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 ...

, a convex uniform honeycomb is a

uniform A uniform is a variety of costume worn by members of an organization while usually participating in that organization's activity. Modern uniforms are most often worn by armed forces and paramilitary organizations such as police, emergency serv ...

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 ...

which fills three-dimensional

Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are ''Euclidean spaces ...

with non-overlapping

convex Convex or convexity may refer to: Science and technology * Convex lens, in optics Mathematics * Convex set, containing the whole line segment that joins points ** Convex polygon, a polygon which encloses a convex set of points ** Convex polytop ...

uniform polyhedral cells. Twenty-eight such honeycombs are known: * the familiar

cubic honeycomb The cubic honeycomb or cubic cellulation is the only proper regular space-filling tessellation (or honeycomb (geometry), honeycomb) in Euclidean 3-space made up of cube, cubic cells. It has 4 cubes around every edge, and 8 cubes around each verte ...

and 7 truncations thereof; * the alternated cubic honeycomb and 4 truncations thereof; * 10 prismatic forms based on the uniform plane tilings (11 if including the cubic honeycomb); * 5 modifications of some of the above by elongation and/or gyration. They can be considered the three-dimensional analogue to the uniform tilings of the plane. The

Voronoi diagram In mathematics, a Voronoi diagram is a partition of a plane into regions close to each of a given set of objects. It can be classified also as a tessellation. In the simplest case, these objects are just finitely many points in the plane (calle ...

of any lattice forms a convex uniform honeycomb in which the cells are

zonohedra In geometry, a zonohedron is a convex polyhedron that is point symmetry, centrally symmetric, every face of which is a polygon that is centrally symmetric (a zonogon). Any zonohedron may equivalently be described as the Minkowski addition, Minkows ...

History

* 1900:

Thorold Gosset John Herbert de Paz Thorold Gosset (16 October 1869 – December 1962) was an English lawyer and an amateur mathematician. In mathematics, he is noted for discovering and classifying the semiregular polytopes in dimensions four and higher, a ...

enumerated the list of semiregular convex polytopes with regular cells (

Platonic solid In geometry, a Platonic solid is a Convex polytope, convex, regular polyhedron in three-dimensional space, three-dimensional Euclidean space. Being a regular polyhedron means that the face (geometry), faces are congruence (geometry), congruent (id ...

s) in his publication ''On the Regular and Semi-Regular Figures in Space of n Dimensions'', including one regular cubic honeycomb, and two semiregular forms with tetrahedra and octahedra. * 1905: Alfredo Andreini enumerated 25 of these tessellations. * 1991: Norman Johnson's manuscript ''Uniform Polytopes'' identified the list of 28. * 1994:

Branko Grünbaum Branko Grünbaum (; 2 October 1929 – 14 September 2018) was a Croatian-born mathematician of Jewish descent4 being missing. Grünbaum states in this paper that Norman Johnson deserves priority for achieving the same enumeration in 1991. He also mentions that I. Alexeyev of Russia had contacted him regarding a putative enumeration of these forms, but that Grünbaum was unable to verify this at the time. * 2006: George Olshevsky, in his manuscript ''Uniform Panoploid Tetracombs'', along with repeating the derived list of 11 convex uniform tilings, and 28 convex uniform honeycombs, expands a further derived list of 143 convex uniform tetracombs (Honeycombs of

uniform 4-polytope In geometry, a uniform 4-polytope (or uniform polychoron) is a 4-dimensional polytope which is vertex-transitive and whose cells are uniform polyhedron, uniform polyhedra, and faces are regular polygons. There are 47 non-Prism (geometry), prism ...

s in 4-space). Only 14 of the convex uniform polyhedra appear in these patterns: * three of the five

s (the

tetrahedron In geometry, a tetrahedron (: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular Face (geometry), faces, six straight Edge (geometry), edges, and four vertex (geometry), vertices. The tet ...

cube A cube or regular hexahedron is a three-dimensional space, three-dimensional solid object in geometry, which is bounded by six congruent square (geometry), square faces, a type of polyhedron. It has twelve congruent edges and eight vertices. It i ...

, and

octahedron In geometry, an octahedron (: octahedra or octahedrons) is any polyhedron with eight faces. One special case is the regular octahedron, a Platonic solid composed of eight equilateral triangles, four of which meet at each vertex. Many types of i ...

), * six of the thirteen

Archimedean solid The Archimedean solids are a set of thirteen convex polyhedra whose faces are regular polygon and are vertex-transitive, although they aren't face-transitive. The solids were named after Archimedes, although he did not claim credit for them. They ...

s (the ones with reflective tetrahedral or octahedral symmetry), and * five of the infinite family of prisms (the 3-, 4-, 6-, 8-, and 12-gonal ones; the 4-gonal prism duplicates the cube). The

icosahedron In geometry, an icosahedron ( or ) is a polyhedron with 20 faces. The name comes . The plural can be either "icosahedra" () or "icosahedrons". There are infinitely many non- similar shapes of icosahedra, some of them being more symmetrical tha ...

, snub cube, and

square antiprism In geometry, the square antiprism is the second in an infinite family of antiprisms formed by an even number, even-numbered sequence of triangle sides closed by two polygon caps. It is also known as an ''anticube''. If all its faces are regular ...

appear in some alternations, but those honeycombs cannot be realised with all edges unit length.

Names

This set can be called the regular and semiregular honeycombs. It has been called the Archimedean honeycombs by analogy with the convex uniform (non-regular) polyhedra, commonly called

s. Recently

Conway Conway may refer to: Places United States * Conway, Arkansas * Conway County, Arkansas * Lake Conway, Arkansas * Conway, Florida * Conway, Iowa * Conway, Kansas * Conway, Louisiana * Conway, Massachusetts * Conway, Michigan * Conway Townshi ...

has suggested naming the set as the Architectonic tessellations and the dual honeycombs as the Catoptric tessellations. The individual honeycombs are listed with names given to them by Norman Johnson. (Some of the terms used below are defined in Uniform 4-polytope#Geometric derivations for 46 nonprismatic Wythoffian uniform 4-polytopes) For cross-referencing, they are given with list indices from Andreini (1-22), Williams(1–2,9-19), Johnson (11–19, 21–25, 31–34, 41–49, 51–52, 61–65), and Grünbaum(1-28). Coxeter uses δ₄ for a

, hδ₄ for an alternated cubic honeycomb, qδ₄ for a quarter cubic honeycomb, with subscripts for other forms based on the ring patterns of the Coxeter diagram.

Compact Euclidean uniform tessellations (by their infinite Coxeter group families)

Coxeter diagram affine rank4 correspondence

The fundamental infinite

Coxeter group In mathematics, a Coxeter group, named after H. S. M. Coxeter, is an abstract group that admits a formal description in terms of reflections (or kaleidoscopic mirrors). Indeed, the finite Coxeter groups are precisely the finite Euclidean ref ...

s for 3-space are: # The

_3

, ,3,4 cubic, (8 unique forms plus one alternation) # The

_3

, ,3^1,1 alternated cubic, (11 forms, 3 new) # The

_3

cyclic group, 3,3,3,3)or ^[4/sup>">.html" ;"title="^{[4">^{[4/sup> (5 forms, one new)

There is a correspondence between all three families. Removing one mirror from $_3$ produces $_3$ , and removing one mirror from $_3$ produces $_3$ . This allows multiple constructions of the same honeycombs. If cells are colored based on unique positions within each Wythoff construction, these different symmetries can be shown.

In addition there are 5 special honeycombs which don't have pure reflectional symmetry and are constructed from reflectional forms with ''elongation'' and ''gyration'' operations.

The total unique honeycombs above are 18.

The prismatic stacks from infinite Coxeter groups for 3-space are:
# The $_2$ × $_1$ , [4,4,2,∞] prismatic group, (2 new forms)
# The $_2$ × $_1$ , [6,3,2,∞] prismatic group, (7 unique forms)
# The $_2$ × $_1$ , [(3,3,3),2,∞] prismatic group, (No new forms)
# The $_1$ × $_1$ × $_1$ , ��,2,∞,2,∞prismatic group, (These all become a ''cubic honeycomb'')

In addition there is one special ''elongated'' form of the triangular prismatic honeycomb.

The total unique prismatic honeycombs above (excluding the cubic counted previously) are 10.

Combining these counts, 18 and 10 gives us the total 28 uniform honeycombs.

The C̃₃, ,3,4group (cubic)

The regular cubic honeycomb, represented by Schläfli symbol , offers seven unique derived uniform honeycombs via truncation operations. (One redundant form, the ''runcinated cubic honeycomb'', is included for completeness though identical to the cubic honeycomb.) The reflectional symmetry is the affine Coxeter group
In mathematics, a Coxeter group, named after H. S. M. Coxeter, is an abstract group that admits a formal description in terms of reflections (or kaleidoscopic mirrors). Indeed, the finite Coxeter groups are precisely the finite Euclidean ref ...
,3,4 There are four index 2 subgroups that generate alternations: ⁺,4,3,4 4,3,4,2⁺) ,3⁺,4 and ,3,4sup>+}, with the first two generated repeated forms, and the last two are nonuniform.

B̃₃, ,3^1,1group

The $_3$ , ,3group offers 11 derived forms via truncation operations, four being unique uniform honeycombs. There are 3 index 2 subgroups that generate alternations: ⁺,4,3^1,1 ,(3^1,1)⁺ and ,3^1,1sup>+}. The first generates repeated honeycomb, and the last two are nonuniform but included for completeness. The honeycombs from this group are called ''alternated cubic'' because the first form can be seen as a ''cubic honeycomb'' with alternate vertices removed, reducing cubic cells to tetrahedra and creating octahedron cells in the gaps. Nodes are indexed left to right as ''0,1,0',3'' with 0' being below and interchangeable with ''0''. The ''alternate cubic'' names given are based on this ordering.

Ã₃, ^[4/sup>">.html" ;"title="^{[4">^[4/sup>group}

There are 5 forms constructed from the

_3

, ^[4/sup>">.html" ;"title="^{[4">^{[4/sup>Coxeter group
In mathematics, a Coxeter group, named after H. S. M. Coxeter, is an abstract group that admits a formal description in terms of reflections (or kaleidoscopic mirrors). Indeed, the finite Coxeter groups are precisely the finite Euclidean ref ...
, of which only the ''quarter cubic honeycomb'' is unique. There is one index 2 subgroup ^[4/sup>]⁺ which generates the snub form, which is not uniform, but included for completeness.

Nonwythoffian forms (gyrated and elongated)

Three more uniform honeycombs are generated by breaking one or another of the above honeycombs where its faces form a continuous plane, then rotating alternate layers by 60 or 90 degrees (''gyration'') and/or inserting a layer of prisms (''elongation'').

The elongated and gyroelongated alternated cubic tilings have the same vertex figure, but are not alike. In the ''elongated'' form, each prism meets a tetrahedron at one triangular end and an octahedron at the other. In the ''gyroelongated'' form, prisms that meet tetrahedra at both ends alternate with prisms that meet octahedra at both ends.

The gyroelongated triangular prismatic tiling has the same vertex figure as one of the plain prismatic tilings; the two may be derived from the gyrated and plain triangular prismatic tilings, respectively, by inserting layers of cubes.

Prismatic stacks

Eleven prismatic tilings are obtained by stacking the eleven uniform plane tilings, shown below, in parallel layers. (One of these honeycombs is the cubic, shown above.) The 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 ...
of each is an irregular bipyramid

In geometry, a bipyramid, dipyramid, or double pyramid is a polyhedron formed by fusing two Pyramid (geometry), pyramids together base (geometry), base-to-base. The polygonal base of each pyramid must therefore be the same, and unless otherwise ...
whose faces are isosceles triangle

In geometry, an isosceles triangle () is a triangle that has two Edge (geometry), sides of equal length and two angles of equal measure. Sometimes it is specified as having ''exactly'' two sides of equal length, and sometimes as having ''at le ...
s.

The C̃₂×Ĩ₁(∞), ,4,2,∞ prismatic group

There are only 3 unique honeycombs from the square tiling, but all 6 tiling truncations are listed below for completeness, and tiling images are shown by colors corresponding to each form.

The G̃₂xĨ₁(∞), ,3,2,∞prismatic group

Enumeration of Wythoff forms

All nonprismatic Wythoff constructions by Coxeter groups are given below, along with their alternations. Uniform solutions are indexed with Branko Grünbaum

Branko Grünbaum (; 2 October 1929 – 14 September 2018) was a Croatian-born mathematician of Jewish descentalternated cubic honeycomb is of special importance since its vertices form a cubic close-packing

In geometry, close-packing of equal spheres is a dense arrangement of congruent spheres in an infinite, regular arrangement (or lattice). Carl Friedrich Gauss proved that the highest average density – that is, the greatest fraction of space occ ...
of spheres. The space-filling truss

A truss is an assembly of ''members'' such as Beam (structure), beams, connected by ''nodes'', that creates a rigid structure.

In engineering, a truss is a structure that "consists of two-force members only, where the members are organized so ...
of packed octahedra and tetrahedra was apparently first discovered by Alexander Graham Bell

Alexander Graham Bell (; born Alexander Bell; March 3, 1847 – August 2, 1922) was a Scottish-born Canadian Americans, Canadian-American inventor, scientist, and engineer who is credited with patenting the first practical telephone. He als ...
and independently re-discovered by Buckminster Fuller

Richard Buckminster Fuller (; July 12, 1895 – July 1, 1983) was an American architect, systems theorist, writer, designer, inventor, philosopher, and futurist. He styled his name as R. Buckminster Fuller in his writings, publishing more t ...
(who called it the octet truss and patented it in the 1940s)

Octet trusses are now among the most common types of truss used in construction.

Frieze forms

If Cell (mathematics), cells are allowed to be uniform tilings, more uniform honeycombs can be defined:

Families:
* $_2$ × $A_1$ : ,4,2 ''Cubic slab honeycombs'' (3 forms)
* $_2$ × $A_1$ : ,3,2 ''Tri-hexagonal slab honeycombs'' (8 forms)
* $_2$ × $A_1$ : 3,3,3),2 ''Triangular slab honeycombs'' (No new forms)
* $_1$ × $A_1$ × $A_1$ : ��,2,2 = ''Cubic column honeycombs'' (1 form)
* $I_2(p)$ × $_1$ : ,2,∞ ''Polygonal column honeycombs'' (analogous to duoprism

In geometry of 4 dimensions or higher, a double prism or duoprism is a polytope resulting from the Cartesian product of two polytopes, each of two dimensions or higher. The Cartesian product of an -polytope and an -polytope is an -polytope, wher ...
s: these look like a single infinite tower of p-gonal prisms, with the remaining space filled with apeirogonal prisms)
* $_1$ × $_1$ × $A_1$ : ��,2,∞,2= ,4,2- = (Same as cubic slab honeycomb family)

The first two forms shown above are semiregular (uniform with only regular facets), and were listed by Thorold Gosset

John Herbert de Paz Thorold Gosset (16 October 1869 – December 1962) was an English lawyer and an amateur mathematician. In mathematics, he is noted for discovering and classifying the semiregular polytopes in dimensions four and higher, a ...
in 1900 respectively as the ''3-ic semi-check'' and ''tetroctahedric semi-check''.

Scaliform honeycomb

A scaliform honeycomb is vertex-transitive

In geometry, a polytope (e.g. a polygon or polyhedron) or a tiling is isogonal or vertex-transitive if all its vertices are equivalent under the symmetries of the figure. This implies that each vertex is surrounded by the same kinds of face i ...
, like a ''uniform honeycomb'', with regular polygon faces while cells and higher elements are only required to be ''orbiforms'', equilateral, with their vertices lying on hyperspheres. For 3D honeycombs, this allows a subset of Johnson solid

In geometry, a Johnson solid, sometimes also known as a Johnson–Zalgaller solid, is a convex polyhedron whose faces are regular polygons. They are sometimes defined to exclude the uniform polyhedrons. There are ninety-two Solid geometry, s ...
s along with the uniform polyhedra. Some scaliforms can be generated by an alternation process, leaving, for example, pyramid

A pyramid () is a structure whose visible surfaces are triangular in broad outline and converge toward the top, making the appearance roughly a pyramid in the geometric sense. The base of a pyramid can be of any polygon shape, such as trian ...
and cupola

In architecture, a cupola () is a relatively small, usually dome-like structure on top of a building often crowning a larger roof or dome. Cupolas often serve as a roof lantern to admit light and air or as a lookout.

The word derives, via Ital ...
gaps.

Hyperbolic forms

There are 9 Coxeter group
In mathematics, a Coxeter group, named after H. S. M. Coxeter, is an abstract group that admits a formal description in terms of reflections (or kaleidoscopic mirrors). Indeed, the finite Coxeter groups are precisely the finite Euclidean ref ...
families of compact uniform honeycombs in hyperbolic 3-space, generated as Wythoff constructions, and represented by ring permutations of the Coxeter-Dynkin diagrams for each family.

From these 9 families, there are a total of 76 unique honeycombs generated:
* ,5,3: - 9 forms
* ,3,4: - 15 forms
* ,3,5: - 9 forms
* ,3^1,1: - 11 forms (7 overlap with ,3,4family, 4 are unique)
* 4,3,3,3): - 9 forms
* 4,3,4,3): - 6 forms
* 5,3,3,3): - 9 forms
* 5,3,4,3): - 9 forms
* 5,3,5,3): - 6 forms

Several non-Wythoffian forms outside the list of 76 are known; it is not known how many there are.

Paracompact hyperbolic forms

There are also 23 paracompact Coxeter groups of rank 4. These families can produce uniform honeycombs with unbounded facets or vertex figure, including ideal vertices at infinity:

References

* John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, (2008) ''The Symmetries of Things'', (Chapter 21, Naming the Archimedean and Catalan polyhedra and tilings, Architectonic and Catoptric tessellations, p 292–298, includes all the nonprismatic forms)
* Branko Grünbaum

Branko Grünbaum (; 2 October 1929 – 14 September 2018) was a Croatian-born mathematician of Jewish descentGeombinatorics

Alexander Soifer is a Russian-born American mathematician and mathematics author.

Soifer obtained his Ph.D. in 1973 and has been a professor of mathematics at the University of Colorado since 1979. He was visiting fellow at Princeton University ...
4, 49 - 56.
* Norman Johnson (1991) ''Uniform Polytopes'', Manuscript
* (Chapter 5: Polyhedra packing and space filling)
*
* Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995,

** (Paper 22) H.S.M. Coxeter, ''Regular and Semi Regular Polytopes I'', ath. Zeit. 46 (1940) 380–407, MR 2,10(1.9 Uniform space-fillings)
* A. Andreini, (1905) ''Sulle reti di poliedri regolari e semiregolari e sulle corrispondenti reti correlative'' (On the regular and semiregular nets of polyhedra and on the corresponding correlative nets), Mem. Società Italiana della Scienze, Ser.3, 14 75–129. PDF

Portable document format (PDF), standardized as ISO 32000, is a file format developed by Adobe Inc., Adobe in 1992 to present documents, including text formatting and images, in a manner independent of application software, computer hardware, ...
br>
* Duncan MacLaren Young Sommerville, D. M. Y. Sommerville, (1930) ''An Introduction to the Geometry of n Dimensions.'' New York, E. P. Dutton, . 196 pp. (Dover Publications edition, 1958) Chapter X: The Regular Polytopes
* Chapter 5. Joining polyhedra

Crystallography of Quasicrystals: Concepts, Methods and Structures
by Walter Steurer, Sofia Deloudi (2009), p. 54-55. 12 packings of 2 or more uniform polyhedra with cubic symmetry

External links

*

VRML models

Vertex transitive space filling honeycombs with non-uniform cells.

Uniform partitions of 3-space, their relatives and embedding
1999
The Uniform Polyhedra
The Encyclopedia of Polyhedra
octet truss animationReview: A. F. Wells, Three-dimensional nets and polyhedra, H. S. M. Coxeter (Source: Bull. Amer. Math. Soc. Volume 84, Number 3 (1978), 466-470.)
*
*

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Honeycombs (geometry)

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History

Names

Compact Euclidean uniform tessellations (by their infinite Coxeter group families)

The C̃3, ,3,4group (cubic)

B̃3, ,31,1group

Ã3, [4/sup>">.html" ;"title="[4">[4/sup>group

Nonwythoffian forms (gyrated and elongated)

Prismatic stacks

The C̃2×Ĩ1(∞), ,4,2,∞ prismatic group

The G̃2xĨ1(∞), ,3,2,∞prismatic group

Enumeration of Wythoff forms

Frieze forms

Scaliform honeycomb

Hyperbolic forms

Paracompact hyperbolic forms

References

External links

The C̃₃, ,3,4group (cubic)

B̃₃, ,3^1,1group

Ã₃, ^[4/sup>">.html" ;"title="^{[4">^[4/sup>group}

The C̃₂×Ĩ₁(∞), ,4,2,∞ prismatic group

The G̃₂xĨ₁(∞), ,3,2,∞prismatic group