In the
geometry
Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...
of
hyperbolic 3-space
In mathematics, hyperbolic space of dimension ''n'' is the unique simply connected, ''n''-dimensional Riemannian manifold of constant sectional curvature equal to −1.
It is homogeneous, and satisfies the stronger property of being a symme ...
, the order-3-7 heptagonal honeycomb a regular space-filling
tessellation
A tessellation or tiling is the covering of a surface, often a plane, using one or more geometric shapes, called ''tiles'', with no overlaps and no gaps. In mathematics, tessellation can be generalized to higher dimensions and a variety ...
(or
honeycomb
A honeycomb is a mass of Triangular prismatic honeycomb#Hexagonal prismatic honeycomb, hexagonal prismatic cells built from beeswax by honey bees in their beehive, nests to contain their brood (eggs, larvae, and pupae) and stores of honey and pol ...
) with
Schläfli symbol
In geometry, the Schläfli symbol is a notation of the form \ that defines List of regular polytopes and compounds, regular polytopes and tessellations.
The Schläfli symbol is named after the 19th-century Swiss mathematician Ludwig Schläfli, wh ...
.
Geometry
All vertices are ultra-ideal (existing beyond the ideal boundary) with seven heptagonal tilings existing around each edge and with an
order-7 triangular tiling
In geometry, the order-7 triangular tiling is a regular tiling of the hyperbolic plane with a Schläfli symbol of .
Hurwitz surfaces
The symmetry group of the tiling is the (2,3,7) triangle group, and a fundamental domain for this action is th ...
vertex figure
In geometry, a vertex figure, broadly speaking, is the figure exposed when a corner of a general -polytope is sliced off.
Definitions
Take some corner or Vertex (geometry), vertex of a polyhedron. Mark a point somewhere along each connected ed ...
.
Related polytopes and honeycombs
It a part of a sequence of
regular polychora
In mathematics, a regular 4-polytope or regular polychoron is a regular polytope, regular 4-polytope, four-dimensional polytope. They are the four-dimensional analogues of the Regular polyhedron, regular polyhedra in three dimensions and the regul ...
and honeycombs :
Order-3-8 octagonal honeycomb
In the
geometry
Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...
of
hyperbolic 3-space
In mathematics, hyperbolic space of dimension ''n'' is the unique simply connected, ''n''-dimensional Riemannian manifold of constant sectional curvature equal to −1.
It is homogeneous, and satisfies the stronger property of being a symme ...
, the order-3-8 octagonal honeycomb is a regular space-filling
tessellation
A tessellation or tiling is the covering of a surface, often a plane, using one or more geometric shapes, called ''tiles'', with no overlaps and no gaps. In mathematics, tessellation can be generalized to higher dimensions and a variety ...
(or
honeycomb
A honeycomb is a mass of Triangular prismatic honeycomb#Hexagonal prismatic honeycomb, hexagonal prismatic cells built from beeswax by honey bees in their beehive, nests to contain their brood (eggs, larvae, and pupae) and stores of honey and pol ...
) with
Schläfli symbol
In geometry, the Schläfli symbol is a notation of the form \ that defines List of regular polytopes and compounds, regular polytopes and tessellations.
The Schläfli symbol is named after the 19th-century Swiss mathematician Ludwig Schläfli, wh ...
. It has eight
octagonal tilings, , around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many octagonal tilings existing around each vertex in an
order-8 triangular tiling
In geometry, the order-8 triangular tiling is a regular tiling of the hyperbolic plane. It is represented by Schläfli symbol of ', having eight regular triangles around each vertex.
Uniform colorings
The half symmetry +,8,3= 4,3,3)can be sho ...
vertex arrangement
In geometry, a vertex arrangement is a set of points in space described by their relative positions. They can be described by their use in polytopes.
For example, a ''square vertex arrangement'' is understood to mean four points in a plane, equa ...
.
It has a second construction as a uniform honeycomb,
Schläfli symbol
In geometry, the Schläfli symbol is a notation of the form \ that defines List of regular polytopes and compounds, regular polytopes and tessellations.
The Schläfli symbol is named after the 19th-century Swiss mathematician Ludwig Schläfli, wh ...
, Coxeter diagram, , with alternating types or colors of cells. In Coxeter notation the half symmetry is
,3,8,1+=
,((3,4,3))
Order-3-infinite apeirogonal honeycomb
In the
geometry
Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...
of
hyperbolic 3-space
In mathematics, hyperbolic space of dimension ''n'' is the unique simply connected, ''n''-dimensional Riemannian manifold of constant sectional curvature equal to −1.
It is homogeneous, and satisfies the stronger property of being a symme ...
, the order-3-infinite apeirogonal honeycomb is a regular space-filling
tessellation
A tessellation or tiling is the covering of a surface, often a plane, using one or more geometric shapes, called ''tiles'', with no overlaps and no gaps. In mathematics, tessellation can be generalized to higher dimensions and a variety ...
(or
honeycomb
A honeycomb is a mass of Triangular prismatic honeycomb#Hexagonal prismatic honeycomb, hexagonal prismatic cells built from beeswax by honey bees in their beehive, nests to contain their brood (eggs, larvae, and pupae) and stores of honey and pol ...
) with
Schläfli symbol
In geometry, the Schläfli symbol is a notation of the form \ that defines List of regular polytopes and compounds, regular polytopes and tessellations.
The Schläfli symbol is named after the 19th-century Swiss mathematician Ludwig Schläfli, wh ...
. It has infinitely many
order-3 apeirogonal tiling
In geometry, the order-3 apeirogonal tiling is a regular hyperbolic tiling, regular tiling of the Hyperbolic geometry, hyperbolic plane. It is represented by the Schläfli symbol , having three regular Apeirogon#Hyperbolic geometry, apeirogons aro ...
around each edge. All vertices are ultra-ideal (Existing beyond the ideal boundary) with infinitely many apeirogonal tilings existing around each vertex in an
infinite-order triangular tiling
In geometry, the infinite-order triangular tiling is a regular tiling of the hyperbolic plane with a Schläfli symbol of . All vertices are ''ideal'', located at "infinity" and seen on the boundary of the Poincaré hyperbolic disk projection.
...
vertex arrangement
In geometry, a vertex arrangement is a set of points in space described by their relative positions. They can be described by their use in polytopes.
For example, a ''square vertex arrangement'' is understood to mean four points in a plane, equa ...
.
It has a second construction as a uniform honeycomb,
Schläfli symbol
In geometry, the Schläfli symbol is a notation of the form \ that defines List of regular polytopes and compounds, regular polytopes and tessellations.
The Schläfli symbol is named after the 19th-century Swiss mathematician Ludwig Schläfli, wh ...
, Coxeter diagram, , with alternating types or colors of apeirogonal tiling cells.
See also
*
Convex uniform honeycombs in hyperbolic space
In hyperbolic geometry, a uniform honeycomb in hyperbolic space is a uniform tessellation of uniform polyhedral cells. In 3-dimensional hyperbolic space there are nine Coxeter group families of compact convex uniform honeycombs, generated as W ...
*
List of regular polytopes
This article lists the regular polytopes in Euclidean, spherical and hyperbolic spaces.
Overview
This table shows a summary of regular polytope counts by rank.
There are no Euclidean regular star tessellations in any number of dimensions.
...
*
Infinite-order dodecahedral honeycomb
References
*
Coxeter
Harold Scott MacDonald "Donald" Coxeter (9 February 1907 – 31 March 2003) was a British-Canadian geometer and mathematician. He is regarded as one of the greatest geometers of the 20th century.
Coxeter was born in England and educated ...
, ''
Regular Polytopes
''Regular Polytopes'' is a geometry book on regular polytopes written by Harold Scott MacDonald Coxeter. It was originally published by Methuen in 1947 and by Pitman Publishing in 1948, with a second edition published by Macmillan in 1963 and a th ...
'', 3rd. ed., Dover Publications, 1973. . (Tables I and II: Regular polytopes and honeycombs, pp. 294–296)
* ''The Beauty of Geometry: Twelve Essays'' (1999), Dover Publications, , (Chapter 10
Regular Honeycombs in Hyperbolic Space Table III
*
Jeffrey R. Weeks ''The Shape of Space, 2nd edition'' (Chapters 16–17: Geometries on Three-manifolds I, II)
* George Maxwell, ''Sphere Packings and Hyperbolic Reflection Groups'', JOURNAL OF ALGEBRA 79,78-97 (1982
* Hao Chen, Jean-Philippe Labbé, ''Lorentzian Coxeter groups and Boyd-Maxwell ball packings'', (201
Visualizing Hyperbolic Honeycombs arXiv:1511.02851Roice Nelson,
Henry Segerman
Henry Segerman (born 1979 in Manchester, UK) is an Associate Professor of mathematics at Oklahoma State University in Stillwater, Oklahoma who does research in three-dimensional geometry and topology, especially three-manifolds, triangulation ...
(2015)
External links
*
John Baez
John Carlos Baez ( ; born June 12, 1961) is an American mathematical physicist and a professor of mathematics at the University of California, Riverside (UCR) in Riverside, California. He has worked on spin foams in loop quantum gravity, ap ...
, ''Visual insights''
Honeycomb(2014/08/01
{7,3,3} Honeycomb Meets Plane at Infinity(2014/08/14)
*
Danny Calegari
Danny Matthew Cornelius Calegari is a mathematician and, , a professor of mathematics at the University of Chicago. His research interests include geometry, dynamical systems, low-dimensional topology, and geometric group theory.
Education an ...
Kleinian, a tool for visualizing Kleinian groups, Geometry and the Imagination4 March 2014
Heptagonal tilings
Infinite-order tilings
3-honeycombs
Regular 3-honeycombs