoid tetrahedral honeycomb Others: Weaire–Phelan structure
4-polytope
Four-dimensional analog of the octahedron
16-cell is also related to the cubic honeycomb, order-4 dodecahedral honeycomb, and order-4 hexagonal tiling honeycomb which all have octahedral vertex figures
16-cell
Type of uniform 4-polytope in four-dimensional geography
page 38 und 39, 1965 N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D. Dissertation, University of Toronto, 1966 Four-dimensional Archimedean
Prismatic_uniform_4-polytope
+1/60[IxI].23). It is called the icosahedral pyramidal group and is the 3D icosahedral group, [5,3]. A regular dodecahedral pyramid can have this symmetry
Point groups in four dimensions
Point_groups_in_four_dimensions
Polytope with highest degree of symmetry
sometimes named after regular polytopes, for example, the tetrahedral and icosahedral symmetries. The idea of a polytope is sometimes generalised to include
Regular_polytope
Uniform 4-polytope
truncation of the regular 120-cell 4-polytope. It is made of 120 truncated dodecahedral and 600 tetrahedral cells. It has 3120 faces: 2400 being triangles and
Truncated_120-cells
Convex polyhedron with 90 rhombic faces
achieved by putting a rhombicuboctahedron in each cell of the rhombic dodecahedral honeycomb, and it cannot be surpassed, since otherwise the optimal packing
Rhombic_enneacontahedron
Solid made by joining an n- and 2n-gon with triangles and squares
Name Tetrahedral cupola Cubic cupola Octahedral cupola Dodecahedral cupola Hexagonal tiling cupola Schläfli symbol {3,3} || rr{3,3} {4,3} || rr{4,3} {3
Cupola_(geometry)
geometry, runcination is an operation that cuts a regular polytope (or honeycomb) simultaneously along the faces, edges, and vertices, creating new facets
Runcination
Mathematical space
as a hyperbolic manifold. It is a quotient space of the order-5 dodecahedral honeycomb, a regular tessellation of hyperbolic 3-space by dodecahedra with
3-manifold
Flat-sided three-dimensional shape
biological creatures, as in The Braarudosphaera bigelowii has a regular dodecahedral structure. Ernst Haeckel described a number of species of radiolarians
Polyhedron
Value determined from a polyhedron
edges with dodecahedral dihedral angles. ( 0 , 0 , 30 s ) {\displaystyle (0,0,30s)} for the icosahedron. It has 30 edges with icosahedral dihedral angles
Dehn_invariant
travel, tourism, insurance
DODECAHEDRAL ICOSAHEDRAL-HONEYCOMB
DODECAHEDRAL ICOSAHEDRAL-HONEYCOMB
Biblical
beholder; honeycomb; garment
Biblical
viol; honeycomb
Boy/Male
Biblical Hebrew
A wood, honeycomb, watching closely.
Girl/Female
Biblical
Honeycomb, anything that distills or drops.
Biblical
honeycomb; anything that distills or drops
Boy/Male
Indian, Sanskrit
Little; Honeycomb; Cake
Boy/Male
Biblical
Viol, honeycomb.
Biblical
a wood; honeycomb; watching closely
Boy/Male
Biblical
Beholder, honeycomb, garment.
Girl/Female
Arabic, Muslim
Honeycomb
DODECAHEDRAL ICOSAHEDRAL-HONEYCOMB
DODECAHEDRAL ICOSAHEDRAL-HONEYCOMB
DODECAHEDRAL ICOSAHEDRAL-HONEYCOMB
DODECAHEDRAL ICOSAHEDRAL-HONEYCOMB
DODECAHEDRAL ICOSAHEDRAL-HONEYCOMB
DODECAHEDRAL ICOSAHEDRAL-HONEYCOMB
DODECAHEDRAL ICOSAHEDRAL-HONEYCOMB
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