Convex uniform honeycomb

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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:

They can be considered the three-dimensional analogue to the uniform tilings of the plane.

Contents

[edit] 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.

Only 14 of the convex uniform polyhedra appear in these patterns:

[edit] 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, the are give 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).

[edit] Tessellations listed by category

[edit] Cubic forms {4,3,4}

Symmetry domain and points within cubic cell.
Enlarge
Symmetry domain and points within cubic cell.

The regular cubic honeycomb, represented by Schläfli symbol {4,3,4}, offers seven unque derived uniform honeycombs via truncation operations. (One redundant form, the runcinated cubic honeycomb, is included for completeness though identical to the cubic honeycomb.)


Reference
Indices
Extended
Schläfli
symbol
name Cell counts/vertex
and positions in cubic honeycomb
Cell
(3)
Face
(2)
Edge
(1)
Vertex
(0)
Solids
(Partial)
Frames
(Perspective)
vertex figure
J11,15
A1
W1
G22
t0{4,3,4} cubic 8

(4.4.4)




octahedron
J12,32
A15
W14
G7
t1{4,3,4} rectified cubic 4

(3.4.3.4)


2

(3.3.3.3)

cuboid
J13
A14
W15
G8
t0,1{4,3,4} truncated cubic 4

(3.8.8)


1

(3.3.3.3)

square pyramid
J14
A17
W12
G9
t0,2{4,3,4} cantellated cubic 2

(3.4.4.4)

2

(4.4.4)
1

(3.4.3.4)

wedge
J11,15 t0,3{4,3,4} runcinated cubic
(same as regular cubic)
1

(4.4.4)
3

(4.4.4)
3

(4.4.4)
1

(4.4.4)

octahedron
J16
A3
W2
G28
t1,2{4,3,4} bitruncated cubic 2

(4.6.6)


2

(4.6.6)

isosceles tetrahedron
J17
A18
W13
G25
t0,1,2{4,3,4} cantitruncated cubic 2

(4.6.8)

1

(4.4.4)
1

(4.6.6)

irregular tetrahedron
J18
A19
W19
G20
t0,1,3{4,3,4} runcitruncated cubic 1

(3.8.8)
2

(4.4.8)
1

(4.4.4)
1

(3.4.4.4)

oblique trapezoidal pyramid
J19
A22
W18
G27
t0,1,2,3{4,3,4} omnitruncated cubic 1

(4.6.8)
1

(4.4.8)
1

(4.4.8)
1

(4.6.8)

irregular tetrahedron

[edit] Alternated cubic forms: h{4,3,4}

The tet-oct (alternated cubic) honeycomb offers four derived uniform honeycombs via truncation operations.

This set is called alternated cubic because they can be derived from the cubic honeycomb by deleting alternate vertices. The cubic cells degenerate into tetrahedral cells, and the deleted vertices form voids in which new octahedral cells appear.

Referenced
indices
name cell types (# at each vertex) # families of continuous face planes Solids
(Partial)
Frames
(Perspective)
vertex figure
J21,31,51
A2
W9
G1
alternated cubic tetrahedron (8)
octahedron (6)
4 triangle
cuboctahedron
J22,34
A21
W17
G10
truncated alternated cubic truncated octahedron (2)
truncated tetrahedron (2)
cuboctahedron (1)
0
J23
A16
W11
G5
runcinated alternated cubic rhombicuboctahedron (3)
cube (1)
tetrahedron (1)
0
J24
A20
W16
G21
cantitruncated alternated cubic truncated cuboctahedron (2)
truncated cube (1)
truncated tetrahedron (1)
0
J25,33
A13
W10
G6
quarter cubic honeycomb truncated tetrahedron (6)
tetrahedron (2)
4 tri-hex

[edit] 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).

Referenced
indices
name cell types (# at each vertex) # families of continuous face planes Solids
(Partial)
Frames
(Perspective)
vertex figure
J52
A2'
G2
gyrated alternated cubic tetrahedron (8)
octahedron (6)
1 triangle
J61
A?
G3
gyroelongated alternated cubic triangular prism (6)
tetrahedron (4)
octahedron (3)
1 triangle
J62
A?
G4
elongated alternated cubic triangular prism (6)
tetrahedron (4)
octahedron (3)
1 triangle

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.

[edit] Prismatic forms

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.

[edit] Square prismatics: {}x{4,4}

Indices Schläfli symbol Name Picture
J11,15, A1, G22 {} x {4,4} Cubic honeycomb
(square prismatic honeycomb)

square tiling
J45, A6, G24 {} x t1{4,4} Truncated square prismatic honeycomb
4.8.8 tiling
J44, A11, G14 {} x s{4,4} Snub square prismatic honeycomb
3.3.4.3.4 tiling

[edit] Trihexagonal prismatics: {}x{3,6}

Indices Schläfli symbol Name Picture
J41, A4, G11 {} x {3,6} Triangular prismatic honeycomb
triangular tiling
J42, A5, G26 {} x t2{3,6} Hexagonal prismatic honeycomb
hexagonal tiling
J43, A8, G18 {} x t1{3,6} Triangular-hexagonal prismatic honeycomb
3.6.3.6 tiling
J46, A7, G19 {} x t1,2{3,6} Truncated hexagonal prismatic honeycomb
3.12.12 tiling
J47, A9, G16 {} x t0,2{3,6} Rhombitriangular-hexagonal prismatic honeycomb
3.4.6.4 tiling
J48, A12, G17 {} x s{3,6} Snub triangular-hexagonal prismatic honeycomb
3.3.3.3.6 tiling
J49, A10, G23 {} x t0,1,2{3,6} Omnitruncated triangular-hexagonal prismatic honeycomb
4.6.12 tiling
J65, A11', G13 {} x {3,6}e Elongated triangular prismatic honeycomb
3.3.3.4.4 tiling

[edit] Gyrated prismatic forms

Referenced
indices
name cell types (# at each vertex) # families of continuous face planes Solids
(Partial)
Frames
(Perspective)
vertex figure
J63
A?
G12
gyrated triangular prismatic triangular prism (12) 1 square
J64
A?
G15
gyroelongated triangular prismatic triangular prism (6)
cube (4)
1 square

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.

[edit] 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.

[edit] External links

[edit] 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)
  • Williams, Robert (1979). The Geometrical Foundation of Natural Structure: A Source Book of Design. Dover Publications, Inc. ISBN 0-486-23729-X.
  • Critchlow, Keith (1970). Order in Space: A design source book. Viking Press. ISBN 0-500-34033-1.
  • 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.