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EspañolCONVEX UNIFORM HONEYCOMB
In geometry, a 'convex uniform honeycomb' is a uniform space-filling tessellation in three-dimensional Euclidean space with non-overlapping convex uniform polyhedral cells.
Twenty-eight such honeycombs exist:
★ the familiar cubic honeycomb 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.
★ '1900': Thorold Gosset enumerated the list of semiregular convex polytopes with regular cells (Platonic solids) 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 complete list of 28.
★ '1994': Branko Grünbaum, in his paper ''Uniform tilings of 3-space'', also independently enumerated all 28, after discovering errors in Andreini's publication. He found the 1905 paper, which listed 25, had 1 wrong, and 4 being missing. Grünbaum also states that I. Alexeyev of Russia also independently enumerated these forms around the same 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 polychorons in 4-space).
Only 14 of the convex uniform polyhedra appear in these patterns:
★ three of the five Platonic solids,
★ six of the thirteen Archimedean solids, and
★ five of the infinite family of prisms.
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 Archimedean solids. Recently Conway 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 polychoron#Geometric derivations.)
For cross-referencing, they are given with list indices from [A]ndreini (1-22), [W]illiams(1-2,9-19), [J]ohnson (11-19, 21-25, 31-34, 41-49, 51-52, 61-65), and [G]runbaum(1-28).
The fundamental infinite Coxeter groups for 3-space are:
# The R4, [4,3,4], cubic,
(8 unique forms plus one alternation)
# The S4, h[4,3,4], alternated cubic,
(11 forms, 3 new)
# The P4 cyclic group,
(5 forms, one new)
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 R3xW2, [4,4]x[∞] prismatic group,
(2 new forms)
# The V3xW2, [6,3]x[∞] prismatic group,
(7 unique forms)
# The P3xW2, [Δ]x[∞] prismatic group,
(No new forms)
# The W2xW2xW2, [∞]x[∞]x[∞] 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 regular cubic honeycomb, represented by Schläfli symbol {4,3,4}, 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.)
{|class="wikitable"
!rowspan=2|Reference
Indices
!rowspan=2|Coxeter-Dynkin
and Schläfli
symbols
!rowspan=2|Honeycomb name
! colspan=4|Cell counts/vertex
and positions in cubic honeycomb
!
!
!
|-
!(0)
!(1)
!(2)
!(3)
!Solids
(Partial)
!Frames
(Perspective)
!vertex figure
|-
|J11,15
A1
W1
G22
|align=center|
t0{4,3,4}
|align=center|cubic
|align=center|
|align=center|
|align=center|
|align=center|8
(4.4.4)
|
|
|
octahedron
|-
|J12,32
A15
W14
G7
|align=center|
t1{4,3,4}
|align=center|rectified cubic
|align=center|2
(3.3.3.3)
|align=center|
|align=center|
|align=center|4
(3.4.3.4)
|
|
|
cuboid
|-
|J13
A14
W15
G8
|align=center|
t0,1{4,3,4}
|align=center|truncated cubic
|align=center|1
(3.3.3.3)
|align=center|
|align=center|
|align=center|4
(3.8.8)
|
|
|
square pyramid
|-
|J14
A17
W12
G9
|align=center|
t0,2{4,3,4}
|align=center|cantellated cubic
|align=center|1
(3.4.3.4)
|align=center|2
(4.4.4)
|align=center|
|align=center|2
(3.4.4.4)
|
|
|
wedge
|-
|J11,15
|align=center|
t0,3{4,3,4}
|align=center|'runcinated cubic'
(same as regular cubic)
|align=center|1
(4.4.4)
|align=center|3
(4.4.4)
|align=center|3
(4.4.4)
|align=center|1
(4.4.4)
|
|
|
octahedron
|-
|J16
A3
W2
G28
|align=center|
t1,2{4,3,4}
|align=center|bitruncated cubic
|align=center|2
(4.6.6)
|align=center|
|align=center|
|align=center|2
(4.6.6)
|
|
|
isosceles tetrahedron
|-
|J17
A18
W13
G25
|align=center|
t0,1,2{4,3,4}
|align=center|cantitruncated cubic
|align=center|1
(4.6.6)
|align=center|1
(4.4.4)
|align=center|
|align=center|2
(4.6.8)
|
|
|
irregular tetrahedron
|-
|J18
A19
W19
G20
|align=center|
t0,1,3{4,3,4}
|align=center|runcitruncated cubic
|align=center|1
(3.4.4.4)
|align=center|1
(4.4.4)
|align=center|2
(4.4.8)
|align=center|1
(3.8.8)
|
|
|
oblique trapezoidal pyramid
|-
|J19
A22
W18
G27
|align=center|
t0,1,2,3{4,3,4}
|align=center|omnitruncated cubic
|align=center|1
(4.6.8)
|align=center|1
(4.4.8)
|align=center|1
(4.4.8)
|align=center|1
(4.6.8)
|
|
|
irregular tetrahedron
|-
|-
|J21,31,51
A2
W9
G1
|align=center|
h0{4,3,4}
|align=center|alternated cubic
|align=center|6
3.3.3.3
|align=center|
|align=center|
|align=center|8
3.3.3
|
|
|
cuboctahedron
|}
The S4 group offers 11 derived forms via truncation operations, four being unique uniform honeycombs.
The honeycombs from this group are called ''alternated cubic'' because the first form can be seen as a ''cubic honeycomb'' with alternate vertices cubic cells to tetrahedra, and creating octahedron cells in the deleted 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.
{|class="wikitable"
!rowspan=2|Referenced
indices
!rowspan=2|Coxeter-Dynkin
diagram
!rowspan=2|Honeycomb name
!colspan=4|Cells by location
(and count around each vertex)
!rowspan=2|Solids
(Partial)
!rowspan=2|Frames
(Perspective)
!rowspan=2|vertex figure
|-
!(0)
!(1)
!(0')
!(3)
|-
|J21,31,51
A2
W9
G1
|align=center|
|align=center|alternated cubic
|align=center|
|align=center|
|align=center| (6)
3.3.3.3
|align=center|(8)
3.3.3
|
|
|
cuboctahedron
|-
|J22,34
A21
W17
G10
|align=center|
|align=center|truncated alternated cubic
|align=center|
|align=center|
(1)
3.4.3.4
|align=center|
(2)
4.6.6
|align=center|
(2)
3.6.6
||
|
|
|-
|J12,32
A15
W14
G7
|align=center|
|align=center|rectified cubic
(rectified alternate cubic)
|align=center| (2)
(3.4.3.4)
|align=center|
|align=center| (2)
(3.4.3.4)
|align=center|
(2)
(3.3.3.3)
|
|
|
cuboid
|-
|J12,32
A15
W14
G7
|align=center|
|align=center|rectified cubic
(cantellated alternate cubic)
|align=center| (1)
(3.3.3.3)
|align=center|
|align=center| (1)
(3.3.3.3)
|align=center|
(4)
(3.4.3.4)
|
|
|
cuboid
|-
|J16
A3
W2
G28
|align=center|
|align=center|bitruncated cubic
(cantitruncated alternate cubic)
|align=center| (1)
(4.6.6)
|align=center|
|align=center| (1)
(4.6.6)
|align=center|
(2)
(4.6.6)
|
|
|
isosceles tetrahedron
|-
|J13
A14
W15
G8
|align=center|
|align=center|truncated cubic
(bicantellated alternate cubic)
|align=center| (2)
(3.8.8)
|align=center|
|align=center| (2)
(3.8.8)
|align=center| (1)
(3.3.3.3)
|
|
|
square pyramid
|-
|J11,15
A1
W1
G22
|align=center|
|align=center|cubic
(trirectified alternate cubic)
|align=center| (4)
(4.4.4)
|align=center|
|align=center| (4)
(4.4.4)
|align=center|
|
|
|
octahedron
|-
|J23
A16
W11
G5
|align=center|
|align=center|runcinated alternated cubic
|align=center| (1)
cube
|align=center|
|align=center|
(3)
3.4.4.4
|align=center|
(1)
3.3.3
|
|
|
|-
|J14
A17
W12
G9
|align=center|
|align=center|cantellated cubic
(runcicantellated alternate cubic)
|align=center| (1)
(3.4.4.4)
|align=center|
(2)
(4.4.4)
|align=center| (1)
(3.4.4.4)
|align=center| (1)
(3.4.3.4)
|
|
|
wedge
|-
|J24
A20
W16
G21
|align=center|
|align=center|cantitruncated alternated cubic
(or runcitruncated alternate cubic)
|align=center|
|align=center| (1)
3.8.8
|align=center|
(2)
4.6.8
|align=center|
(1)
3.6.6
|
|
|
|-
|J17
A18
W13
G25
|align=center|
|align=center|cantitruncated cubic
(omnitruncated alternated cubic)
|align=center| (1)
(4.6.8)
|align=center|
(1)
(4.4.4)
|align=center| (1)
(4.6.8)
|align=center|(1)
(4.6.6)
|
|
|
irregular tetrahedron
|}
There are 5 forms constructed from the P4 group, only the ''quarter cubic honeycomb'' is unique.
{|class="wikitable"
!rowspan=2|Referenced
indices
!rowspan=2|Coxeter-Dynkin
diagram
!rowspan=2|Honeycomb name
!colspan=4|Cells by location
(and count around each vertex)
!rowspan=2|Solids
(Partial)
!rowspan=2|Frames
(Perspective)
!rowspan=2|vertex figure
|-
!(0)
!(1)
!(2)
!(3)
|-
|J21,31,51
A2
W9
G1
|align=center|
|align=center|alternated cubic
|align=center|
|align=center|
(4)
3.3.3
|align=center| (6)
3.3.3.3
|align=center|
(4)
3.3.3
|
|
|
cuboctahedron
|-
|J12,32
A15
W14
G7
|align=center|
|align=center|rectified cubic
|align=center| (2)
(3.4.3.4)
|align=center| (1)
(3.3.3.3)
|align=center| (2)
(3.4.3.4)
|align=center| (1)
(3.3.3.3)
|
|
|
cuboid
|-
|J25,33
A13
W10
G6
|align=center|
|align=center|quarter cubic
|align=center| (1)
3.3.3
|align=center| (1)
3.3.3
|align=center| (3)
3.6.6
|align=center| (3)
3.6.6
|
|
|
|-
|J22,34
A21
W17
G10
|align=center|
|align=center|truncated alternated cubic
|align=center| (1)
3.6.6
|align=center| (1)
3.4.3.4
|align=center| (1)
3.6.6
|align=center| (2)
4.6.6
||
|
|
|-
|J16
A3
W2
G28
|align=center|
|align=center|bitruncated cubic
|align=center| (1)
(4.6.6)
|align=center| (1)
(4.6.6)
|align=center| (1)
(4.6.6)
|align=center| (1)
(4.6.6)
|
|
|
isosceles tetrahedron
|}
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.
{|class="wikitable"
!Referenced
indices
!symbol
!Honeycomb name
!cell types (# at each vertex)
!Solids
(Partial)
!Frames
(Perspective)
!vertex figure
|-
|J52
A2'
G2
|h{4,3,4}:g
|align=center|gyrated alternated cubic
|align=center|tetrahedron (8)
octahedron (6)
|
|
|
|-
|J61
A?
G3
|h{4,3,4}:ge
|align=center|gyroelongated alternated cubic
|align=center|triangular prism (6)
tetrahedron (4)
octahedron (3)
|
|
| -
|-
|J62
A?
G4
|h{4,3,4}:e
|align=center|elongated alternated cubic
|align=center|triangular prism (6)
tetrahedron (4)
octahedron (3)
|
|
|
|-
|J63
A?
G12
|{3,6}:g x {∞}
|align=center|gyrated triangular prismatic
|align=center|triangular prism (12)
|
|
|
|-
|J64
A?
G15
|{3,6}:ge x {∞}
|align=center|gyroelongated triangular prismatic
|align=center|triangular prism (6)
cube (4)
|
|
|
|}
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 of each is an irregular bipyramid whose faces are isosceles triangles.
There's 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.
{|class="wikitable"
!Indices
!Coxeter-Dynkin
and Schläfli
symbols
!Honeycomb name
!Plane
tiling
!Solids
(Partial)
!Tiling
|-
|J11,15
A1
G22
|align=center|
{4,4} x {∞}
|align=center|Cubic
(Square prismatic)
|(4.4.4.4)
|
|
|-
|J45
A6
G24
|align=center|
t0,1{4,4} x {∞}
|align=center|Truncated/Bitruncated square prismatic
|(4.8.8)
|
|
|-
|J11,15
A1
G22
|align=center|
t1{4,4} x {∞}
|align=center|Cubic
(Rectified square prismatic)
|(4.4.4.4)
|
|
|-
|J11,15
A1
G22
|align=center|
t0,2{4,4} x {∞}
|align=center|Cubic
(Cantellated square prismatic)
|(4.4.4.4)
|
|
|-
|J45
A6
G24
|align=center|
t0,1,2{4,4} x {∞}
|align=center|Truncated square prismatic
(Omnitruncated square prismatic)
|(4.8.8)
|
|
|-
|J44
A11
G14
|align=center|
s{4,4} x {∞}
|align=center|Snub square prismatic
|(3.3.4.3.4)
|
|
|}
{|class="wikitable"
!Indices
!Coxeter-Dynkin
and Schläfli
symbols
!Honeycomb name
!Plane
tiling
!Solids
(Partial)
!Tiling
|-
|J42
A5
G26
|align=center|
t0{6,3} x {∞}
|align=center|Hexagonal prismatic
|(63)
|
|
|-
|J46
A7
G19
|align=center|
t0,1{6,3} x {∞}
|align=center|Truncated hexagonal prismatic
|(3.12.12)
|
|
|-
|J43
A8
G18
|align=center|
t1{6,3} x {∞}
|align=center|Trihexagonal prismatic
|(3.6.3.6)
|
|
|-
|J42
A5
G26
|align=center|
t1,2{6,3} x {∞}
|align=center|''Truncated triangular prismatic''
Hexagonal prismatic
|(6.6.6)
||
|
|-
|J41
A4
G11
|align=center|
t2{6,3} x {∞}
|align=center|Triangular prismatic
|(36)
|
|
|-
|J47
A9
G16
|align=center|
t0,2{6,3} x {∞}
|align=center|Rhombi-trihexagonal prismatic
|(3.4.6.4)
|
|
|-
|J49
A10
G23
|align=center|
t0,1,2{6,3} x {∞}
|align=center|Omnitruncated trihexagonal prismatic
|(4.6.12)
|
|
|-
|J48
A12
G17
|align=center|
s{6,3} x {∞}
|align=center|Snub trihexagonal prismatic
|(3.3.3.3.6)
|
|
|-
|J65
A11'
G13
|{3,6}:e x {∞}
|align=center|elongated triangular prismatic
|align=center|3.3.3.4.4
|
|
|}
All 28 of these tessellations are found in crystal arrangements.
The alternated cubic honeycomb is of special importance since its vertices form a cubic close-packing of spheres. The space-filling truss of packed octahedra and tetrahedra was apparently first discovered by Alexander Graham Bell and independently re-discovered by Buckminster Fuller (who called it the octet truss and patented it in the 1940s).
[1]
[2]
[3]
[4]. Octet trusses are now among the most common types of truss used in construction.
★
★ Uniform Honeycombs in 3-Space VRML models
★ Elementary Honeycombs
★ Uniform partitions of 3-space, their relatives and embedding PDF, 1999
★ The Uniform Polyhedra
★ Virtual Reality Polyhedra The Encyclopedia of Polyhedra
★ George Olshevsky, ''Uniform Panoploid Tetracombs'', Manuscript (2006) ''(Complete list of 11 convex uniform tilings, 28 convex uniform honeycombs, and 143 convex uniform tetracombs)''
★ Branko Grünbaum, Uniform tilings of 3-space. Geombinatorics 4(1994), 49 - 56.
★ Norman Johnson ''Uniform Polytopes'', Manuscript (1991)
★ The Geometrical Foundation of Natural Structure: A Source Book of Design, , Robert, Williams, Dover Publications, Inc, 1979, ISBN 0-486-23729-X
★ Order in Space: A design source book, , Keith, Critchlow, Viking Press, 1970, ISBN 0-500-34033-1
★ 'Kaleidoscopes: Selected Writings of H.S.M. Coxeter', editied by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, ISBN 978-0-471-01003-6 [5]
★
★ (Paper 22) H.S.M. Coxeter, ''Regular and Semi Regular Polytopes I'', [Math. Zeit. 46 (1940) 380-407, MR 2,10] (1.9 Uniform space-fillings)
★ A. Andreini, ''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 (1905) 75–129.
Twenty-eight such honeycombs exist:
★ the familiar cubic honeycomb 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.
History
★ '1900': Thorold Gosset enumerated the list of semiregular convex polytopes with regular cells (Platonic solids) 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 complete list of 28.
★ '1994': Branko Grünbaum, in his paper ''Uniform tilings of 3-space'', also independently enumerated all 28, after discovering errors in Andreini's publication. He found the 1905 paper, which listed 25, had 1 wrong, and 4 being missing. Grünbaum also states that I. Alexeyev of Russia also independently enumerated these forms around the same 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 polychorons in 4-space).
Only 14 of the convex uniform polyhedra appear in these patterns:
★ three of the five Platonic solids,
★ six of the thirteen Archimedean solids, and
★ five of the infinite family of prisms.
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 Archimedean solids. Recently Conway 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 polychoron#Geometric derivations.)
For cross-referencing, they are given with list indices from [A]ndreini (1-22), [W]illiams(1-2,9-19), [J]ohnson (11-19, 21-25, 31-34, 41-49, 51-52, 61-65), and [G]runbaum(1-28).
Tessellations listed by infinite Coxeter group families
Fundamental domains in a cubic element of three groups.
The fundamental infinite Coxeter groups for 3-space are:
# The R4, [4,3,4], cubic,
CDW_dot.png
CDW_4.png
CDW_dot.png
CDW_3.png
CDW_dot.png
CDW_4.png
CDW_dot.png
# The S4, h[4,3,4], alternated cubic,
CD_dot.png
CD_3b.png
CD_downbranch-00.png
CD_3b.png
CD_4.png
CD_dot.png
# The P4 cyclic group,
CD downbranch-00.png
CD downbranch-33.png
CD downbranch-00.png
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 R3xW2, [4,4]x[∞] prismatic group,
CDW_dot.png
CDW_4.png
CDW_dot.png
CDW_4.png
CDW_dot.png
CDW_2.png
CDW_dot.png
CDW_infin.png
CDW_dot.png
# The V3xW2, [6,3]x[∞] prismatic group,
CDW_dot.png
CDW_6.png
CDW_dot.png
CDW_3.png
CDW_dot.png
CDW_2.png
CDW_dot.png
CDW_infin.png
CDW_dot.png
# The P3xW2, [Δ]x[∞] prismatic group,
CD_2.png
CD_dot.png
CD_infin.png
CD_dot.png
# The W2xW2xW2, [∞]x[∞]x[∞] prismatic group,
CDW_dot.png
CDW_infin.png
CDW_dot.png
CDW_2.png
CDW_dot.png
CDW_infin.png
CDW_dot.png
CDW_2.png
CDW_dot.png
CDW_infin.png
CDW_dot.png
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 R4, [4,3,4] group (cubic)
The regular cubic honeycomb, represented by Schläfli symbol {4,3,4}, 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.)
{|class="wikitable"
!rowspan=2|Reference
Indices
!rowspan=2|Coxeter-Dynkin
and Schläfli
symbols
!rowspan=2|Honeycomb name
! colspan=4|Cell counts/vertex
and positions in cubic honeycomb
!
!
!
|-
!(0)
!(1)
!(2)
!(3)
!Solids
(Partial)
!Frames
(Perspective)
!vertex figure
|-
|J11,15
A1
W1
G22
|align=center|
CDW_ring.png
CDW_4.png
CDW_dot.png
CDW_3.png
CDW_dot.png
CDW_4.png
CDW_dot.png
t0{4,3,4}
|align=center|cubic
|align=center|
|align=center|
|align=center|
|align=center|8
(4.4.4)
|
|
|
octahedron
|-
|J12,32
A15
W14
G7
|align=center|
CDW_dot.png
CDW_4.png
CDW_ring.png
CDW_3.png
CDW_dot.png
CDW_4.png
CDW_dot.png
t1{4,3,4}
|align=center|rectified cubic
|align=center|2
(3.3.3.3)
|align=center|
|align=center|
|align=center|4
(3.4.3.4)
|
|
|
cuboid
|-
|J13
A14
W15
G8
|align=center|
CDW_ring.png
CDW_4.png
CDW_ring.png
CDW_3.png
CDW_dot.png
CDW_4.png
CDW_dot.png
t0,1{4,3,4}
|align=center|truncated cubic
|align=center|1
(3.3.3.3)
|align=center|
|align=center|
|align=center|4
(3.8.8)
|
|
|
square pyramid
|-
|J14
A17
W12
G9
|align=center|
CDW_ring.png
CDW_4.png
CDW_dot.png
CDW_3.png
CDW_ring.png
CDW_4.png
CDW_dot.png
t0,2{4,3,4}
|align=center|cantellated cubic
|align=center|1
(3.4.3.4)
|align=center|2
(4.4.4)
|align=center|
|align=center|2
(3.4.4.4)
|
|
|
wedge
|-
|J11,15
|align=center|
CDW_ring.png
CDW_4.png
CDW_dot.png
CDW_3.png
CDW_dot.png
CDW_4.png
CDW_ring.png
t0,3{4,3,4}
|align=center|'runcinated cubic'
(same as regular cubic)
|align=center|1
(4.4.4)
|align=center|3
(4.4.4)
|align=center|3
(4.4.4)
|align=center|1
(4.4.4)
|
|
|
octahedron
|-
|J16
A3
W2
G28
|align=center|
CDW_dot.png
CDW_4.png
CDW_ring.png
CDW_3.png
CDW_ring.png
CDW_4.png
CDW_dot.png
t1,2{4,3,4}
|align=center|bitruncated cubic
|align=center|2
(4.6.6)
|align=center|
|align=center|
|align=center|2
(4.6.6)
|
|
|
isosceles tetrahedron
|-
|J17
A18
W13
G25
|align=center|
CDW_ring.png
CDW_4.png
CDW_ring.png
CDW_3.png
CDW_ring.png
CDW_4.png
CDW_dot.png
t0,1,2{4,3,4}
|align=center|cantitruncated cubic
|align=center|1
(4.6.6)
|align=center|1
(4.4.4)
|align=center|
|align=center|2
(4.6.8)
|
|
|
irregular tetrahedron
|-
|J18
A19
W19
G20
|align=center|
CDW_ring.png
CDW_4.png
CDW_ring.png
CDW_3.png
CDW_dot.png
CDW_4.png
CDW_ring.png
t0,1,3{4,3,4}
|align=center|runcitruncated cubic
|align=center|1
(3.4.4.4)
|align=center|1
(4.4.4)
|align=center|2
(4.4.8)
|align=center|1
(3.8.8)
|
|
|
oblique trapezoidal pyramid
|-
|J19
A22
W18
G27
|align=center|
CDW_ring.png
CDW_4.png
CDW_ring.png
CDW_3.png
CDW_ring.png
CDW_4.png
CDW_ring.png
t0,1,2,3{4,3,4}
|align=center|omnitruncated cubic
|align=center|1
(4.6.8)
|align=center|1
(4.4.8)
|align=center|1
(4.4.8)
|align=center|1
(4.6.8)
|
|
|
irregular tetrahedron
|-
|-
|J21,31,51
A2
W9
G1
|align=center|
CDW_hole.png
CDW_4.png
CDW_dot.png
CDW_3.png
CDW_dot.png
CDW_4.png
CDW_dot.png
h0{4,3,4}
|align=center|alternated cubic
|align=center|6
3.3.3.3
|align=center|
|align=center|
|align=center|8
3.3.3
|
|
|
cuboctahedron
|}
S4, h[4,3,4] group
The S4 group offers 11 derived forms via truncation operations, four being unique uniform honeycombs.
The honeycombs from this group are called ''alternated cubic'' because the first form can be seen as a ''cubic honeycomb'' with alternate vertices cubic cells to tetrahedra, and creating octahedron cells in the deleted 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.
{|class="wikitable"
!rowspan=2|Referenced
indices
!rowspan=2|Coxeter-Dynkin
diagram
!rowspan=2|Honeycomb name
!colspan=4|Cells by location
(and count around each vertex)
!rowspan=2|Solids
(Partial)
!rowspan=2|Frames
(Perspective)
!rowspan=2|vertex figure
|-
!(0)
!(1)
!(0')
!(3)
|-
|J21,31,51
A2
W9
G1
|align=center|
CD_ring.png
CD_3b.png
CD_downbranch-00.png
CD_3b.png
CD_4.png
CD_dot.png
|align=center|alternated cubic
|align=center|
|align=center|
|align=center|
3.3.3.3
|align=center|
3.3.3
|
|
|
cuboctahedron
|-
|J22,34
A21
W17
G10
|align=center|
CD_ring.png
CD_3b.png
CD_downbranch-10.png
CD_3b.png
CD_4.png
CD_dot.png
|align=center|truncated alternated cubic
|align=center|
|align=center|
3.4.3.4
|align=center|
4.6.6
|align=center|
3.6.6
||
|
|
|-
|J12,32
A15
W14
G7
|align=center|
CD_dot.png
CD_3b.png
CD_downbranch-10.png
CD_3b.png
CD_4.png
CD_dot.png
|align=center|rectified cubic
(rectified alternate cubic)
|align=center|
(3.4.3.4)
|align=center|
|align=center|
(3.4.3.4)
|align=center|
(3.3.3.3)
|
|
|
cuboid
|-
|J12,32
A15
W14
G7
|align=center|
CD_ring.png
CD_3b.png
CD_downbranch-01.png
CD_3b.png
CD_4.png
CD_dot.png
|align=center|rectified cubic
(cantellated alternate cubic)
|align=center|
(3.3.3.3)
|align=center|
|align=center|
(3.3.3.3)
|align=center|
(3.4.3.4)
|
|
|
cuboid
|-
|J16
A3
W2
G28
|align=center|
CD_ring.png
CD_3b.png
CD_downbranch-11.png
CD_3b.png
CD_4.png
CD_dot.png
|align=center|bitruncated cubic
(cantitruncated alternate cubic)
|align=center|
(4.6.6)
|align=center|
|align=center|
(4.6.6)
|align=center|
(4.6.6)
|
|
|
isosceles tetrahedron
|-
|J13
A14
W15
G8
|align=center|
CD_dot.png
CD_3b.png
CD_downbranch-10.png
CD_3b.png
CD_4.png
CD_ring.png
|align=center|truncated cubic
(bicantellated alternate cubic)
|align=center|
(3.8.8)
|align=center|
|align=center|
(3.8.8)
|align=center|
(3.3.3.3)
|
|
|
square pyramid
|-
|J11,15
A1
W1
G22
|align=center|
CD_dot.png
CD_3b.png
CD_downbranch-00.png
CD_3b.png
CD_4.png
CD_ring.png
|align=center|cubic
(trirectified alternate cubic)
|align=center|
(4.4.4)
|align=center|
|align=center|
(4.4.4)
|align=center|
|
|
|
octahedron
|-
|J23
A16
W11
G5
|align=center|
CD_ring.png
CD_3b.png
CD_downbranch-00.png
CD_3b.png
CD_4.png
CD_ring.png
|align=center|runcinated alternated cubic
|align=center|
cube
|align=center|
|align=center|
3.4.4.4
|align=center|
3.3.3
|
|
|
|-
|J14
A17
W12
G9
|align=center|
CD_ring.png
CD_3b.png
CD_downbranch-01.png
CD_3b.png
CD_4.png
CD_ring.png
|align=center|cantellated cubic
(runcicantellated alternate cubic)
|align=center|
(3.4.4.4)
|align=center|
(4.4.4)
|align=center|
(3.4.4.4)
|align=center|
(3.4.3.4)
|
|
|
wedge
|-
|J24
A20
W16
G21
|align=center|
CD_ring.png
CD_3b.png
CD_downbranch-10.png
CD_3b.png
CD_4.png
CD_ring.png
|align=center|cantitruncated alternated cubic
(or runcitruncated alternate cubic)
|align=center|
|align=center|
3.8.8
|align=center|
4.6.8
|align=center|
3.6.6
|
|
|
|-
|J17
A18
W13
G25
|align=center|
CD_ring.png
CD_3b.png
CD_downbranch-11.png
CD_3b.png
CD_4.png
CD_ring.png
|align=center|cantitruncated cubic
(omnitruncated alternated cubic)
|align=center|
(4.6.8)
|align=center|
(4.4.4)
|align=center|
(4.6.8)
|align=center|
(4.6.6)
|
|
|
irregular tetrahedron
|}
P4 group
There are 5 forms constructed from the P4 group, only the ''quarter cubic honeycomb'' is unique.
{|class="wikitable"
!rowspan=2|Referenced
indices
!rowspan=2|Coxeter-Dynkin
diagram
!rowspan=2|Honeycomb name
!colspan=4|Cells by location
(and count around each vertex)
!rowspan=2|Solids
(Partial)
!rowspan=2|Frames
(Perspective)
!rowspan=2|vertex figure
|-
!(0)
!(1)
!(2)
!(3)
|-
|J21,31,51
A2
W9
G1
|align=center|
CD_p4-1000.png
|align=center|alternated cubic
|align=center|
|align=center|
3.3.3
|align=center|
3.3.3.3
|align=center|
3.3.3
|
|
|
cuboctahedron
|-
|J12,32
A15
W14
G7
|align=center|
CD_p4-1010.png
|align=center|rectified cubic
|align=center|
(3.4.3.4)
|align=center|
(3.3.3.3)
|align=center|
(3.4.3.4)
|align=center|
(3.3.3.3)
|
|
|
cuboid
|-
|J25,33
A13
W10
G6
|align=center|
CD_p4-1100.png
|align=center|quarter cubic
|align=center|
3.3.3
|align=center|
3.3.3
|align=center|
3.6.6
|align=center|
3.6.6
|
|
|
|-
|J22,34
A21
W17
G10
|align=center|
CD_p4-1110.png
|align=center|truncated alternated cubic
|align=center|
3.6.6
|align=center|
3.4.3.4
|align=center|
3.6.6
|align=center|
4.6.6
||
|
|
|-
|J16
A3
W2
G28
|align=center|
CD_p4-1111.png
|align=center|bitruncated cubic
|align=center|
(4.6.6)
|align=center|
(4.6.6)
|align=center|
(4.6.6)
|align=center|
(4.6.6)
|
|
|
isosceles tetrahedron
|}
Gyrated and elongated forms
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.
{|class="wikitable"
!Referenced
indices
!symbol
!Honeycomb name
!cell types (# at each vertex)
!Solids
(Partial)
!Frames
(Perspective)
!vertex figure
|-
|J52
A2'
G2
|h{4,3,4}:g
|align=center|gyrated alternated cubic
|align=center|tetrahedron (8)
octahedron (6)
|
|
|
|-
|J61
A?
G3
|h{4,3,4}:ge
|align=center|gyroelongated alternated cubic
|align=center|triangular prism (6)
tetrahedron (4)
octahedron (3)
|
|
| -
|-
|J62
A?
G4
|h{4,3,4}:e
|align=center|elongated alternated cubic
|align=center|triangular prism (6)
tetrahedron (4)
octahedron (3)
|
|
|
|-
|J63
A?
G12
|{3,6}:g x {∞}
|align=center|gyrated triangular prismatic
|align=center|triangular prism (12)
|
|
|
|-
|J64
A?
G15
|{3,6}:ge x {∞}
|align=center|gyroelongated triangular prismatic
|align=center|triangular prism (6)
cube (4)
|
|
|
|}
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 of each is an irregular bipyramid whose faces are isosceles triangles.
The R3xW2, [4,4] x [∞], prismatic group
There's 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.
{|class="wikitable"
!Indices
!Coxeter-Dynkin
and Schläfli
symbols
!Honeycomb name
!Plane
tiling
!Solids
(Partial)
!Tiling
|-
|J11,15
A1
G22
|align=center|
CDW_ring.png
CDW_4.png
CDW_dot.png
CDW_4.png
CDW_dot.png
CDW_2.png
CDW_ring.png
CDW_infin.png
CDW_dot.png
{4,4} x {∞}
|align=center|Cubic
(Square prismatic)
|(4.4.4.4)
|
|
|-
|J45
A6
G24
|align=center|
CDW_ring.png
CDW_4.png
CDW_ring.png
CDW_4.png
CDW_dot.png
CDW_2.png
CDW_ring.png
CDW_infin.png
CDW_dot.png
t0,1{4,4} x {∞}
|align=center|Truncated/Bitruncated square prismatic
|(4.8.8)
|
|
|-
|J11,15
A1
G22
|align=center|
CDW_dot.png
CDW_4.png
CDW_ring.png
CDW_4.png
CDW_dot.png
CDW_2.png
CDW_ring.png
CDW_infin.png
CDW_dot.png
t1{4,4} x {∞}
|align=center|Cubic
(Rectified square prismatic)
|(4.4.4.4)
|
|
|-
|J11,15
A1
G22
|align=center|
CDW_ring.png
CDW_4.png
CDW_dot.png
CDW_4.png
CDW_ring.png
CDW_2.png
CDW_ring.png
CDW_infin.png
CDW_dot.png
t0,2{4,4} x {∞}
|align=center|Cubic
(Cantellated square prismatic)
|(4.4.4.4)
|
|
|-
|J45
A6
G24
|align=center|
CDW_ring.png
CDW_4.png
CDW_ring.png
CDW_4.png
CDW_ring.png
CDW_2.png
CDW_ring.png
CDW_infin.png
CDW_dot.png
t0,1,2{4,4} x {∞}
|align=center|Truncated square prismatic
(Omnitruncated square prismatic)
|(4.8.8)
|
|
|-
|J44
A11
G14
|align=center|
CDW_hole.png
CDW_4.png
CDW_hole.png
CDW_4.png
CDW_hole.png
CDW_2.png
CDW_ring.png
CDW_infin.png
CDW_dot.png
s{4,4} x {∞}
|align=center|Snub square prismatic
|(3.3.4.3.4)
|
|
|}
The V3xW2, [6,3] x [∞] prismatic group
{|class="wikitable"
!Indices
!Coxeter-Dynkin
and Schläfli
symbols
!Honeycomb name
!Plane
tiling
!Solids
(Partial)
!Tiling
|-
|J42
A5
G26
|align=center|
CDW_ring.png
CDW_6.png
CDW_dot.png
CDW_3.png
CDW_dot.png
CDW_2.png
CDW_ring.png
CDW_infin.png
CDW_dot.png
t0{6,3} x {∞}
|align=center|Hexagonal prismatic
|(63)
|
|
|-
|J46
A7
G19
|align=center|
CDW_ring.png
CDW_6.png
CDW_ring.png
CDW_3.png
CDW_dot.png
CDW_2.png
CDW_ring.png
CDW_infin.png
CDW_dot.png
t0,1{6,3} x {∞}
|align=center|Truncated hexagonal prismatic
|(3.12.12)
|
|
|-
|J43
A8
G18
|align=center|
CDW_dot.png
CDW_6.png
CDW_ring.png
CDW_3.png
CDW_dot.png
CDW_2.png
CDW_ring.png
CDW_infin.png
CDW_dot.png
t1{6,3} x {∞}
|align=center|Trihexagonal prismatic
|(3.6.3.6)
|
|
|-
|J42
A5
G26
|align=center|
CDW_dot.png
CDW_6.png
CDW_ring.png
CDW_3.png
CDW_ring.png
CDW_2.png
CDW_ring.png
CDW_infin.png
CDW_dot.png
t1,2{6,3} x {∞}
|align=center|''Truncated triangular prismatic''
Hexagonal prismatic
|(6.6.6)
||
|
|-
|J41
A4
G11
|align=center|
CDW_dot.png
CDW_6.png
CDW_dot.png
CDW_3.png
CDW_ring.png
CDW_2.png
CDW_ring.png
CDW_infin.png
CDW_dot.png
t2{6,3} x {∞}
|align=center|Triangular prismatic
|(36)
|
|
|-
|J47
A9
G16
|align=center|
CDW_ring.png
CDW_6.png
CDW_dot.png
CDW_3.png
CDW_ring.png
CDW_2.png
CDW_ring.png
CDW_infin.png
CDW_dot.png
t0,2{6,3} x {∞}
|align=center|Rhombi-trihexagonal prismatic
|(3.4.6.4)
|
|
|-
|J49
A10
G23
|align=center|
CDW_ring.png
CDW_6.png
CDW_ring.png
CDW_3.png
CDW_ring.png
CDW_2.png
CDW_ring.png
CDW_infin.png
CDW_dot.png
t0,1,2{6,3} x {∞}
|align=center|Omnitruncated trihexagonal prismatic
|(4.6.12)
|
|
|-
|J48
A12
G17
|align=center|
CDW_hole.png
CDW_6.png
CDW_hole.png
CDW_3.png
CDW_hole.png
CDW_2.png
CDW_ring.png
CDW_infin.png
CDW_dot.png
s{6,3} x {∞}
|align=center|Snub trihexagonal prismatic
|(3.3.3.3.6)
|
|
|-
|J65
A11'
G13
|{3,6}:e x {∞}
|align=center|elongated triangular prismatic
|align=center|3.3.3.4.4
|
|
|}
Examples
All 28 of these tessellations are found in crystal arrangements.
The alternated cubic honeycomb is of special importance since its vertices form a cubic close-packing of spheres. The space-filling truss of packed octahedra and tetrahedra was apparently first discovered by Alexander Graham Bell and independently re-discovered by Buckminster Fuller (who called it the octet truss and patented it in the 1940s).
[1]
[2]
[3]
[4]. Octet trusses are now among the most common types of truss used in construction.
External links
★
★ Uniform Honeycombs in 3-Space VRML models
★ Elementary Honeycombs
★ Uniform partitions of 3-space, their relatives and embedding PDF, 1999
★ The Uniform Polyhedra
★ Virtual Reality Polyhedra The Encyclopedia of Polyhedra
References
★ George Olshevsky, ''Uniform Panoploid Tetracombs'', Manuscript (2006) ''(Complete list of 11 convex uniform tilings, 28 convex uniform honeycombs, and 143 convex uniform tetracombs)''
★ Branko Grünbaum, Uniform tilings of 3-space. Geombinatorics 4(1994), 49 - 56.
★ Norman Johnson ''Uniform Polytopes'', Manuscript (1991)
★ The Geometrical Foundation of Natural Structure: A Source Book of Design, , Robert, Williams, Dover Publications, Inc, 1979, ISBN 0-486-23729-X
★ Order in Space: A design source book, , Keith, Critchlow, Viking Press, 1970, ISBN 0-500-34033-1
★ 'Kaleidoscopes: Selected Writings of H.S.M. Coxeter', editied by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, ISBN 978-0-471-01003-6 [5]
★
★ (Paper 22) H.S.M. Coxeter, ''Regular and Semi Regular Polytopes I'', [Math. Zeit. 46 (1940) 380-407, MR 2,10] (1.9 Uniform space-fillings)
★ A. Andreini, ''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 (1905) 75–129.
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