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http://en.wikipedia.org/wiki/Convex_uniform_honeycomb
Panales Uniformes Convexos
In geometry, a convex uniform honeycomb is a uniform space-filling tessellation in three-
dimensional Euclidean spacewith non-overlapping convexuniform polyhedral cells.
Twenty-eight such honeycombs exist:
the familiarcubic 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.
Contents
1 History
o 1.1 Names
2 Compact Euclidean uniform tessellations (by their infinite Coxeter group families)
o 2.1 The C~3, [4,3,4] group (cubic)
o 2.2 B~4, h[4,3,4], [4,31,1] group
o 2.3 A~3 group
o 2.4 Nonwythoffian forms (gyrated and elongated)
o 2.5 Prismatic stacks
2.5.1 The C~2xI~1(), [4,4] x [], prismatic group
2.5.2 The H~2xI~1(), [6,3] x [] prismatic group
o 2.6 Examples
3 Noncompact forms
4 Hyperbolic forms
o 4.1 [3,5,3] family
o 4.2 [5,3,4] family
o 4.3 [5,3,5] family
o 4.4 [5,31,1] family
o 4.5 [4,3,3,3:] family
o 4.6 [5,3,3,3:] family
o 4.7 [4,3,4,3:] family
o 4.8 [4,3,5,3:] family
o 4.9 [5,3,5,3:] family
5 Notes
http://en.wikipedia.org/wiki/Geometryhttp://en.wikipedia.org/wiki/Tessellationhttp://en.wikipedia.org/wiki/Euclidean_spacehttp://en.wikipedia.org/wiki/Euclidean_spacehttp://en.wikipedia.org/wiki/Uniform_polyhedronhttp://en.wikipedia.org/wiki/Uniform_polyhedronhttp://en.wikipedia.org/wiki/Convex_uniform_honeycomb#The_R4.2C_.5B4.2C3.2C4.5D_group_.28cubic.29http://en.wikipedia.org/wiki/Convex_uniform_honeycomb#Alternated_cubic_formshttp://en.wikipedia.org/wiki/Convex_uniform_honeycomb#Stacked_prismatic_formshttp://en.wikipedia.org/wiki/List_of_uniform_planar_tilingshttp://en.wikipedia.org/wiki/Convex_uniform_honeycomb#Historyhttp://en.wikipedia.org/wiki/Convex_uniform_honeycomb#Nameshttp://en.wikipedia.org/wiki/Convex_uniform_honeycomb#Compact_Euclidean_uniform_tessellations_.28by_their_infinite_Coxeter_group_families.29http://en.wikipedia.org/wiki/Convex_uniform_honeycomb#The_C.7E3.2C_.5B4.2C3.2C4.5D_group_.28cubic.29http://en.wikipedia.org/wiki/Convex_uniform_honeycomb#The_C.7E3.2C_.5B4.2C3.2C4.5D_group_.28cubic.29http://en.wikipedia.org/wiki/Convex_uniform_honeycomb#The_C.7E3.2C_.5B4.2C3.2C4.5D_group_.28cubic.29http://en.wikipedia.org/wiki/Convex_uniform_honeycomb#The_C.7E3.2C_.5B4.2C3.2C4.5D_group_.28cubic.29http://en.wikipedia.org/wiki/Convex_uniform_honeycomb#The_C.7E3.2C_.5B4.2C3.2C4.5D_group_.28cubic.29http://en.wikipedia.org/wiki/Convex_uniform_honeycomb#B.7E4.2C_h.5B4.2C3.2C4.5D.2C_.5B4.2C31.2C1.5D_grouphttp://en.wikipedia.org/wiki/Convex_uniform_honeycomb#B.7E4.2C_h.5B4.2C3.2C4.5D.2C_.5B4.2C31.2C1.5D_grouphttp://en.wikipedia.org/wiki/Convex_uniform_honeycomb#B.7E4.2C_h.5B4.2C3.2C4.5D.2C_.5B4.2C31.2C1.5D_grouphttp://en.wikipedia.org/wiki/Convex_uniform_honeycomb#B.7E4.2C_h.5B4.2C3.2C4.5D.2C_.5B4.2C31.2C1.5D_grouphttp://en.wikipedia.org/wiki/Convex_uniform_honeycomb#B.7E4.2C_h.5B4.2C3.2C4.5D.2C_.5B4.2C31.2C1.5D_grouphttp://en.wikipedia.org/wiki/Convex_uniform_honeycomb#B.7E4.2C_h.5B4.2C3.2C4.5D.2C_.5B4.2C31.2C1.5D_grouphttp://en.wikipedia.org/wiki/Convex_uniform_honeycomb#B.7E4.2C_h.5B4.2C3.2C4.5D.2C_.5B4.2C31.2C1.5D_grouphttp://en.wikipedia.org/wiki/Convex_uniform_honeycomb#A.7E3_grouphttp://en.wikipedia.org/wiki/Convex_uniform_honeycomb#A.7E3_grouphttp://en.wikipedia.org/wiki/Convex_uniform_honeycomb#A.7E3_grouphttp://en.wikipedia.org/wiki/Convex_uniform_honeycomb#A.7E3_grouphttp://en.wikipedia.org/wiki/Convex_uniform_honeycomb#A.7E3_grouphttp://en.wikipedia.org/wiki/Convex_uniform_honeycomb#Nonwythoffian_forms_.28gyrated_and_elongated.29http://en.wikipedia.org/wiki/Convex_uniform_honeycomb#Prismatic_stackshttp://en.wikipedia.org/wiki/Convex_uniform_honeycomb#The_C.7E2xI.7E1.28.E2.88.9E.29.2C_.5B4.2C4.5D_x_.5B.E2.88.9E.5D.2C_prismatic_grouphttp://en.wikipedia.org/wiki/Convex_uniform_honeycomb#The_C.7E2xI.7E1.28.E2.88.9E.29.2C_.5B4.2C4.5D_x_.5B.E2.88.9E.5D.2C_prismatic_grouphttp://en.wikipedia.org/wiki/Convex_uniform_honeycomb#The_C.7E2xI.7E1.28.E2.88.9E.29.2C_.5B4.2C4.5D_x_.5B.E2.88.9E.5D.2C_prismatic_grouphttp://en.wikipedia.org/wiki/Convex_uniform_honeycomb#The_C.7E2xI.7E1.28.E2.88.9E.29.2C_.5B4.2C4.5D_x_.5B.E2.88.9E.5D.2C_prismatic_grouphttp://en.wikipedia.org/wiki/Convex_uniform_honeycomb#The_C.7E2xI.7E1.28.E2.88.9E.29.2C_.5B4.2C4.5D_x_.5B.E2.88.9E.5D.2C_prismatic_grouphttp://en.wikipedia.org/wiki/Convex_uniform_honeycomb#The_C.7E2xI.7E1.28.E2.88.9E.29.2C_.5B4.2C4.5D_x_.5B.E2.88.9E.5D.2C_prismatic_grouphttp://en.wikipedia.org/wiki/Convex_uniform_honeycomb#The_C.7E2xI.7E1.28.E2.88.9E.29.2C_.5B4.2C4.5D_x_.5B.E2.88.9E.5D.2C_prismatic_grouphttp://en.wikipedia.org/wiki/Convex_uniform_honeycomb#The_C.7E2xI.7E1.28.E2.88.9E.29.2C_.5B4.2C4.5D_x_.5B.E2.88.9E.5D.2C_prismatic_grouphttp://en.wikipedia.org/wiki/Convex_uniform_honeycomb#The_C.7E2xI.7E1.28.E2.88.9E.29.2C_.5B4.2C4.5D_x_.5B.E2.88.9E.5D.2C_prismatic_grouphttp://en.wikipedia.org/wiki/Convex_uniform_honeycomb#The_H.7E2xI.7E1.28.E2.88.9E.29.2C_.5B6.2C3.5D_x_.5B.E2.88.9E.5D_prismatic_grouphttp://en.wikipedia.org/wiki/Convex_uniform_honeycomb#The_H.7E2xI.7E1.28.E2.88.9E.29.2C_.5B6.2C3.5D_x_.5B.E2.88.9E.5D_prismatic_grouphttp://en.wikipedia.org/wiki/Convex_uniform_honeycomb#The_H.7E2xI.7E1.28.E2.88.9E.29.2C_.5B6.2C3.5D_x_.5B.E2.88.9E.5D_prismatic_grouphttp://en.wikipedia.org/wiki/Convex_uniform_honeycomb#The_H.7E2xI.7E1.28.E2.88.9E.29.2C_.5B6.2C3.5D_x_.5B.E2.88.9E.5D_prismatic_grouphttp://en.wikipedia.org/wiki/Convex_uniform_honeycomb#The_H.7E2xI.7E1.28.E2.88.9E.29.2C_.5B6.2C3.5D_x_.5B.E2.88.9E.5D_prismatic_grouphttp://en.wikipedia.org/wiki/Convex_uniform_honeycomb#The_H.7E2xI.7E1.28.E2.88.9E.29.2C_.5B6.2C3.5D_x_.5B.E2.88.9E.5D_prismatic_grouphttp://en.wikipedia.org/wiki/Convex_uniform_honeycomb#The_H.7E2xI.7E1.28.E2.88.9E.29.2C_.5B6.2C3.5D_x_.5B.E2.88.9E.5D_prismatic_grouphttp://en.wikipedia.org/wiki/Convex_uniform_honeycomb#The_H.7E2xI.7E1.28.E2.88.9E.29.2C_.5B6.2C3.5D_x_.5B.E2.88.9E.5D_prismatic_grouphttp://en.wikipedia.org/wiki/Convex_uniform_honeycomb#The_H.7E2xI.7E1.28.E2.88.9E.29.2C_.5B6.2C3.5D_x_.5B.E2.88.9E.5D_prismatic_grouphttp://en.wikipedia.org/wiki/Convex_uniform_honeycomb#Exampleshttp://en.wikipedia.org/wiki/Convex_uniform_honeycomb#Noncompact_formshttp://en.wikipedia.org/wiki/Convex_uniform_honeycomb#Hyperbolic_formshttp://en.wikipedia.org/wiki/Convex_uniform_honeycomb#.5B3.2C5.2C3.5D_familyhttp://en.wikipedia.org/wiki/Convex_uniform_honeycomb#.5B5.2C3.2C4.5D_familyhttp://en.wikipedia.org/wiki/Convex_uniform_honeycomb#.5B5.2C3.2C5.5D_familyhttp://en.wikipedia.org/wiki/Convex_uniform_honeycomb#.5B5.2C31.2C1.5D_familyhttp://en.wikipedia.org/wiki/Convex_uniform_honeycomb#.5B5.2C31.2C1.5D_familyhttp://en.wikipedia.org/wiki/Convex_uniform_honeycomb#.5B5.2C31.2C1.5D_familyhttp://en.wikipedia.org/wiki/Convex_uniform_honeycomb#.5B4.2C3.2C3.2C3:.5D_familyhttp://en.wikipedia.org/wiki/Convex_uniform_honeycomb#.5B5.2C3.2C3.2C3:.5D_familyhttp://en.wikipedia.org/wiki/Convex_uniform_honeycomb#.5B4.2C3.2C4.2C3:.5D_familyhttp://en.wikipedia.org/wiki/Convex_uniform_honeycomb#.5B4.2C3.2C5.2C3:.5D_familyhttp://en.wikipedia.org/wiki/Convex_uniform_honeycomb#.5B5.2C3.2C5.2C3:.5D_familyhttp://en.wikipedia.org/wiki/Convex_uniform_honeycomb#Noteshttp://en.wikipedia.org/wiki/Geometryhttp://en.wikipedia.org/wiki/Tessellationhttp://en.wikipedia.org/wiki/Euclidean_spacehttp://en.wikipedia.org/wiki/Uniform_polyhedronhttp://en.wikipedia.org/wiki/Convex_uniform_honeycomb#The_R4.2C_.5B4.2C3.2C4.5D_group_.28cubic.29http://en.wikipedia.org/wiki/Convex_uniform_honeycomb#Alternated_cubic_formshttp://en.wikipedia.org/wiki/Convex_uniform_honeycomb#Stacked_prismatic_formshttp://en.wikipedia.org/wiki/List_of_uniform_planar_tilingshttp://en.wikipedia.org/wiki/Convex_uniform_honeycomb#Historyhttp://en.wikipedia.org/wiki/Convex_uniform_honeycomb#Nameshttp://en.wikipedia.org/wiki/Convex_uniform_honeycomb#Compact_Euclidean_uniform_tessellations_.28by_their_infinite_Coxeter_group_families.29http://en.wikipedia.org/wiki/Convex_uniform_honeycomb#The_C.7E3.2C_.5B4.2C3.2C4.5D_group_.28cubic.29http://en.wikipedia.org/wiki/Convex_uniform_honeycomb#B.7E4.2C_h.5B4.2C3.2C4.5D.2C_.5B4.2C31.2C1.5D_grouphttp://en.wikipedia.org/wiki/Convex_uniform_honeycomb#A.7E3_grouphttp://en.wikipedia.org/wiki/Convex_uniform_honeycomb#Nonwythoffian_forms_.28gyrated_and_elongated.29http://en.wikipedia.org/wiki/Convex_uniform_honeycomb#Prismatic_stackshttp://en.wikipedia.org/wiki/Convex_uniform_honeycomb#The_C.7E2xI.7E1.28.E2.88.9E.29.2C_.5B4.2C4.5D_x_.5B.E2.88.9E.5D.2C_prismatic_grouphttp://en.wikipedia.org/wiki/Convex_uniform_honeycomb#The_H.7E2xI.7E1.28.E2.88.9E.29.2C_.5B6.2C3.5D_x_.5B.E2.88.9E.5D_prismatic_grouphttp://en.wikipedia.org/wiki/Convex_uniform_honeycomb#Exampleshttp://en.wikipedia.org/wiki/Convex_uniform_honeycomb#Noncompact_formshttp://en.wikipedia.org/wiki/Convex_uniform_honeycomb#Hyperbolic_formshttp://en.wikipedia.org/wiki/Convex_uniform_honeycomb#.5B3.2C5.2C3.5D_familyhttp://en.wikipedia.org/wiki/Convex_uniform_honeycomb#.5B5.2C3.2C4.5D_familyhttp://en.wikipedia.org/wiki/Convex_uniform_honeycomb#.5B5.2C3.2C5.5D_familyhttp://en.wikipedia.org/wiki/Convex_uniform_honeycomb#.5B5.2C31.2C1.5D_familyhttp://en.wikipedia.org/wiki/Convex_uniform_honeycomb#.5B4.2C3.2C3.2C3:.5D_familyhttp://en.wikipedia.org/wiki/Convex_uniform_honeycomb#.5B5.2C3.2C3.2C3:.5D_familyhttp://en.wikipedia.org/wiki/Convex_uniform_honeycomb#.5B4.2C3.2C4.2C3:.5D_familyhttp://en.wikipedia.org/wiki/Convex_uniform_honeycomb#.5B4.2C3.2C5.2C3:.5D_familyhttp://en.wikipedia.org/wiki/Convex_uniform_honeycomb#.5B5.2C3.2C5.2C3:.5D_familyhttp://en.wikipedia.org/wiki/Convex_uniform_honeycomb#Notes 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6 References
7 External links
History
1900:Thorold Gosset enumerated the list of semiregular convex polytopes withregular 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 Grnbaum, in his paperUniform 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.
Grnbaum also states thatI. Alexeyev of Russia also independently enumeratedthese 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 uniformhoneycombs, expands a further derived list of 143 convex uniform tetracombs
(Honeycombs ofuniform polychorons in 4-space).
Only 14 of the convex uniform polyhedra appear in these patterns:
three of the five Platonic solids,
six of the thirteenArchimedean solids, and
five of the infinite family ofprisms.
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 Conwayhas 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 byNorman 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).
Compact Euclidean uniform tessellations (by their
infinite Coxeter group families)
http://en.wikipedia.org/wiki/Convex_uniform_honeycomb#Referenceshttp://en.wikipedia.org/wiki/Convex_uniform_honeycomb#External_linkshttp://en.wikipedia.org/wiki/Thorold_Gossethttp://en.wikipedia.org/wiki/Thorold_Gossethttp://en.wikipedia.org/wiki/Platonic_solidhttp://en.wikipedia.org/wiki/Alfredo_Andreinihttp://en.wikipedia.org/wiki/Alfredo_Andreinihttp://en.wikipedia.org/wiki/Norman_Johnson_(mathematician)http://en.wikipedia.org/wiki/Norman_Johnson_(mathematician)http://en.wikipedia.org/wiki/Branko_Gr%C3%BCnbaumhttp://en.wikipedia.org/wiki/Branko_Gr%C3%BCnbaumhttp://en.wikipedia.org/wiki/Convex_uniform_honeycomb#Gyrated_and_elongated_formshttp://en.wikipedia.org/w/index.php?title=I._Alexeyev&action=edit&redlink=1http://en.wikipedia.org/w/index.php?title=I._Alexeyev&action=edit&redlink=1http://en.wikipedia.org/wiki/George_Olshevskyhttp://en.wikipedia.org/wiki/George_Olshevskyhttp://en.wikipedia.org/wiki/Uniform_polychoronhttp://en.wikipedia.org/wiki/Uniform_polychoronhttp://en.wikipedia.org/wiki/Platonic_solidhttp://en.wikipedia.org/wiki/Archimedean_solidhttp://en.wikipedia.org/wiki/Archimedean_solidhttp://en.wikipedia.org/wiki/Prism_(geometry)http://en.wikipedia.org/wiki/Archimedean_solidhttp://en.wikipedia.org/wiki/John_Horton_Conwayhttp://en.wikipedia.org/wiki/John_Horton_Conwayhttp://en.wikipedia.org/wiki/Norman_Johnsonhttp://en.wikipedia.org/wiki/Norman_Johnsonhttp://en.wikipedia.org/wiki/Uniform_polychoron#Geometric_derivationshttp://en.wikipedia.org/wiki/Convex_uniform_honeycomb#Referenceshttp://en.wikipedia.org/wiki/Convex_uniform_honeycomb#External_linkshttp://en.wikipedia.org/wiki/Thorold_Gossethttp://en.wikipedia.org/wiki/Platonic_solidhttp://en.wikipedia.org/wiki/Alfredo_Andreinihttp://en.wikipedia.org/wiki/Norman_Johnson_(mathematician)http://en.wikipedia.org/wiki/Branko_Gr%C3%BCnbaumhttp://en.wikipedia.org/wiki/Convex_uniform_honeycomb#Gyrated_and_elongated_formshttp://en.wikipedia.org/w/index.php?title=I._Alexeyev&action=edit&redlink=1http://en.wikipedia.org/wiki/George_Olshevskyhttp://en.wikipedia.org/wiki/Uniform_polychoronhttp://en.wikipedia.org/wiki/Platonic_solidhttp://en.wikipedia.org/wiki/Archimedean_solidhttp://en.wikipedia.org/wiki/Prism_(geometry)http://en.wikipedia.org/wiki/Archimedean_solidhttp://en.wikipedia.org/wiki/John_Horton_Conwayhttp://en.wikipedia.org/wiki/Norman_Johnsonhttp://en.wikipedia.org/wiki/Uniform_polychoron#Geometric_derivations -
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Fundamental domains in a cubic element of three groups.
The fundamental infinite Coxeter groups for 3-space are:
1. The C~3, [4,3,4], cubic, (8 unique forms plus one alternation)
2. The B~3, [4,31,1], alternated cubic, (11 forms, 3 new)
3. The A~3 cyclic group, (5 forms, one new)
In addition there are 5 special honeycombs which don't have pure reflectional symmetryand are constructed from reflectional forms with elongation andgyration operations.
The total unique honeycombs above are 18.
The prismatic stacks from infinite Coxeter groups for 3-space are:
1. The C~2xI~
1, [4,4]x[] prismatic group, (2 new forms)
2. The H~2xI~
1, [6,3]x[] prismatic group, (7 unique forms)
3. The A~2xI~
1, []x[] prismatic group, (No new forms)
4. The I~1xI~
1xI~
1, []x[]x[] prismatic group, (These all become a
cubic honeycomb)
In addition there is one special elongatedform of the triangular prismatic honeycomb.
The total unique prismatic honeycombs above (excluding the cubic counted previously) are10.
Combining these counts, 18 and 10 gives us the total 28 uniform honeycombs.
The C~3, [4,3,4] group (cubic)
http://en.wikipedia.org/wiki/Coxeter_grouphttp://en.wikipedia.org/wiki/File:CDW_dot.pnghttp://en.wikipedia.org/wiki/File:CDW_infin.pnghttp://en.wikipedia.org/wiki/File:CDW_dot.pnghttp://en.wikipedia.org/wiki/File:CDW_2.pnghttp://en.wikipedia.org/wiki/File:CDW_dot.pnghttp://en.wikipedia.org/wiki/File:CDW_infin.pnghttp://en.wikipedia.org/wiki/File:CDW_dot.pnghttp://en.wikipedia.org/wiki/File:CDW_2.pnghttp://en.wikipedia.org/wiki/File:CDW_dot.pnghttp://en.wikipedia.org/wiki/File:CDW_infin.pnghttp://en.wikipedia.org/wiki/File:CDW_dot.pnghttp://en.wikipedia.org/wiki/File:CD_dot.pnghttp://en.wikipedia.org/wiki/File:CD_infin.pnghttp://en.wikipedia.org/wiki/File:CD_dot.pnghttp://en.wikipedia.org/wiki/File:CD_2.pnghttp://en.wikipedia.org/wiki/File:CD_righttriangle-000.pnghttp://en.wikipedia.org/wiki/File:CDW_dot.pnghttp://en.wikipedia.org/wiki/File:CDW_infin.pnghttp://en.wikipedia.org/wiki/File:CDW_dot.pnghttp://en.wikipedia.org/wiki/File:CDW_2.pnghttp://en.wikipedia.org/wiki/File:CDW_dot.pnghttp://en.wikipedia.org/wiki/File:CDW_3.pnghttp://en.wikipedia.org/wiki/File:CDW_dot.pnghttp://en.wikipedia.org/wiki/File:CDW_6.pnghttp://en.wikipedia.org/wiki/File:CDW_dot.pnghttp://en.wikipedia.org/wiki/File:CDW_dot.pnghttp://en.wikipedia.org/wiki/File:CDW_infin.pnghttp://en.wikipedia.org/wiki/File:CDW_dot.pnghttp://en.wikipedia.org/wiki/File:CDW_2.pnghttp://en.wikipedia.org/wiki/File:CDW_dot.pnghttp://en.wikipedia.org/wiki/File:CDW_4.pnghttp://en.wikipedia.org/wiki/File:CDW_dot.pnghttp://en.wikipedia.org/wiki/File:CDW_4.pnghttp://en.wikipedia.org/wiki/File:CDW_dot.pnghttp://en.wikipedia.org/wiki/File:CD_downbranch-00.pnghttp://en.wikipedia.org/wiki/File:CD_downbranch-33.pnghttp://en.wikipedia.org/wiki/File:CD_downbranch-00.pnghttp://en.wikipedia.org/wiki/File:CD_dot.pnghttp://en.wikipedia.org/wiki/File:CD_4.pnghttp://en.wikipedia.org/wiki/File:CD_3b.pnghttp://en.wikipedia.org/wiki/File:CD_downbranch-00.pnghttp://en.wikipedia.org/wiki/File:CD_3b.pnghttp://en.wikipedia.org/wiki/File:CD_dot.pnghttp://en.wikipedia.org/wiki/File:CDW_dot.pnghttp://en.wikipedia.org/wiki/File:CDW_4.pnghttp://en.wikipedia.org/wiki/File:CDW_dot.pnghttp://en.wikipedia.org/wiki/File:CDW_3.pnghttp://en.wikipedia.org/wiki/File:CDW_dot.pnghttp://en.wikipedia.org/wiki/File:CDW_4.pnghttp://en.wikipedia.org/wiki/File:CDW_dot.pnghttp://en.wikipedia.org/wiki/File:Coxeter-Dynkin_3-space_groups.pnghttp://en.wikipedia.org/wiki/Coxeter_group -
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The regular cubic honeycomb, represented by Schlfli symbol {4,3,4}, offers seven unique
derived uniform honeycombs via truncation operations. (One redundant form, theruncinated cubic honeycomb, is included for completeness though identical to the cubichoneycomb.)
Reference
Indices
Honeycomb
name
Coxeter-
Dynkin
and Schlfli
symbols
Cell counts/vertex
and positions in cubic
honeycomb
(0)(1) (2)
(3) Solids
(Partial)
Frames
(Perspective)Vertex figure
J11,15A1W1G22
cubic
t0{4,3,4}
(8)
(4.4.4)octahedron
J12,32A15W14G7
rectified cubic
t1{4,3,4}
(2)
(3.3.3.3)
(4)
(3.4.3.4)cuboid
J13A14W15G8
truncated
cubic
t0,1{4,3,4}
(1)
(3.3.3.3)
(4)
(3.8.8) square
pyramid
J14A17W12G9
cantellatedcubic
t0,2{4,3,4}
(1)
(3.4.3.4)
(2)
(4.4.4)
(2)
(3.4.4.4)wedge
J11,15
runcinated
cubic
(same asregularcubic)
t0,3{4,3,4}
(1)
(4.4.4)
(3)
(4.4.4)
(3)
(4.4.4)
(1)
(4.4.4)octahedron
J16A3W2G28
bitruncatedcubic
t1,2{4,3,4}
(2)
(4.6.6)
(2)
(4.6.6)
http://en.wikipedia.org/wiki/Coxeter-Dynkin_diagramhttp://en.wikipedia.org/wiki/Coxeter-Dynkin_diagramhttp://en.wikipedia.org/wiki/Schl%C3%A4fli_symbol#Extended_for_uniform_polychora_and_3-space_honeycombshttp://en.wikipedia.org/wiki/Vertex_figurehttp://en.wikipedia.org/wiki/Cubic_honeycombhttp://en.wikipedia.org/wiki/Cubehttp://en.wikipedia.org/wiki/Octahedronhttp://en.wikipedia.org/wiki/Rectified_cubic_honeycombhttp://en.wikipedia.org/wiki/Octahedronhttp://en.wikipedia.org/wiki/Cuboctahedronhttp://en.wikipedia.org/wiki/Cuboidhttp://en.wikipedia.org/wiki/Truncated_cubic_honeycombhttp://en.wikipedia.org/wiki/Truncated_cubic_honeycombhttp://en.wikipedia.org/wiki/Octahedronhttp://en.wikipedia.org/wiki/Truncated_cubehttp://en.wikipedia.org/wiki/Square_pyramidhttp://en.wikipedia.org/wiki/Square_pyramidhttp://en.wikipedia.org/wiki/Cantellated_cubic_honeycombhttp://en.wikipedia.org/wiki/Cantellated_cubic_honeycombhttp://en.wikipedia.org/wiki/Cuboctahedronhttp://en.wikipedia.org/wiki/Cubehttp://en.wikipedia.org/wiki/Small_rhombicuboctahedronhttp://en.wikipedia.org/wiki/Wedge_(geometry)http://en.wikipedia.org/wiki/Runcination_(geometry)http://en.wikipedia.org/wiki/Cubic_honeycombhttp://en.wikipedia.org/wiki/Cubic_honeycombhttp://en.wikipedia.org/wiki/Cubehttp://en.wikipedia.org/wiki/Cubehttp://en.wikipedia.org/wiki/Cubehttp://en.wikipedia.org/wiki/Cubehttp://en.wikipedia.org/wiki/Octahedronhttp://en.wikipedia.org/wiki/Bitruncated_cubic_honeycombhttp://en.wikipedia.org/wiki/Bitruncated_cubic_honeycombhttp://en.wikipedia.org/wiki/Truncated_octahedronhttp://en.wikipedia.org/wiki/Truncated_octahedronhttp://en.wikipedia.org/wiki/File:Bitruncated_cubic_honeycomb_verf.pnghttp://en.wikipedia.org/wiki/File:Bitruncated_cubic_tiling.pnghttp://en.wikipedia.org/wiki/File:Bitruncated_cubic_honeycomb.pnghttp://en.wikipedia.org/wiki/File:Truncated_octahedron.pnghttp://en.wikipedia.org/wiki/File:Truncated_octahedron.pnghttp://en.wikipedia.org/wiki/File:CDW_dot.pnghttp://en.wikipedia.org/wiki/File:CDW_4.pnghttp://en.wikipedia.org/wiki/File:CDW_ring.pnghttp://en.wikipedia.org/wiki/File:CDW_3.pnghttp://en.wikipedia.org/wiki/File:CDW_ring.pnghttp://en.wikipedia.org/wiki/File:CDW_4.pnghttp://en.wikipedia.org/wiki/File:CDW_dot.pnghttp://en.wikipedia.org/wiki/File:Runcinated_cubic_honeycomb_verf.pnghttp://en.wikipedia.org/wiki/File:Cubic_tiling.pnghttp://en.wikipedia.org/wiki/File:Runcinated_cubic_honeycomb.pnghttp://en.wikipedia.org/wiki/File:Hexahedron.pnghttp://en.wikipedia.org/wiki/File:Hexahedron.pnghttp://en.wikipedia.org/wiki/File:Hexahedron.pnghttp://en.wikipedia.org/wiki/File:Hexahedron.pnghttp://en.wikipedia.org/wiki/File:CDW_ring.pnghttp://en.wikipedia.org/wiki/File:CDW_4.pnghttp://en.wikipedia.org/wiki/File:CDW_dot.pnghttp://en.wikipedia.org/wiki/File:CDW_3.pnghttp://en.wikipedia.org/wiki/File:CDW_dot.pnghttp://en.wikipedia.org/wiki/File:CDW_4.pnghttp://en.wikipedia.org/wiki/File:CDW_ring.pnghttp://en.wikipedia.org/wiki/File:Cantellated_cubic_honeycomb_verf.pnghttp://en.wikipedia.org/wiki/File:Cantellated_cubic_tiling.pnghttp://en.wikipedia.org/wiki/File:Cantellated_cubic_honeycomb.jpghttp://en.wikipedia.org/wiki/File:Small_rhombicuboctahedron.pnghttp://en.wikipedia.org/wiki/File:Hexahedron.pnghttp://en.wikipedia.org/wiki/File:Cuboctahedron.pnghttp://en.wikipedia.org/wiki/File:CDW_dot.pnghttp://en.wikipedia.org/wiki/File:CDW_4.pnghttp://en.wikipedia.org/wiki/File:CDW_ring.pnghttp://en.wikipedia.org/wiki/File:CDW_3.pnghttp://en.wikipedia.org/wiki/File:CDW_dot.pnghttp://en.wikipedia.org/wiki/File:CDW_4.pnghttp://en.wikipedia.org/wiki/File:CDW_ring.pnghttp://en.wikipedia.org/wiki/File:Truncated_cubic_honeycomb_verf.pnghttp://en.wikipedia.org/wiki/File:Truncated_cubic_tiling.pnghttp://en.wikipedia.org/wiki/File:Truncated_cubic_honeycomb.pnghttp://en.wikipedia.org/wiki/File:Truncated_hexahedron.pnghttp://en.wikipedia.org/wiki/File:Octahedron.pnghttp://en.wikipedia.org/wiki/File:CDW_dot.pnghttp://en.wikipedia.org/wiki/File:CDW_4.pnghttp://en.wikipedia.org/wiki/File:CDW_dot.pnghttp://en.wikipedia.org/wiki/File:CDW_3.pnghttp://en.wikipedia.org/wiki/File:CDW_ring.pnghttp://en.wikipedia.org/wiki/File:CDW_4.pnghttp://en.wikipedia.org/wiki/File:CDW_ring.pnghttp://en.wikipedia.org/wiki/File:Rectified_cubic_honeycomb_verf.pnghttp://en.wikipedia.org/wiki/File:Rectified_cubic_tiling.pnghttp://en.wikipedia.org/wiki/File:Rectified_cubic_honeycomb.pnghttp://en.wikipedia.org/wiki/File:Cuboctahedron.pnghttp://en.wikipedia.org/wiki/File:Octahedron.pnghttp://en.wikipedia.org/wiki/File:CDW_dot.pnghttp://en.wikipedia.org/wiki/File:CDW_4.pnghttp://en.wikipedia.org/wiki/File:CDW_dot.pnghttp://en.wikipedia.org/wiki/File:CDW_3.pnghttp://en.wikipedia.org/wiki/File:CDW_ring.pnghttp://en.wikipedia.org/wiki/File:CDW_4.pnghttp://en.wikipedia.org/wiki/File:CDW_dot.pnghttp://en.wikipedia.org/wiki/File:Cubic_honeycomb_verf.pnghttp://en.wikipedia.org/wiki/File:Cubic_honeycomb.pnghttp://en.wikipedia.org/wiki/File:Partial_cubic_honeycomb.pnghttp://en.wikipedia.org/wiki/File:Hexahedron.pnghttp://en.wikipedia.org/wiki/File:CDW_dot.pnghttp://en.wikipedia.org/wiki/File:CDW_4.pnghttp://en.wikipedia.org/wiki/File:CDW_dot.pnghttp://en.wikipedia.org/wiki/File:CDW_3.pnghttp://en.wikipedia.org/wiki/File:CDW_dot.pnghttp://en.wikipedia.org/wiki/File:CDW_4.pnghttp://en.wikipedia.org/wiki/File:CDW_ring.pnghttp://en.wikipedia.org/wiki/File:CDW_dash.pnghttp://en.wikipedia.org/wiki/File:CDW_2c.pnghttp://en.wikipedia.org/wiki/File:CDW_dot.pnghttp://en.wikipedia.org/wiki/File:CDW_3.pnghttp://en.wikipedia.org/wiki/File:CDW_dot.pnghttp://en.wikipedia.org/wiki/File:CDW_4.pnghttp://en.wikipedia.org/wiki/File:CDW_dot.pnghttp://en.wikipedia.org/wiki/File:CDW_dot.pnghttp://en.wikipedia.org/wiki/File:CDW_2c.pnghttp://en.wikipedia.org/wiki/File:CDW_dash.pnghttp://en.wikipedia.org/wiki/File:CDW_2c.pnghttp://en.wikipedia.org/wiki/File:CDW_dot.pnghttp://en.wikipedia.org/wiki/File:CDW_4.pnghttp://en.wikipedia.org/wiki/File:CDW_dot.pnghttp://en.wikipedia.org/wiki/File:CDW_dot.pnghttp://en.wikipedia.org/wiki/File:CDW_4.pnghttp://en.wikipedia.org/wiki/File:CDW_dot.pnghttp://en.wikipedia.org/wiki/File:CDW_2c.pnghttp://en.wikipedia.org/wiki/File:CDW_dash.pnghttp://en.wikipedia.org/wiki/File:CDW_2c.pnghttp://en.wikipedia.org/wiki/File:CDW_dot.pnghttp://en.wikipedia.org/wiki/File:CDW_dot.pnghttp://en.wikipedia.org/wiki/File:CDW_4.pnghttp://en.wikipedia.org/wiki/File:CDW_dot.pnghttp://en.wikipedia.org/wiki/File:CDW_3.pnghttp://en.wikipedia.org/wiki/File:CDW_dot.pnghttp://en.wikipedia.org/wiki/File:CDW_2c.pnghttp://en.wikipedia.org/wiki/File:CDW_dash.pnghttp://en.wikipedia.org/wiki/Coxeter-Dynkin_diagramhttp://en.wikipedia.org/wiki/Coxeter-Dynkin_diagramhttp://en.wikipedia.org/wiki/Schl%C3%A4fli_symbol#Extended_for_uniform_polychora_and_3-space_honeycombshttp://en.wikipedia.org/wiki/Vertex_figurehttp://en.wikipedia.org/wiki/Cubic_honeycombhttp://en.wikipedia.org/wiki/Cubehttp://en.wikipedia.org/wiki/Octahedronhttp://en.wikipedia.org/wiki/Rectified_cubic_honeycombhttp://en.wikipedia.org/wiki/Octahedronhttp://en.wikipedia.org/wiki/Cuboctahedronhttp://en.wikipedia.org/wiki/Cuboidhttp://en.wikipedia.org/wiki/Truncated_cubic_honeycombhttp://en.wikipedia.org/wiki/Truncated_cubic_honeycombhttp://en.wikipedia.org/wiki/Octahedronhttp://en.wikipedia.org/wiki/Truncated_cubehttp://en.wikipedia.org/wiki/Square_pyramidhttp://en.wikipedia.org/wiki/Square_pyramidhttp://en.wikipedia.org/wiki/Cantellated_cubic_honeycombhttp://en.wikipedia.org/wiki/Cantellated_cubic_honeycombhttp://en.wikipedia.org/wiki/Cuboctahedronhttp://en.wikipedia.org/wiki/Cubehttp://en.wikipedia.org/wiki/Small_rhombicuboctahedronhttp://en.wikipedia.org/wiki/Wedge_(geometry)http://en.wikipedia.org/wiki/Runcination_(geometry)http://en.wikipedia.org/wiki/Cubic_honeycombhttp://en.wikipedia.org/wiki/Cubehttp://en.wikipedia.org/wiki/Cubehttp://en.wikipedia.org/wiki/Cubehttp://en.wikipedia.org/wiki/Cubehttp://en.wikipedia.org/wiki/Octahedronhttp://en.wikipedia.org/wiki/Bitruncated_cubic_honeycombhttp://en.wikipedia.org/wiki/Bitruncated_cubic_honeycombhttp://en.wikipedia.org/wiki/Truncated_octahedronhttp://en.wikipedia.org/wiki/Truncated_octahedron -
8/9/2019 PANALES CONVEXOS
5/26
(disphenoid
tetrahedron)
J17A18
W13G25
cantitruncatedcubic
t0,1,2{4,3,4}
(1)
(4.6.6)
(1)
(4.4.4)
(2)
(4.6.8) irregulartetrahedron
J18A19W19G20
runcitruncatedcubic
t0,1,3{4,3,4}
(1)
(3.4.4.4)
(1)
(4.4.4)
(2)
(4.4.8)
(1)
(3.8.8)oblique
trapezoidalpyramid
J19A22W18G27
omnitruncated
cubic
t0,1,2,3{4,3,4}
(1)
(4.6.8)
(1)
(4.4.8)
(1)
(4.4.8)
(1)
(4.6.8) irregular
tetrahedron
J21,31,51A2W9G1
alternated
cubic
h0{4,3,4}
(6)
(3.3.3.3)
(8)
(3.3.3)cuboctahedron
B~
4, h[4,3,4], [4,31,1
] group
The B~4 group offers 11 derived forms via truncation operations, four being unique uniformhoneycombs.
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.
Referenced
indices
Honeycomb
name
Coxeter-
Dynkin
diagram
Cells by location
(and count around each vertex)Solids
(Partial)
Frames
(Perspective)verte(0) (1) (0') (3)
http://en.wikipedia.org/wiki/Disphenoid_tetrahedronhttp://en.wikipedia.org/wiki/Disphenoid_tetrahedronhttp://en.wikipedia.org/wiki/Cantitruncated_cubic_honeycombhttp://en.wikipedia.org/wiki/Cantitruncated_cubic_honeycombhttp://en.wikipedia.org/wiki/Truncated_octahedronhttp://en.wikipedia.org/wiki/Cubehttp://en.wikipedia.org/wiki/Great_rhombicuboctahedronhttp://en.wikipedia.org/wiki/Tetrahedronhttp://en.wikipedia.org/wiki/Runcitruncated_cubic_honeycombhttp://en.wikipedia.org/wiki/Runcitruncated_cubic_honeycombhttp://en.wikipedia.org/wiki/Small_rhombicuboctahedronhttp://en.wikipedia.org/wiki/Cubehttp://en.wikipedia.org/wiki/Octagonal_prismhttp://en.wikipedia.org/wiki/Truncated_cubehttp://en.wikipedia.org/wiki/Omnitruncated_cubic_honeycombhttp://en.wikipedia.org/wiki/Omnitruncated_cubic_honeycombhttp://en.wikipedia.org/wiki/Great_rhombicuboctahedronhttp://en.wikipedia.org/wiki/Octagonal_prismhttp://en.wikipedia.org/wiki/Octagonal_prismhttp://en.wikipedia.org/wiki/Great_rhombicuboctahedronhttp://en.wikipedia.org/wiki/Tetrahedronhttp://en.wikipedia.org/wiki/Tetrahedral-octahedral_honeycombhttp://en.wikipedia.org/wiki/Tetrahedral-octahedral_honeycombhttp://en.wikipedia.org/wiki/Octahedronhttp://en.wikipedia.org/wiki/Tetrahedronhttp://en.wikipedia.org/wiki/Cuboctahedronhttp://en.wikipedia.org/wiki/Coxeter-Dynkin_diagramhttp://en.wikipedia.org/wiki/Coxeter-Dynkin_diagramhttp://en.wikipedia.org/wiki/Coxeter-Dynkin_diagramhttp://en.wikipedia.org/wiki/Vertex_figurehttp://en.wikipedia.org/wiki/File:CD_dash.pnghttp://en.wikipedia.org/wiki/File:CD_2c.pnghttp://en.wikipedia.org/wiki/File:CD_downbranch-00.pnghttp://en.wikipedia.org/wiki/File:CD_3b.pnghttp://en.wikipedia.org/wiki/File:CD_dot.pnghttp://en.wikipedia.org/wiki/File:CD_dot.pnghttp://en.wikipedia.org/wiki/File:CD_4.pnghttp://en.wikipedia.org/wiki/File:CD_downbranch-0dash.pnghttp://en.wikipedia.org/wiki/File:CD_3b.pnghttp://en.wikipedia.org/wiki/File:CD_dot.pnghttp://en.wikipedia.org/wiki/File:CD_dot.pnghttp://en.wikipedia.org/wiki/File:CD_2.pnghttp://en.wikipedia.org/wiki/File:CD_downbranch-dash0.pnghttp://en.wikipedia.org/wiki/File:CD_2c.pnghttp://en.wikipedia.org/wiki/File:CD_dot.pnghttp://en.wikipedia.org/wiki/File:CD_dot.pnghttp://en.wikipedia.org/wiki/File:CD_4.pnghttp://en.wikipedia.org/wiki/File:CD_3b.pnghttp://en.wikipedia.org/wiki/File:CD_downbranch-00.pnghttp://en.wikipedia.org/wiki/File:CD_2c.pnghttp://en.wikipedia.org/wiki/File:CD_dash.pnghttp://en.wikipedia.org/wiki/File:Alternated_cubic_honeycomb_verf.pnghttp://en.wikipedia.org/wiki/File:Alternated_cubic_tiling.pnghttp://en.wikipedia.org/wiki/File:Tetrahedral-octahedral_honeycomb.pnghttp://en.wikipedia.org/wiki/File:Tetrahedron.pnghttp://en.wikipedia.org/wiki/File:Octahedron.pnghttp://en.wikipedia.org/wiki/File:CDW_dot.pnghttp://en.wikipedia.org/wiki/File:CDW_4.pnghttp://en.wikipedia.org/wiki/File:CDW_dot.pnghttp://en.wikipedia.org/wiki/File:CDW_3.pnghttp://en.wikipedia.org/wiki/File:CDW_dot.pnghttp://en.wikipedia.org/wiki/File:CDW_4.pnghttp://en.wikipedia.org/wiki/File:CDW_hole.pnghttp://en.wikipedia.org/wiki/File:Omnitruncated_cubic_honeycomb_verf.pnghttp://en.wikipedia.org/wiki/File:Omnitruncated_cubic_tiling.pnghttp://en.wikipedia.org/wiki/File:Omnitruncated_cubic_honeycomb.jpghttp://en.wikipedia.org/wiki/File:Great_rhombicuboctahedron.pnghttp://en.wikipedia.org/wiki/File:Octagonal_prism.pnghttp://en.wikipedia.org/wiki/File:Octagonal_prism.pnghttp://en.wikipedia.org/wiki/File:Great_rhombicuboctahedron.pnghttp://en.wikipedia.org/wiki/File:CDW_ring.pnghttp://en.wikipedia.org/wiki/File:CDW_4.pnghttp://en.wikipedia.org/wiki/File:CDW_ring.pnghttp://en.wikipedia.org/wiki/File:CDW_3.pnghttp://en.wikipedia.org/wiki/File:CDW_ring.pnghttp://en.wikipedia.org/wiki/File:CDW_4.pnghttp://en.wikipedia.org/wiki/File:CDW_ring.pnghttp://en.wikipedia.org/wiki/File:Runcitruncated_cubic_honeycomb_verf.pnghttp://en.wikipedia.org/wiki/File:Runcitruncated_cubic_tiling.pnghttp://en.wikipedia.org/wiki/File:Runcitruncated_cubic_honeycomb.jpghttp://en.wikipedia.org/wiki/File:Truncated_hexahedron.pnghttp://en.wikipedia.org/wiki/File:Octagonal_prism.pnghttp://en.wikipedia.org/wiki/File:Hexahedron.pnghttp://en.wikipedia.org/wiki/File:Small_rhombicuboctahedron.pnghttp://en.wikipedia.org/wiki/File:CDW_ring.pnghttp://en.wikipedia.org/wiki/File:CDW_4.pnghttp://en.wikipedia.org/wiki/File:CDW_dot.pnghttp://en.wikipedia.org/wiki/File:CDW_3.pnghttp://en.wikipedia.org/wiki/File:CDW_ring.pnghttp://en.wikipedia.org/wiki/File:CDW_4.pnghttp://en.wikipedia.org/wiki/File:CDW_ring.pnghttp://en.wikipedia.org/wiki/File:Cantitruncated_cubic_honeycomb_verf.pnghttp://en.wikipedia.org/wiki/File:Cantitruncated_cubic_tiling.pnghttp://en.wikipedia.org/wiki/File:Cantitruncated_Cubic_Honeycomb.svghttp://en.wikipedia.org/wiki/File:Great_rhombicuboctahedron.pnghttp://en.wikipedia.org/wiki/File:Hexahedron.pnghttp://en.wikipedia.org/wiki/File:Truncated_octahedron.pnghttp://en.wikipedia.org/wiki/File:CDW_dot.pnghttp://en.wikipedia.org/wiki/File:CDW_4.pnghttp://en.wikipedia.org/wiki/File:CDW_ring.pnghttp://en.wikipedia.org/wiki/File:CDW_3.pnghttp://en.wikipedia.org/wiki/File:CDW_ring.pnghttp://en.wikipedia.org/wiki/File:CDW_4.pnghttp://en.wikipedia.org/wiki/File:CDW_ring.pnghttp://en.wikipedia.org/wiki/Disphenoid_tetrahedronhttp://en.wikipedia.org/wiki/Disphenoid_tetrahedronhttp://en.wikipedia.org/wiki/Cantitruncated_cubic_honeycombhttp://en.wikipedia.org/wiki/Cantitruncated_cubic_honeycombhttp://en.wikipedia.org/wiki/Truncated_octahedronhttp://en.wikipedia.org/wiki/Cubehttp://en.wikipedia.org/wiki/Great_rhombicuboctahedronhttp://en.wikipedia.org/wiki/Tetrahedronhttp://en.wikipedia.org/wiki/Runcitruncated_cubic_honeycombhttp://en.wikipedia.org/wiki/Runcitruncated_cubic_honeycombhttp://en.wikipedia.org/wiki/Small_rhombicuboctahedronhttp://en.wikipedia.org/wiki/Cubehttp://en.wikipedia.org/wiki/Octagonal_prismhttp://en.wikipedia.org/wiki/Truncated_cubehttp://en.wikipedia.org/wiki/Omnitruncated_cubic_honeycombhttp://en.wikipedia.org/wiki/Omnitruncated_cubic_honeycombhttp://en.wikipedia.org/wiki/Great_rhombicuboctahedronhttp://en.wikipedia.org/wiki/Octagonal_prismhttp://en.wikipedia.org/wiki/Octagonal_prismhttp://en.wikipedia.org/wiki/Great_rhombicuboctahedronhttp://en.wikipedia.org/wiki/Tetrahedronhttp://en.wikipedia.org/wiki/Tetrahedral-octahedral_honeycombhttp://en.wikipedia.org/wiki/Tetrahedral-octahedral_honeycombhttp://en.wikipedia.org/wiki/Octahedronhttp://en.wikipedia.org/wiki/Tetrahedronhttp://en.wikipedia.org/wiki/Cuboctahedronhttp://en.wikipedia.org/wiki/Coxeter-Dynkin_diagramhttp://en.wikipedia.org/wiki/Coxeter-Dynkin_diagramhttp://en.wikipedia.org/wiki/Coxeter-Dynkin_diagramhttp://en.wikipedia.org/wiki/Vertex_figure -
8/9/2019 PANALES CONVEXOS
6/26
J21,31,51A2W9G1
alternated cubic (6)
(3.3.3.3)(8)
(3.3.3)
cuboc
J22,34A21W17G10
truncatedalternated cubic (1)
(3.4.3.4) (2)
(4.6.6)(2)
(3.6.6)
J12,32A15W14G7
rectified cubic
(rectifiedalternate cubic) (2)
(3.4.3.4) (2)
(3.4.3.4)
(2)
(3.3.3.3)
cu
J12,32
A15W14G7
rectified cubic(cantellated
alternate cubic) (1)(3.3.3.3)
(1)(3.3.3.3)
(4)(3.4.3.4)
cu
J16A3W2G28
bitruncatedcubic
(cantitruncated
alternate cubic)(1)
(4.6.6) (1)
(4.6.6)(2)
(4.6.6)iso
tetra
J13
A14W15G8
truncated cubic
(bicantellatedalternate cubic) (2)
(3.8.8) (2)
(3.8.8)
(1)
(3.3.3.3)sq
pyr
J11,15A1W1G22
cubic
(trirectified
alternate cubic) (4)(4.4.4)
(4)(4.4.4)
octa
J23A16W11G5
runcinated
alternated cubic (1)cube
(3)(3.4.4.4)
(1)(3.3.3)
J14A17W12G9
cantellated cubic(runcicantellated
alternate cubic) (1)
(3.4.4.4)
(2
)
(4.4.4)
(1)
(3.4.4.4)
(1)
(3.4.3.4)
w
http://en.wikipedia.org/wiki/Tetrahedral-octahedral_honeycombhttp://en.wikipedia.org/wiki/Octahedronhttp://en.wikipedia.org/wiki/Tetrahedronhttp://en.wikipedia.org/wiki/Cuboctahedronhttp://en.wikipedia.org/wiki/Truncated_alternated_cubic_honeycombhttp://en.wikipedia.org/wiki/Truncated_alternated_cubic_honeycombhttp://en.wikipedia.org/wiki/Cuboctahedronhttp://en.wikipedia.org/wiki/Truncated_octahedronhttp://en.wikipedia.org/wiki/Truncated_tetrahedronhttp://en.wikipedia.org/wiki/Rectified_cubic_honeycombhttp://en.wikipedia.org/wiki/Cuboctahedronhttp://en.wikipedia.org/wiki/Cuboctahedronhttp://en.wikipedia.org/wiki/Octahedronhttp://en.wikipedia.org/wiki/Cuboidhttp://en.wikipedia.org/wiki/Rectified_cubic_honeycombhttp://en.wikipedia.org/wiki/Octahedronhttp://en.wikipedia.org/wiki/Octahedronhttp://en.wikipedia.org/wiki/Cuboctahedronhttp://en.wikipedia.org/wiki/Cuboidhttp://en.wikipedia.org/wiki/Bitruncated_cubic_honeycombhttp://en.wikipedia.org/wiki/Bitruncated_cubic_honeycombhttp://en.wikipedia.org/wiki/Truncated_octahedronhttp://en.wikipedia.org/wiki/Truncated_octahedronhttp://en.wikipedia.org/wiki/Truncated_octahedronhttp://en.wikipedia.org/wiki/Tetrahedronhttp://en.wikipedia.org/wiki/Truncated_cubic_honeycombhttp://en.wikipedia.org/wiki/Truncated_cubehttp://en.wikipedia.org/wiki/Truncated_cubehttp://en.wikipedia.org/wiki/Octahedronhttp://en.wikipedia.org/wiki/Square_pyramidhttp://en.wikipedia.org/wiki/Square_pyramidhttp://en.wikipedia.org/wiki/Cubic_honeycombhttp://en.wikipedia.org/wiki/Cubehttp://en.wikipedia.org/wiki/Cubehttp://en.wikipedia.org/wiki/Octahedronhttp://en.wikipedia.org/wiki/Runcinated_alternated_cubic_honeycombhttp://en.wikipedia.org/wiki/Runcinated_alternated_cubic_honeycombhttp://en.wikipedia.org/wiki/Cubehttp://en.wikipedia.org/wiki/Rhombicuboctahedronhttp://en.wikipedia.org/wiki/Tetrahedronhttp://en.wikipedia.org/wiki/Cantellated_cubic_honeycombhttp://en.wikipedia.org/wiki/Small_rhombicuboctahedronhttp://en.wikipedia.org/wiki/Cubehttp://en.wikipedia.org/wiki/Small_rhombicuboctahedronhttp://en.wikipedia.org/wiki/Cuboctahedronhttp://en.wikipedia.org/wiki/Wedge_(geometry)http://en.wikipedia.org/wiki/File:Runcicantellated_alternate_cubic_honeycomb_verf.pnghttp://en.wikipedia.org/wiki/File:Cantellated_cubic_tiling.pnghttp://en.wikipedia.org/wiki/File:Cantellated_cubic_honeycomb.jpghttp://en.wikipedia.org/wiki/File:Uniform_polyhedron-33-t02.pnghttp://en.wikipedia.org/wiki/File:Small_rhombicuboctahedron.pnghttp://en.wikipedia.org/wiki/File:Uniform_polyhedron_222-t012.pnghttp://en.wikipedia.org/wiki/File:Small_rhombicuboctahedron.pnghttp://en.wikipedia.org/wiki/File:CD_ring.pnghttp://en.wikipedia.org/wiki/File:CD_4.pnghttp://en.wikipedia.org/wiki/File:CD_3b.pnghttp://en.wikipedia.org/wiki/File:CD_downbranch-01.pnghttp://en.wikipedia.org/wiki/File:CD_3b.pnghttp://en.wikipedia.org/wiki/File:CD_ring.pnghttp://en.wikipedia.org/wiki/File:Runcinated_alternated_cubic_honeycomb_verf.pnghttp://en.wikipedia.org/wiki/File:Runcinated_alternated_cubic_tiling.pnghttp://en.wikipedia.org/wiki/File:Runcinated_alternated_cubic_honeycomb.jpghttp://en.wikipedia.org/wiki/File:Tetrahedron.pnghttp://en.wikipedia.org/wiki/File:Small_rhombicuboctahedron.pnghttp://en.wikipedia.org/wiki/File:Hexahedron.pnghttp://en.wikipedia.org/wiki/File:CD_ring.pnghttp://en.wikipedia.org/wiki/File:CD_4.pnghttp://en.wikipedia.org/wiki/File:CD_3b.pnghttp://en.wikipedia.org/wiki/File:CD_downbranch-00.pnghttp://en.wikipedia.org/wiki/File:CD_3b.pnghttp://en.wikipedia.org/wiki/File:CD_ring.pnghttp://en.wikipedia.org/wiki/File:Cubic_honeycomb_verf.pnghttp://en.wikipedia.org/wiki/File:Cubic_tiling.pnghttp://en.wikipedia.org/wiki/File:Bicolor_cubic_honeycomb.pnghttp://en.wikipedia.org/wiki/File:Hexahedron.pnghttp://en.wikipedia.org/wiki/File:Hexahedron.pnghttp://en.wikipedia.org/wiki/File:CD_ring.pnghttp://en.wikipedia.org/wiki/File:CD_4.pnghttp://en.wikipedia.org/wiki/File:CD_3b.pnghttp://en.wikipedia.org/wiki/File:CD_downbranch-00.pnghttp://en.wikipedia.org/wiki/File:CD_3b.pnghttp://en.wikipedia.org/wiki/File:CD_dot.pnghttp://en.wikipedia.org/wiki/File:Bicantellated_alternate_cubic_honeycomb_verf.pnghttp://en.wikipedia.org/wiki/File:Truncated_cubic_tiling.pnghttp://en.wikipedia.org/wiki/File:Truncated_cubic_honeycomb2.pnghttp://en.wikipedia.org/wiki/File:Uniform_polyhedron-33-t1.pnghttp://en.wikipedia.org/wiki/File:Truncated_hexahedron.pnghttp://en.wikipedia.org/wiki/File:Truncated_hexahedron.pnghttp://en.wikipedia.org/wiki/File:CD_ring.pnghttp://en.wikipedia.org/wiki/File:CD_4.pnghttp://en.wikipedia.org/wiki/File:CD_3b.pnghttp://en.wikipedia.org/wiki/File:CD_downbranch-10.pnghttp://en.wikipedia.org/wiki/File:CD_3b.pnghttp://en.wikipedia.org/wiki/File:CD_dot.pnghttp://en.wikipedia.org/wiki/File:Cantitruncated_alternate_cubic_honeycomb_verf.pnghttp://en.wikipedia.org/wiki/File:Bitruncated_cubic_tiling.pnghttp://en.wikipedia.org/wiki/File:Bitruncated_cubic_honeycomb3.pnghttp://en.wikipedia.org/wiki/File:Uniform_polyhedron-33-t012.pnghttp://en.wikipedia.org/wiki/File:Truncated_octahedron.pnghttp://en.wikipedia.org/wiki/File:Truncated_octahedron.pnghttp://en.wikipedia.org/wiki/File:CD_dot.pnghttp://en.wikipedia.org/wiki/File:CD_4.pnghttp://en.wikipedia.org/wiki/File:CD_3b.pnghttp://en.wikipedia.org/wiki/File:CD_downbranch-11.pnghttp://en.wikipedia.org/wiki/File:CD_3b.pnghttp://en.wikipedia.org/wiki/File:CD_ring.pnghttp://en.wikipedia.org/wiki/File:Cantellated_alternate_cubic_honeycomb_verf.pnghttp://en.wikipedia.org/wiki/File:Rectified_cubic_tiling.pnghttp://en.wikipedia.org/wiki/File:Rectified_cubic_honeycomb3.pnghttp://en.wikipedia.org/wiki/File:Uniform_polyhedron-33-t02.pnghttp://en.wikipedia.org/wiki/File:Octahedron.pnghttp://en.wikipedia.org/wiki/File:Octahedron.pnghttp://en.wikipedia.org/wiki/File:CD_dot.pnghttp://en.wikipedia.org/wiki/File:CD_4.pnghttp://en.wikipedia.org/wiki/File:CD_3b.pnghttp://en.wikipedia.org/wiki/File:CD_downbranch-01.pnghttp://en.wikipedia.org/wiki/File:CD_3b.pnghttp://en.wikipedia.org/wiki/File:CD_ring.pnghttp://en.wikipedia.org/wiki/File:Rectified_alternate_cubic_honeycomb_verf.pnghttp://en.wikipedia.org/wiki/File:Rectified_cubic_tiling.pnghttp://en.wikipedia.org/wiki/File:Rectified_cubic_honeycomb4.pnghttp://en.wikipedia.org/wiki/File:Uniform_polyhedron-33-t1.pnghttp://en.wikipedia.org/wiki/File:Cuboctahedron.pnghttp://en.wikipedia.org/wiki/File:Cuboctahedron.pnghttp://en.wikipedia.org/wiki/File:CD_dot.pnghttp://en.wikipedia.org/wiki/File:CD_4.pnghttp://en.wikipedia.org/wiki/File:CD_3b.pnghttp://en.wikipedia.org/wiki/File:CD_downbranch-10.pnghttp://en.wikipedia.org/wiki/File:CD_3b.pnghttp://en.wikipedia.org/wiki/File:CD_dot.pnghttp://en.wikipedia.org/wiki/File:Truncated_alternated_cubic_honeycomb_verf.pnghttp://en.wikipedia.org/wiki/File:Truncated_alternated_cubic_tiling.pnghttp://en.wikipedia.org/wiki/File:Truncated_Alternated_Cubic_Honeycomb.svghttp://en.wikipedia.org/wiki/File:Truncated_tetrahedron.pnghttp://en.wikipedia.org/wiki/File:Truncated_octahedron.pnghttp://en.wikipedia.org/wiki/File:Cuboctahedron.pnghttp://en.wikipedia.org/wiki/File:CD_dot.pnghttp://en.wikipedia.org/wiki/File:CD_4.pnghttp://en.wikipedia.org/wiki/File:CD_3b.pnghttp://en.wikipedia.org/wiki/File:CD_downbranch-10.pnghttp://en.wikipedia.org/wiki/File:CD_3b.pnghttp://en.wikipedia.org/wiki/File:CD_ring.pnghttp://en.wikipedia.org/wiki/File:Alternated_cubic_honeycomb_verf.pnghttp://en.wikipedia.org/wiki/File:Alternated_cubic_tiling.pnghttp://en.wikipedia.org/wiki/File:Tetrahedral-octahedral_honeycomb.pnghttp://en.wikipedia.org/wiki/File:Tetrahedron.pnghttp://en.wikipedia.org/wiki/File:Octahedron.pnghttp://en.wikipedia.org/wiki/File:CD_dot.pnghttp://en.wikipedia.org/wiki/File:CD_4.pnghttp://en.wikipedia.org/wiki/File:CD_3b.pnghttp://en.wikipedia.org/wiki/File:CD_downbranch-00.pnghttp://en.wikipedia.org/wiki/File:CD_3b.pnghttp://en.wikipedia.org/wiki/File:CD_ring.pnghttp://en.wikipedia.org/wiki/Tetrahedral-octahedral_honeycombhttp://en.wikipedia.org/wiki/Octahedronhttp://en.wikipedia.org/wiki/Tetrahedronhttp://en.wikipedia.org/wiki/Cuboctahedronhttp://en.wikipedia.org/wiki/Truncated_alternated_cubic_honeycombhttp://en.wikipedia.org/wiki/Truncated_alternated_cubic_honeycombhttp://en.wikipedia.org/wiki/Cuboctahedronhttp://en.wikipedia.org/wiki/Truncated_octahedronhttp://en.wikipedia.org/wiki/Truncated_tetrahedronhttp://en.wikipedia.org/wiki/Rectified_cubic_honeycombhttp://en.wikipedia.org/wiki/Cuboctahedronhttp://en.wikipedia.org/wiki/Cuboctahedronhttp://en.wikipedia.org/wiki/Octahedronhttp://en.wikipedia.org/wiki/Cuboidhttp://en.wikipedia.org/wiki/Rectified_cubic_honeycombhttp://en.wikipedia.org/wiki/Octahedronhttp://en.wikipedia.org/wiki/Octahedronhttp://en.wikipedia.org/wiki/Cuboctahedronhttp://en.wikipedia.org/wiki/Cuboidhttp://en.wikipedia.org/wiki/Bitruncated_cubic_honeycombhttp://en.wikipedia.org/wiki/Bitruncated_cubic_honeycombhttp://en.wikipedia.org/wiki/Truncated_octahedronhttp://en.wikipedia.org/wiki/Truncated_octahedronhttp://en.wikipedia.org/wiki/Truncated_octahedronhttp://en.wikipedia.org/wiki/Tetrahedronhttp://en.wikipedia.org/wiki/Truncated_cubic_honeycombhttp://en.wikipedia.org/wiki/Truncated_cubehttp://en.wikipedia.org/wiki/Truncated_cubehttp://en.wikipedia.org/wiki/Octahedronhttp://en.wikipedia.org/wiki/Square_pyramidhttp://en.wikipedia.org/wiki/Square_pyramidhttp://en.wikipedia.org/wiki/Cubic_honeycombhttp://en.wikipedia.org/wiki/Cubehttp://en.wikipedia.org/wiki/Cubehttp://en.wikipedia.org/wiki/Octahedronhttp://en.wikipedia.org/wiki/Runcinated_alternated_cubic_honeycombhttp://en.wikipedia.org/wiki/Runcinated_alternated_cubic_honeycombhttp://en.wikipedia.org/wiki/Cubehttp://en.wikipedia.org/wiki/Rhombicuboctahedronhttp://en.wikipedia.org/wiki/Tetrahedronhttp://en.wikipedia.org/wiki/Cantellated_cubic_honeycombhttp://en.wikipedia.org/wiki/Small_rhombicuboctahedronhttp://en.wikipedia.org/wiki/Cubehttp://en.wikipedia.org/wiki/Small_rhombicuboctahedronhttp://en.wikipedia.org/wiki/Cuboctahedronhttp://en.wikipedia.org/wiki/Wedge_(geometry) -
8/9/2019 PANALES CONVEXOS
7/26
J24A20W16G21
cantitruncated
alternated cubic
(orruncitruncated
alternate cubic)
(1)
(3.8.8) (2)
(4.6.8)
(1)
(3.6.6)
J17A18W13G25
cantitruncated
cubic
(omnitruncated
alternated cubic)(1)
(4.6.8)
(1)(4.4.4)
(1)(4.6.8)
(1)(4.6.6)
irretetra
[edit] A~3 group
There are 5 forms constructed from the A
~
3 group, only the quarter cubic honeycomb isunique.
Referenced
indices
Honeycomb
name
Coxeter-
Dynkin
diagram
Cells by location
(and count around each vertex)Solids
(Partial)
Frames
(Perspective)vertex figu(0) (1) (2) (3)
J21,31,51A2W9
G1
alternated
cubic (4)
(3.3.3)
(6)
(3.3.3.3)
(4)
(3.3.3)cuboctahedr
J12,32A15W14G7
rectified
cubic (2)(3.4.3.4)
(1)(3.3.3.3)
(2)(3.4.3.4)
(1)(3.3.3.3)
cuboid
J25,33A13
W10
G6
quartercubic (1)
(3.3.3)
(1)
(3.3.3)
(3)
(3.6.6)
(3)
(3.6.6) Elongatedtriangularantiprism
J22,34A21W17G10
truncated
alternated
cubic
(1)
(3.6.6)
(1)
(3.4.3.4)
(1)
(3.6.6)
(2)
(4.6.6)
Rectangula
http://en.wikipedia.org/wiki/Cantitruncated_alternated_cubic_honeycombhttp://en.wikipedia.org/wiki/Cantitruncated_alternated_cubic_honeycombhttp://en.wikipedia.org/wiki/Truncated_cubehttp://en.wikipedia.org/wiki/Truncated_cuboctahedronhttp://en.wikipedia.org/wiki/Truncated_tetrahedronhttp://en.wikipedia.org/wiki/Cantitruncated_cubic_honeycombhttp://en.wikipedia.org/wiki/Cantitruncated_cubic_honeycombhttp://en.wikipedia.org/wiki/Great_rhombicuboctahedronhttp://en.wikipedia.org/wiki/Cubehttp://en.wikipedia.org/wiki/Great_rhombicuboctahedronhttp://en.wikipedia.org/wiki/Truncated_octahedronhttp://en.wikipedia.org/wiki/Tetrahedronhttp://en.wikipedia.org/w/index.php?title=Convex_uniform_honeycomb&action=edit§ion=6http://en.wikipedia.org/wiki/Coxeter-Dynkin_diagramhttp://en.wikipedia.org/wiki/Coxeter-Dynkin_diagramhttp://en.wikipedia.org/wiki/Coxeter-Dynkin_diagramhttp://en.wikipedia.org/wiki/Vertex_figurehttp://en.wikipedia.org/wiki/Tetrahedral-octahedral_honeycombhttp://en.wikipedia.org/wiki/Tetrahedral-octahedral_honeycombhttp://en.wikipedia.org/wiki/Tetrahedronhttp://en.wikipedia.org/wiki/Octahedronhttp://en.wikipedia.org/wiki/Tetrahedronhttp://en.wikipedia.org/wiki/Cuboctahedronhttp://en.wikipedia.org/wiki/Rectified_cubic_honeycombhttp://en.wikipedia.org/wiki/Rectified_cubic_honeycombhttp://en.wikipedia.org/wiki/Cuboctahedronhttp://en.wikipedia.org/wiki/Octahedronhttp://en.wikipedia.org/wiki/Cuboctahedronhttp://en.wikipedia.org/wiki/Octahedronhttp://en.wikipedia.org/wiki/Cuboidhttp://en.wikipedia.org/wiki/Quarter_cubic_honeycombhttp://en.wikipedia.org/wiki/Quarter_cubic_honeycombhttp://en.wikipedia.org/wiki/Tetrahedronhttp://en.wikipedia.org/wiki/Tetrahedronhttp://en.wikipedia.org/wiki/Truncated_tetrahedronhttp://en.wikipedia.org/wiki/Truncated_tetrahedronhttp://en.wikipedia.org/wiki/Truncated_alternated_cubic_honeycombhttp://en.wikipedia.org/wiki/Truncated_alternated_cubic_honeycombhttp://en.wikipedia.org/wiki/Truncated_alternated_cubic_honeycombhttp://en.wikipedia.org/wiki/Truncated_tetrahedronhttp://en.wikipedia.org/wiki/Cuboctahedronhttp://en.wikipedia.org/wiki/Truncated_tetrahedronhttp://en.wikipedia.org/wiki/Truncated_octahedronhttp://en.wikipedia.org/wiki/File:T012_quarter_cubic_honeycomb_verf.pnghttp://en.wikipedia.org/wiki/File:Truncated_alternated_cubic_tiling.pnghttp://en.wikipedia.org/wiki/File:Truncated_Alternated_Cubic_Honeycomb.svghttp://en.wikipedia.org/wiki/File:Uniform_polyhedron-33-t012.pnghttp://en.wikipedia.org/wiki/File:Truncated_tetrahedron.pnghttp://en.wikipedia.org/wiki/File:Uniform_polyhedron-33-t02.pnghttp://en.wikipedia.org/wiki/File:Truncated_tetrahedron.pnghttp://en.wikipedia.org/wiki/File:CD_downbranch-10-right-3.pnghttp://en.wikipedia.org/wiki/File:CD_downbranch-33b.pnghttp://en.wikipedia.org/wiki/File:CD_downbranch-11-left-3.pnghttp://en.wikipedia.org/wiki/File:T01_quarter_cubic_honeycomb_verf.pnghttp://en.wikipedia.org/wiki/File:Bitruncated_alternated_cubic_tiling.pnghttp://en.wikipedia.org/wiki/File:Quarter_cubic_honeycomb.pnghttp://en.wikipedia.org/wiki/File:Truncated_tetrahedron.pnghttp://en.wikipedia.org/wiki/File:Truncated_tetrahedron.pnghttp://en.wikipedia.org/wiki/File:Tetrahedron.pnghttp://en.wikipedia.org/wiki/File:Tetrahedron.pnghttp://en.wikipedia.org/wiki/File:CD_downbranch-10-right-3.pnghttp://en.wikipedia.org/wiki/File:CD_downbranch-33b.pnghttp://en.wikipedia.org/wiki/File:CD_downbranch-10-left-3.pnghttp://en.wikipedia.org/wiki/File:T02_quarter_cubic_honeycomb_verf.pnghttp://en.wikipedia.org/wiki/File:Rectified_cubic_tiling.pnghttp://en.wikipedia.org/wiki/File:Rectified_cubic_honeycomb2.pnghttp://en.wikipedia.org/wiki/File:Uniform_polyhedron-33-t1.pnghttp://en.wikipedia.org/wiki/File:Uniform_polyhedron-33-t02.pnghttp://en.wikipedia.org/wiki/File:Uniform_polyhedron-33-t1.pnghttp://en.wikipedia.org/wiki/File:Uniform_polyhedron-33-t02.pnghttp://en.wikipedia.org/wiki/File:CD_downbranch-01-right-3.pnghttp://en.wikipedia.org/wiki/File:CD_downbranch-33b.pnghttp://en.wikipedia.org/wiki/File:CD_downbranch-10-left-3.pnghttp://en.wikipedia.org/wiki/File:Alternated_cubic_honeycomb_verf.pnghttp://en.wikipedia.org/wiki/File:Alternated_cubic_tiling.pnghttp://en.wikipedia.org/wiki/File:Tetrahedral-octahedral_honeycomb2.pnghttp://en.wikipedia.org/wiki/File:Uniform_polyhedron-33-t2.pnghttp://en.wikipedia.org/wiki/File:Uniform_polyhedron-33-t1.pnghttp://en.wikipedia.org/wiki/File:Uniform_polyhedron-33-t0.pnghttp://en.wikipedia.org/wiki/File:CD_downbranch-00.pnghttp://en.wikipedia.org/wiki/File:CD_downbranch-33b.pnghttp://en.wikipedia.org/wiki/File:CD_downbranch-10-left-3.pnghttp://en.wikipedia.org/wiki/File:CD_downbranch-00.pnghttp://en.wikipedia.org/wiki/File:CD_3.pnghttp://en.wikipedia.org/wiki/File:CD_downbranch-0dash.pnghttp://en.wikipedia.org/wiki/File:CD_downbranch-0dash.pnghttp://en.wikipedia.org/wiki/File:CD_3.pnghttp://en.wikipedia.org/wiki/File:CD_downbranch-00.pnghttp://en.wikipedia.org/wiki/File:CD_downbranch-dash0.pnghttp://en.wikipedia.org/wiki/File:CD_downbranch-23.pnghttp://en.wikipedia.org/wiki/File:CD_downbranch-00.pnghttp://en.wikipedia.org/wiki/File:CD_downbranch-00.pnghttp://en.wikipedia.org/wiki/File:CD_downbranch-23.pnghttp://en.wikipedia.org/wiki/File:CD_downbranch-dash0.pnghttp://en.wikipedia.org/wiki/File:Omnitruncated_alternated_cubic_honeycomb_verf.pnghttp://en.wikipedia.org/wiki/File:Cantitruncated_cubic_tiling.pnghttp://en.wikipedia.org/wiki/File:Cantitruncated_Cubic_Honeycomb.svghttp://en.wikipedia.org/wiki/File:Uniform_polyhedron-33-t012.pnghttp://en.wikipedia.org/wiki/File:Great_rhombicuboctahedron.pnghttp://en.wikipedia.org/wiki/File:Uniform_polyhedron_222-t012.pnghttp://en.wikipedia.org/wiki/File:Great_rhombicuboctahedron.pnghttp://en.wikipedia.org/wiki/File:CD_ring.pnghttp://en.wikipedia.org/wiki/File:CD_4.pnghttp://en.wikipedia.org/wiki/File:CD_3b.pnghttp://en.wikipedia.org/wiki/File:CD_downbranch-11.pnghttp://en.wikipedia.org/wiki/File:CD_3b.pnghttp://en.wikipedia.org/wiki/File:CD_ring.pnghttp://en.wikipedia.org/wiki/File:Runcitruncated_alternate_cubic_honeycomb_verf.pnghttp://en.wikipedia.org/wiki/File:Cantitruncated_alternated_cubic_tiling.pnghttp://en.wikipedia.org/wiki/File:Cantitruncated_alternated_cubic_honeycomb.jpghttp://en.wikipedia.org/wiki/File:Truncated_tetrahedron.pnghttp://en.wikipedia.org/wiki/File:Great_rhombicuboctahedron.pnghttp://en.wikipedia.org/wiki/File:Truncated_hexahedron.pnghttp://en.wikipedia.org/wiki/File:CD_ring.pnghttp://en.wikipedia.org/wiki/File:CD_4.pnghttp://en.wikipedia.org/wiki/File:CD_3b.pnghttp://en.wikipedia.org/wiki/File:CD_downbranch-10.pnghttp://en.wikipedia.org/wiki/File:CD_3b.pnghttp://en.wikipedia.org/wiki/File:CD_ring.pnghttp://en.wikipedia.org/wiki/Cantitruncated_alternated_cubic_honeycombhttp://en.wikipedia.org/wiki/Cantitruncated_alternated_cubic_honeycombhttp://en.wikipedia.org/wiki/Truncated_cubehttp://en.wikipedia.org/wiki/Truncated_cuboctahedronhttp://en.wikipedia.org/wiki/Truncated_tetrahedronhttp://en.wikipedia.org/wiki/Cantitruncated_cubic_honeycombhttp://en.wikipedia.org/wiki/Cantitruncated_cubic_honeycombhttp://en.wikipedia.org/wiki/Great_rhombicuboctahedronhttp://en.wikipedia.org/wiki/Cubehttp://en.wikipedia.org/wiki/Great_rhombicuboctahedronhttp://en.wikipedia.org/wiki/Truncated_octahedronhttp://en.wikipedia.org/wiki/Tetrahedronhttp://en.wikipedia.org/w/index.php?title=Convex_uniform_honeycomb&action=edit§ion=6http://en.wikipedia.org/wiki/Coxeter-Dynkin_diagramhttp://en.wikipedia.org/wiki/Coxeter-Dynkin_diagramhttp://en.wikipedia.org/wiki/Coxeter-Dynkin_diagramhttp://en.wikipedia.org/wiki/Vertex_figurehttp://en.wikipedia.org/wiki/Tetrahedral-octahedral_honeycombhttp://en.wikipedia.org/wiki/Tetrahedral-octahedral_honeycombhttp://en.wikipedia.org/wiki/Tetrahedronhttp://en.wikipedia.org/wiki/Octahedronhttp://en.wikipedia.org/wiki/Tetrahedronhttp://en.wikipedia.org/wiki/Cuboctahedronhttp://en.wikipedia.org/wiki/Rectified_cubic_honeycombhttp://en.wikipedia.org/wiki/Rectified_cubic_honeycombhttp://en.wikipedia.org/wiki/Cuboctahedronhttp://en.wikipedia.org/wiki/Octahedronhttp://en.wikipedia.org/wiki/Cuboctahedronhttp://en.wikipedia.org/wiki/Octahedronhttp://en.wikipedia.org/wiki/Cuboidhttp://en.wikipedia.org/wiki/Quarter_cubic_honeycombhttp://en.wikipedia.org/wiki/Quarter_cubic_honeycombhttp://en.wikipedia.org/wiki/Tetrahedronhttp://en.wikipedia.org/wiki/Tetrahedronhttp://en.wikipedia.org/wiki/Truncated_tetrahedronhttp://en.wikipedia.org/wiki/Truncated_tetrahedronhttp://en.wikipedia.org/wiki/Truncated_alternated_cubic_honeycombhttp://en.wikipedia.org/wiki/Truncated_alternated_cubic_honeycombhttp://en.wikipedia.org/wiki/Truncated_alternated_cubic_honeycombhttp://en.wikipedia.org/wiki/Truncated_tetrahedronhttp://en.wikipedia.org/wiki/Cuboctahedronhttp://en.wikipedia.org/wiki/Truncated_tetrahedronhttp://en.wikipedia.org/wiki/Truncated_octahedron -
8/9/2019 PANALES CONVEXOS
8/26
pyramid
J16A3W2
G28
bitruncatedcubic (1)
(4.6.6)
(1)
(4.6.6)
(1)
(4.6.6)
(1)
(4.6.6)
isoscelestetrahedron
Nonwythoffian forms (gyrated and elongated)
Three more uniform honeycombs are generated by breaking one or another of the abovehoneycombs 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 elongatedform, each prism meets a tetrahedron at one triangular end
and an octahedron at the other. In thegyroelongatedform, prisms that meet tetrahedra atboth 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 prismatictilings, respectively, by inserting layers of cubes.
Referenced
indicessymbol
Honeycomb
name
cell types
(# at each
vertex)
Solids
(Partial)
Frames
(Perspective)vertex figure
J52A2'G2
h{4,3,4}:ggyrated
alternated
cubic
tetrahedron(8)
octahedron
(6) triangular
orthobicupola
J61A?G3
h{4,3,4}:ge
gyroelongated
alternated
cubic
triangular
prism (6)tetrahedron
(4)
octahedron(3)
-
J62A?G4
h{4,3,4}:eelongatedalternated
cubic
triangular
prism (6)tetrahedron
(4)
octahedron
(3)
http://en.wikipedia.org/wiki/Bitruncated_cubic_honeycombhttp://en.wikipedia.org/wiki/Bitruncated_cubic_honeycombhttp://en.wikipedia.org/wiki/Truncated_octahedronhttp://en.wikipedia.org/wiki/Truncated_octahedronhttp://en.wikipedia.org/wiki/Truncated_octahedronhttp://en.wikipedia.org/wiki/Truncated_octahedronhttp://en.wikipedia.org/wiki/Tetrahedronhttp://en.wikipedia.org/wiki/Vertex_figurehttp://en.wikipedia.org/wiki/Gyrated_alternated_cubic_honeycombhttp://en.wikipedia.org/wiki/Gyrated_alternated_cubic_honeycombhttp://en.wikipedia.org/wiki/Gyrated_alternated_cubic_honeycombhttp://en.wikipedia.org/wiki/Tetrahedronhttp://en.wikipedia.org/wiki/Octahedronhttp://en.wikipedia.org/wiki/Triangular_orthobicupolahttp://en.wikipedia.org/wiki/Triangular_orthobicupolahttp://en.wikipedia.org/wiki/Gyroelongated_alternated_cubic_honeycombhttp://en.wikipedia.org/wiki/Gyroelongated_alternated_cubic_honeycombhttp://en.wikipedia.org/wiki/Gyroelongated_alternated_cubic_honeycombhttp://en.wikipedia.org/wiki/Triangular_prismhttp://en.wikipedia.org/wiki/Triangular_prismhttp://en.wikipedia.org/wiki/Tetrahedronhttp://en.wikipedia.org/wiki/Octahedronhttp://en.wikipedia.org/wiki/Elongated_alternated_cubic_honeycombhttp://en.wikipedia.org/wiki/Elongated_alternated_cubic_honeycombhttp://en.wikipedia.org/wiki/Elongated_alternated_cubic_honeycombhttp://en.wikipedia.org/wiki/Triangular_prismhttp://en.wikipedia.org/wiki/Triangular_prismhttp://en.wikipedia.org/wiki/Tetrahedronhttp://en.wikipedia.org/wiki/File:Elongated_alternated_cubic_honeycomb.pnghttp://en.wikipedia.org/wiki/Octahedronhttp://en.wikipedia.org/wiki/File:VF-extended_alternated_cubic.pnghttp://en.wikipedia.org/wiki/File:Elongated_alternated_cubic_tiling.pnghttp://en.wikipedia.org/wiki/File:Elongated_alternated_cubic_honeycomb.pnghttp://en.wikipedia.org/wiki/File:Gyroelongated_alternated_cubic_tiling.pnghttp://en.wikipedia.org/wiki/File:Gyroelongated_alternated_cubic_honeycomb.pnghttp://en.wikipedia.org/wiki/File:VF-gyrated_alternated_cubic.pnghttp://en.wikipedia.org/wiki/File:Gyrated_alternated_cubic.pnghttp://en.wikipedia.org/wiki/File:Gyrated_alternated_cubic.jpghttp://en.wikipedia.org/wiki/File:T0123_quarter_cubic_honeycomb_verf.pnghttp://en.wikipedia.org/wiki/File:Bitruncated_cubic_tiling.pnghttp://en.wikipedia.org/wiki/File:Bitruncated_cubic_honeycomb2.pnghttp://en.wikipedia.org/wiki/File:Uniform_polyhedron-33-t012.pnghttp://en.wikipedia.org/wiki/File:Uniform_polyhedron-33-t012.pnghttp://en.wikipedia.org/wiki/File:Uniform_polyhedron-33-t012.pnghttp://en.wikipedia.org/wiki/File:Uniform_polyhedron-33-t012.pnghttp://en.wikipedia.org/wiki/File:CD_downbranch-11-right-3.pnghttp://en.wikipedia.org/wiki/File:CD_downbranch-33b.pnghttp://en.wikipedia.org/wiki/File:CD_downbranch-11-left-3.pnghttp://en.wikipedia.org/wiki/Bitruncated_cubic_honeycombhttp://en.wikipedia.org/wiki/Bitruncated_cubic_honeycombhttp://en.wikipedia.org/wiki/Truncated_octahedronhttp://en.wikipedia.org/wiki/Truncated_octahedronhttp://en.wikipedia.org/wiki/Truncated_octahedronhttp://en.wikipedia.org/wiki/Truncated_octahedronhttp://en.wikipedia.org/wiki/Tetrahedronhttp://en.wikipedia.org/wiki/Vertex_figurehttp://en.wikipedia.org/wiki/Gyrated_alternated_cubic_honeycombhttp://en.wikipedia.org/wiki/Gyrated_alternated_cubic_honeycombhttp://en.wikipedia.org/wiki/Gyrated_alternated_cubic_honeycombhttp://en.wikipedia.org/wiki/Tetrahedronhttp://en.wikipedia.org/wiki/Octahedronhttp://en.wikipedia.org/wiki/Triangular_orthobicupolahttp://en.wikipedia.org/wiki/Triangular_orthobicupolahttp://en.wikipedia.org/wiki/Gyroelongated_alternated_cubic_honeycombhttp://en.wikipedia.org/wiki/Gyroelongated_alternated_cubic_honeycombhttp://en.wikipedia.org/wiki/Gyroelongated_alternated_cubic_honeycombhttp://en.wikipedia.org/wiki/Triangular_prismhttp://en.wikipedia.org/wiki/Triangular_prismhttp://en.wikipedia.org/wiki/Tetrahedronhttp://en.wikipedia.org/wiki/Octahedronhttp://en.wikipedia.org/wiki/Elongated_alternated_cubic_honeycombhttp://en.wikipedia.org/wiki/Elongated_alternated_cubic_honeycombhttp://en.wikipedia.org/wiki/Elongated_alternated_cubic_honeycombhttp://en.wikipedia.org/wiki/Triangular_prismhttp://en.wikipedia.org/wiki/Triangular_prismhttp://en.wikipedia.org/wiki/Tetrahedronhttp://en.wikipedia.org/wiki/Octahedron -
8/9/2019 PANALES CONVEXOS
9/26
J63A?G12
{3,6}:g x{}
gyrated
triangularprismatic
triangularprism (12)
J64A?G15
{3,6}:ge x{}
gyroelongatedtriangularprismatic
triangularprism (6)cube (4)
[edit] 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 vertexfigure of each is an irregularbipyramid whose faces areisosceles triangles.
[edit] The C
~
2xI
~
1(), [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 eachform.
Indices
Coxeter-
Dynkin
and Schlfli
symbols
Honeycomb namePlane
tiling
Solids
(Partial)Tiling
J11,15A1G22
{4,4} x {}Cubic
(Square prismatic)(4.4.4.4)
J45A6G24
t0,1{4,4} x
{}
Truncated/Bitruncated square
prismatic(4.8.8)
J11,15A1G22
t1{4,4} x {}
Cubic
(Rectified square prismatic)(4.4.4.4)
J11,15A1G22
t0,2{4,4} x
{}
Cubic
(Cantellated square prismatic)(4.4.4.4)
http://en.wikipedia.org/wiki/Gyrated_triangular_prismatic_honeycombhttp://en.wikipedia.org/wiki/Gyrated_triangular_prismatic_honeycombhttp://en.wikipedia.org/wiki/Gyrated_triangular_prismatic_honeycombhttp://en.wikipedia.org/wiki/Triangular_prismhttp://en.wikipedia.org/wiki/Triangular_prismhttp://en.wikipedia.org/wiki/Gyroelongated_triangular_prismatic_honeycombhttp://en.wikipedia.org/wiki/Gyroelongated_triangular_prismatic_honeycombhttp://en.wikipedia.org/wiki/Gyroelongated_triangular_prismatic_honeycombhttp://en.wikipedia.org/wiki/Triangular_prismhttp://en.wikipedia.org/wiki/Triangular_prismhttp://en.wikipedia.org/wiki/Cubehttp://en.wikipedia.org/w/index.php?title=Convex_uniform_honeycomb&action=edit§ion=8http://en.wikipedia.org/wiki/Tiling_by_regular_polygonshttp://en.wikipedia.org/wiki/Vertex_figurehttp://en.wikipedia.org/wiki/Vertex_figurehttp://en.wikipedia.org/wiki/Bipyramidhttp://en.wikipedia.org/wiki/Isosceles_trianglehttp://en.wikipedia.org/wiki/Isosceles_trianglehttp://en.wikipedia.org/w/index.php?title=Convex_uniform_honeycomb&action=edit§ion=9http://en.wikipedia.org/wiki/Coxeter-Dynkin_diagramhttp://en.wikipedia.org/wiki/Coxeter-Dynkin_diagramhttp://en.wikipedia.org/wiki/Schl%C3%A4fli_symbol#Extended_for_uniform_polychora_and_3-space_honeycombshttp://en.wikipedia.org/wiki/Cubic_honeycombhttp://en.wikipedia.org/wiki/Square_tilinghttp://en.wikipedia.org/wiki/Truncated_square_prismatic_honeycombhttp://en.wikipedia.org/wiki/Truncated_square_prismatic_honeycombhttp://en.wikipedia.org/wiki/Truncated_square_tilinghttp://en.wikipedia.org/wiki/Cubic_honeycombhttp://en.wikipedia.org/wiki/Square_tilinghttp://en.wikipedia.org/wiki/Cubic_honeycombhttp://en.wikipedia.org/wiki/Square_tilinghttp://en.wikipedia.org/wiki/File:Uniform_tiling_44-t02.pnghttp://en.wikipedia.org/wiki/File:Partial_cubic_honeycomb.pnghttp://en.wikipedia.org/wiki/File:CDW_dot.pnghttp://en.wikipedia.org/wiki/File:CDW_infin.pnghttp://en.wikipedia.org/wiki/File:CDW_ring.pnghttp://en.wikipedia.org/wiki/File:CDW_2.pnghttp://en.wikipedia.org/wiki/File:CDW_ring.pnghttp://en.wikipedia.org/wiki/File:CDW_4.pnghttp://en.wikipedia.org/wiki/File:CDW_dot.pnghttp://en.wikipedia.org/wiki/File:CDW_4.pnghttp://en.wikipedia.org/wiki/File:CDW_ring.pnghttp://en.wikipedia.org/wiki/File:Uniform_tiling_44-t1.pnghttp://en.wikipedia.org/wiki/File:Square_prismatic_2-color_honeycomb.pnghttp://en.wikipedia.org/wiki/File:CDW_dot.pnghttp://en.wikipedia.org/wiki/File:CDW_infin.pnghttp://en.wikipedia.org/wiki/File:CDW_ring.pnghttp://en.wikipedia.org/wiki/File:CDW_2.pnghttp://en.wikipedia.org/wiki/File:CDW_dot.pnghttp://en.wikipedia.org/wiki/File:CDW_4.pnghttp://en.wikipedia.org/wiki/File:CDW_ring.pnghttp://en.wikipedia.org/wiki/File:CDW_4.pnghttp://en.wikipedia.org/wiki/File:CDW_dot.pnghttp://en.wikipedia.org/wiki/File:Uniform_tiling_44-t01.pnghttp://en.wikipedia.org/wiki/File:Truncated_square_prismatic_honeycomb.pnghttp://en.wikipedia.org/wiki/File:CDW_dot.pnghttp://en.wikipedia.org/wiki/File:CDW_infin.pnghttp://en.wikipedia.org/wiki/File:CDW_ring.pnghttp://en.wikipedia.org/wiki/File:CDW_2.pnghttp://en.wikipedia.org/wiki/File:CDW_dot.pnghttp://en.wikipedia.org/wiki/File:CDW_4.pnghttp://en.wikipedia.org/wiki/File:CDW_ring.pnghttp://en.wikipedia.org/wiki/File:CDW_4.pnghttp://en.wikipedia.org/wiki/File:CDW_ring.pnghttp://en.wikipedia.org/wiki/File:Uniform_tiling_44-t0.pnghttp://en.wikipedia.org/wiki/File:Partial_cubic_honeycomb.pnghttp://en.wikipedia.org/wiki/File:CDW_dot.pnghttp://en.wikipedia.org/wiki/File:CDW_infin.pnghttp://en.wikipedia.org/wiki/File:CDW_ring.pnghttp://en.wikipedia.org/wiki/File:CDW_2.pnghttp://en.wikipedia.org/wiki/File:CDW_dot.pnghttp://en.wikipedia.org/wiki/File:CDW_4.pnghttp://en.wikipedia.org/wiki/File:CDW_dot.pnghttp://en.wikipedia.org/wiki/File:CDW_4.pnghttp://en.wikipedia.org/wiki/File:CDW_ring.pnghttp://en.wikipedia.org/wiki/File:VF-prismatic_extended_triangular.pnghttp://en.wikipedia.org/wiki/File:Gyroelongated_triangular_prismatic_tiling.pnghttp://en.wikipedia.org/wiki/File:Gyroelongated_triangular_prismatic_honeycomb.pnghttp://en.wikipedia.org/wiki/File:VF-gyrated_prismatic_triangular.pnghttp://en.wikipedia.org/wiki/File:Gyrated_triangular_prismatic_tiling.pnghttp://en.wikipedia.org/wiki/File:Gyrated_triangular_prismatic_honeycomb.pnghttp://en.wikipedia.org/wiki/Gyrated_triangular_prismatic_honeycombhttp://en.wikipedia.org/wiki/Gyrated_triangular_prismatic_honeycombhttp://en.wikipedia.org/wiki/Gyrated_triangular_prismatic_honeycombhttp://en.wikipedia.org/wiki/Triangular_prismhttp://en.wikipedia.org/wiki/Triangular_prismhttp://en.wikipedia.org/wiki/Gyroelongated_triangular_prismatic_honeycombhttp://en.wikipedia.org/wiki/Gyroelongated_triangular_prismatic_honeycombhttp://en.wikipedia.org/wiki/Gyroelongated_triangular_prismatic_honeycombhttp://en.wikipedia.org/wiki/Triangular_prismhttp://en.wikipedia.org/wiki/Triangular_prismhttp://en.wikipedia.org/wiki/Cubehttp://en.wikipedia.org/w/index.php?title=Convex_uniform_honeycomb&action=edit§ion=8http://en.wikipedia.org/wiki/Tiling_by_regular_polygonshttp://en.wikipedia.org/wiki/Vertex_figurehttp://en.wikipedia.org/wiki/Vertex_figurehttp://en.wikipedia.org/wiki/Bipyramidhttp://en.wikipedia.org/wiki/Isosceles_trianglehttp://en.wikipedia.org/w/index.php?title=Convex_uniform_honeycomb&action=edit§ion=9http://en.wikipedia.org/wiki/Coxeter-Dynkin_diagramhttp://en.wikipedia.org/wiki/Coxeter-Dynkin_diagramhttp://en.wikipedia.org/wiki/Schl%C3%A4fli_symbol#Extended_for_uniform_polychora_and_3-space_honeycombshttp://en.wikipedia.org/wiki/Cubic_honeycombhttp://en.wikipedia.org/wiki/Square_tilinghttp://en.wikipedia.org/wiki/Truncated_square_prismatic_honeycombhttp://en.wikipedia.org/wiki/Truncated_square_prismatic_honeycombhttp://en.wikipedia.org/wiki/Truncated_square_tilinghttp://en.wikipedia.org/wiki/Cubic_honeycombhttp://en.wikipedia.org/wiki/Square_tilinghttp://en.wikipedia.org/wiki/Cubic_honeycombhttp://en.wikipedia.org/wiki/Square_tiling -
8/9/2019 PANALES CONVEXOS
10/26
J45A6G24
t0,1,2{4,4} x
{}
Truncated square prismatic
(Omnitruncated squareprismatic)
(4.8.8)
J44A11G14 s{4,4} x {}
Snub square prismatic (3.3.4.3.4)
[edit] The H~2xI~1(), [6,3] x [] prismatic group
Indices
Coxeter-
Dynkin
and Schlfli
symbols
Honeycomb namePlane
tiling
Solids
(Partial)Tiling
J42A5G26
t0{6,3} x {}Hexagonal prismatic (63)
J46A7G19
t0,1{6,3} x {}Truncated hexagonal prismatic (3.12.12)
J43A8
G18 t1{6,3} x {}
Trihexagonal prismatic (3.6.3.6)
J42A5G26
t1,2{6,3} x {}
Truncated triangular
prismatic
Hexagonal prismatic
(6.6.6)
J41A4G11
t2{6,3} x {}Triangular prismatic (36)
J47A9G16
t0,2{6,3} x {}
Rhombi-trihexagonal
prismatic(3.4.6.4)
http://en.wikipedia.org/wiki/Truncated_square_prismatic_honeycombhttp://en.wikipedia.org/wiki/Truncated_square_tilinghttp://en.wikipedia.org/wiki/Snub_square_prismatic_honeycombhttp://en.wikipedia.org/wiki/Snub_square_tilinghttp://en.wikipedia.org/w/index.php?title=Convex_uniform_honeycomb&action=edit§ion=10http://en.wikipedia.org/wiki/Coxeter-Dynkin_diagramhttp://en.wikipedia.org/wiki/Coxeter-Dynkin_diagramhttp://en.wikipedia.org/wiki/Schl%C3%A4fli_symbol#Extended_for_uniform_polychora_and_3-space_honeycombshttp://en.wikipedia.org/wiki/Hexagonal_prismatic_honeycombhttp://en.wikipedia.org/wiki/Hexagonal_tilinghttp://en.wikipedia.org/wiki/Hexagonal_tilinghttp://en.wikipedia.org/wiki/Hexagonal_tilinghttp://en.wikipedia.org/wiki/Hexagonal_tilinghttp://en.wikipedia.org/wiki/Truncated_hexagonal_prismatic_honeycombhttp://en.wikipedia.org/wiki/Truncated_hexagonal_tilinghttp://en.wikipedia.org/wiki/Triangular-hexagonal_prismatic_honeycombhttp://en.wikipedia.org/wiki/Trihexagonal_tilinghttp://en.wikipedia.org/wiki/Hexagonal_prismatic_honeycombhttp://en.wikipedia.org/wiki/Hexagonal_tilinghttp://en.wikipedia.org/wiki/Triangular_prismatic_honeycombhttp://en.wikipedia.org/wiki/Triangular_tilinghttp://en.wikipedia.org/wiki/Triangular_tilinghttp://en.wikipedia.org/wiki/Triangular_tilinghttp://en.wikipedia.org/wiki/Triangular_tilinghttp://en.wikipedia.org/wiki/Rhombitriangular-hexagonal_prismatic_honeycombhttp://en.wikipedia.org/wiki/Rhombitriangular-hexagonal_prismatic_honeycombhttp://en.wikipedia.org/wiki/Small_rhombitrihexagonal_tilinghttp://en.wikipedia.org/wiki/File:Uniform_tiling_63-t02.pnghttp://en.wikipedia.org/wiki/File:Rhombitriangular-hexagonal_prismatic_honeycomb.pnghttp://en.wikipedia.org/wiki/File:CDW_dot.pnghttp://en.wikipedia.org/wiki/File:CDW_infin.pnghttp://en.wikipedia.org/wiki/File:CDW_ring.pnghttp://en.wikipedia.org/wiki/File:CDW_2.pnghttp://en.wikipedia.org/wiki/File:CDW_ring.pnghttp://en.wikipedia.org/wiki/File:CDW_3.pnghttp://en.wikipedia.org/wiki/File:CDW_dot.pnghttp://en.wikipedia.org/wiki/File:CDW_6.pnghttp://en.wikipedia.org/wiki/File:CDW_ring.pnghttp://en.wikipedia.org/wiki/File:Uniform_tiling_63-t2.pnghttp://en.wikipedia.org/wiki/File:Triangular_prismatic_honeycomb.pnghttp://en.wikipedia.org/wiki/File:CDW_dot.pnghttp://en.wikipedia.org/wiki/File:CDW_infin.pnghttp://en.wikipedia.org/wiki/File:CDW_ring.pnghttp://en.wikipedia.org/wiki/File:CDW_2.pnghttp://en.wikipedia.org/wiki/File:CDW_ring.pnghttp://en.wikipedia.org/wiki/File:CDW_3.pnghttp://en.wikipedia.org/wiki/File:CDW_dot.pnghttp://en.wikipedia.org/wiki/File:CDW_6.pnghttp://en.wikipedia.org/wiki/File:CDW_dot.pnghttp://en.wikipedia.org/wiki/File:Uniform_tiling_63-t12.pnghttp://en.wikipedia.org/wiki/File:Truncated_triangular_prismatic_honeycomb.pnghttp://en.wikipedia.org/wiki/File:CDW_dot.pnghttp://en.wikipedia.org/wiki/File:CDW_infin.pnghttp://en.wikipedia.org/wiki/File:CDW_ring.pnghttp://en.wikipedia.org/wiki/File:CDW_2.pnghttp://en.wikipedia.org/wiki/File:CDW_ring.pnghttp://en.wikipedia.org/wiki/File:CDW_3.pnghttp://en.wikipedia.org/wiki/File:CDW_ring.pnghttp://en.wikipedia.org/wiki/File:CDW_6.pnghttp://en.wikipedia.org/wiki/File:CDW_dot.pnghttp://en.wikipedia.org/wiki/File:Uniform_tiling_63-t1.pnghttp://en.wikipedia.org/wiki/File:Triangular-hexagonal_prismatic_honeycomb.pnghttp://en.wikipedia.org/wiki/File:CDW_dot.pnghttp://en.wikipedia.org/wiki/File:CDW_infin.pnghttp://en.wikipedia.org/wiki/File:CDW_ring.pnghttp://en.wikipedia.org/wiki/File:CDW_2.pnghttp://en.wikipedia.org/wiki/File:CDW_dot.pnghttp://en.wikipedia.org/wiki/File:CDW_3.pnghttp://en.wikipedia.org/wiki/File:CDW_ring.pnghttp://en.wikipedia.org/wiki/File:CDW_6.pnghttp://en.wikipedia.org/wiki/File:CDW_dot.pnghttp://en.wikipedia.org/wiki/File:Uniform_tiling_63-t01.pnghttp://en.wikipedia.org/wiki/File:Truncated_hexagonal_prismatic_honeycomb.pnghttp://en.wikipedia.org/wiki/File:CDW_dot.pnghttp://en.wikipedia.org/wiki/File:CDW_infin.pnghttp://en.wikipedia.org/wiki/File:CDW_ring.pnghttp://en.wikipedia.org/wiki/File:CDW_2.pnghttp://en.wikipedia.org/wiki/File:CDW_dot.pnghttp://en.wikipedia.org/wiki/File:CDW_3.pnghttp://en.wikipedia.org/wiki/File:CDW_ring.pnghttp://en.wikipedia.org/wiki/File:CDW_6.pnghttp://en.wikipedia.org/wiki/File:CDW_ring.pnghttp://en.wikipedia.org/wiki/File:Uniform_tiling_63-t0.pnghttp://en.wikipedia.org/wiki/File:Hexagonal_prismatic_honeycomb.pnghttp://en.wikipedia.org/wiki/File:CDW_dot.pnghttp://en.wikipedia.org/wiki/File:CDW_infin.pnghttp://en.wikipedia.org/wiki/File:CDW_ring.pnghttp://en.wikipedia.org/wiki/File:CDW_2.pnghttp://en.wikipedia.org/wiki/File:CDW_dot.pnghttp://en.wikipedia.org/wiki/File:CDW_3.pnghttp://en.wikipedia.org/wiki/File:CDW_dot.pnghttp://en.wikipedia.org/wiki/File:CDW_6.pnghttp://en.wikipedia.org/wiki/File:CDW_ring.pnghttp://en.wikipedia.org/wiki/File:Uniform_tiling_44-snub.pnghttp://en.wikipedia.org/wiki/File:Snub_square_prismatic_honeycomb.pnghttp://en.wikipedia.org/wiki/File:CDW_dot.pnghttp://en.wikipedia.org/wiki/File:CDW_infin.pnghttp://en.wikipedia.org/wiki/File:CDW_ring.pnghttp://en.wikipedia.org/wiki/File:CDW_2.pnghttp://en.wikipedia.org/wiki/File:CDW_hole.pnghttp://en.wikipedia.org/wiki/File:CDW_4.pnghttp://en.wikipedia.org/wiki/File:CDW_hole.pnghttp://en.wikipedia.org/wiki/File:CDW_4.pnghttp://en.wikipedia.org/wiki/File:CDW_hole.pnghttp://en.wikipedia.org/wiki/File:Uniform_tiling_44-t012.pnghttp://en.wikipedia.org/wiki/File:Truncated_square_prismatic_honeycomb.pnghttp://en.wikipedia.org/wiki/File:CDW_dot.pnghttp://en.wikipedia.org/wiki/File:CDW_infin.pnghttp://en.wikipedia.org/wiki/File:CDW_ring.pnghttp://en.wikipedia.org/wiki/File:CDW_2.pnghttp://en.wikipedia.org/wiki/File:CDW_ring.pnghttp://en.wikipedia.org/wiki/File:CDW_4.pnghttp://en.wikipedia.org/wiki/File:CDW_ring.pnghttp://en.wikipedia.org/wiki/File:CDW_4.pnghttp://en.wikipedia.org/wiki/File:CDW_ring.pnghttp://en.wikipedia.org/wiki/Truncated_square_prismatic_honeycombhttp://en.wikipedia.org/wiki/Truncated_square_tilinghttp://en.wikipedia.org/wiki/Snub_square_prismatic_honeycombhttp://en.wikipedia.org/wiki/Snub_square_tilinghttp://en.wikipedia.org/w/index.php?title=Convex_uniform_honeycomb&action=edit§ion=10http://en.wikipedia.org/wiki/Coxeter-Dynkin_diagramhttp://en.wikipedia.org/wiki/Coxeter-Dynkin_diagramhttp://en.wikipedia.org/wiki/Schl%C3%A4fli_symbol#Extended_for_uniform_polychora_and_3-space_honeycombshttp://en.wikipedia.org/wiki/Hexagonal_prismatic_honeycombhttp://en.wikipedia.org/wiki/Hexagonal_tilinghttp://en.wikipedia.org/wiki/Truncated_hexagonal_prismatic_honeycombhttp://en.wikipedia.org/wiki/Truncated_hexagonal_tilinghttp://en.wikipedia.org/wiki/Triangular-hexagonal_prismatic_honeycombhttp://en.wikipedia.org/wiki/Trihexagonal_tilinghttp://en.wikipedia.org/wiki/Hexagonal_prismatic_honeycombhttp://en.wikipedia.org/wiki/Hexagonal_tilinghttp://en.wikipedia.org/wiki/Triangular_prismatic_honeycombhttp://en.wikipedia.org/wiki/Triangular_tilinghttp://en.wikipedia.org/wiki/Rhombitriangular-hexagonal_prismatic_honeycombhttp://en.wikipedia.org/wiki/Rhombitriangular-hexagonal_prismatic_honeycombhttp://en.wikipedia.org/wiki/Small_rhombitrihexagonal_tiling -
8/9/2019 PANALES CONVEXOS
11/26
J49A10G23
t0,1,2{6,3} x
{}
Omnitruncated trihexagonalprismatic
(4.6.12)
J48A12G17
s{6,3} x {}Snub trihexagonal prismatic (3.3.3.3.6)
J65A11'G13
{3,6}:e x {} elongated triangular prismatic (3.3.3.4.4)
[edit] Examples
All 28 of these tessellations are found in crystal arrangements.[citation needed]
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 byBuckminster Fuller(who called it the octet truss and patented it in the 1940s). [2][3][4]
[5]. Octet trusses are now among the most common types of truss used in construction.
[edit] Noncompact forms
Ifcells are allowed to beuniform tilings, more uniform honeycombs can be defined:
Families:
C~2xA1: [4,4]x[ ] Cubic prismatic slab honeycomb (3 forms)
H~2xA1: [6,3]x[ ] Tri-hexagonal prismatic slab honeycomb (8 forms)
A~2xA1: []x[ ] triangular prismatic slab(No new forms)
I~1xA1xA1: []x[ ]x[ ] = Cubic column honeycomb (1 form)
I2(p)xI~
1: [p]x[] Prismatic column honeycomb
I
~
1xI
~
1xA1: []x[]x[ ] = [4,4]x[ ] - = (Same as cubic slabhoneycomb family)
http://en.wikipedia.org/wiki/Omnitruncated_triangular-hexagonal_prismatic_honeycombhttp://en.wikipedia.org/wiki/Omnitruncated_triangular-hexagonal_prismatic_honeycombhttp://en.wikipedia.org/wiki/Great_rhombitrihexagonal_tilinghttp://en.wikipedia.org/wiki/Snub_triangular-hexagonal_prismatic_honeycombhttp://en.wikipedia.org/wiki/Snub_hexagonal_tilinghttp://en.wikipedia.org/wiki/Elongated_triangular_prismatic_honeycombhttp://en.wikipedia.org/wiki/Elongated_triangular_tilinghttp://en.wikipedia.org/w/index.php?title=Convex_uniform_honeycomb&action=edit§ion=11http://en.wikipedia.org/wiki/Crystalhttp://en.wikipedia.org/wiki/Wikipedia:Citation_neededhttp://en.wikipedia.org/wiki/Wikipedia:Citation_neededhttp://en.wikipedia.org/wiki/Wikipedia:Citation_neededhttp://en.wikipedia.org/wiki/Tetrahedral-octahedral_honeycombhttp://en.wikipedia.org/wiki/Close-packinghttp://en.wikipedia.org/wiki/Trusshttp://en.wikipedia.org/wiki/Alexander_Graham_Bellhttp://en.wikipedia.org/wiki/Buckminster_Fullerhttp://en.wikipedia.org/wiki/Octet_trusshttp://tabletoptelephone.com/~hopspage/Fuller.htmlhttp://members.cruzio.com/~devarco/energy.htmhttp://www.n55.dk/MANUALS/DISCUSSIONS/OTHER_TEXTS/CM_TEXT.htmlhttp://www.cjfearnley.com/fuller-faq-2.htmlhttp://en.wikipedia.org/w/index.php?title=Convex_uniform_honeycomb&action=edit§ion=12http://en.wikipedia.org/wiki/Cell_(mathematics)http://en.wikipedia.org/wiki/Uniform_tilinghttp://en.wikipedia.org/wiki/Uniform_tilinghttp://en.wikipedia.org/wiki/File:CDW_dot.pnghttp://en.wikipedia.org/w/index.php?title=Cubic_prismatic_slab_honeycomb&action=edit&redlink=1http://en.wikipedia.org/w/index.php?title=Tri-hexagonal_prismatic_slab_honeycomb&action=edit&redlink=1http://en.wikipedia.org/w/index.php?title=Triangular_prismatic_slab&action=edit&redlink=1http://en.wikipedia.org/w/index.php?title=Triangular_prismatic_slab&action=edit&redlink=1http://en.wikipedia.org/w/index.php?title=Cubic_column_honeycomb&action=edit&redlink=1http://en.wikipedia.org/w/index.php?title=Prismatic_column_honeycomb&action=edit&redlink=1http://en.wikipedia.org/wiki/File:CDW_dot.pnghttp://en.wikipedia.org/wiki/File:CDW_2.pnghttp://en.wikipedia.org/wiki/File:CDW_dot.pnghttp://en.wikipedia.org/wiki/File:CDW_4.pnghttp://en.wikipedia.org/wiki/File:CDW_dot.pnghttp://en.wikipedia.org/wiki/File:CDW_4.pnghttp://en.wikipedia.org/wiki/File:CDW_dot.pnghttp://en.wikipedia.org/wiki/File:CDW_dot.pnghttp://en.wikipedia.org/wiki/File:CDW_2.pnghttp://en.wikipedia.org/wiki/File:CDW_dot.pnghttp://en.wikipedia.org/wiki/File:CDW_infin.pnghttp://en.wikipedia.org/wiki/File:CDW_dot.pnghttp://en.wikipedia.org/wiki/File:CDW_2.pnghttp://en.wikipedia.org/wiki/File:CDW_dot.pnghttp://en.wikipedia.org/wiki/File:CDW_infin.pnghttp://en.wikipedia.org/wiki/File:CDW_dot.pnghttp://en.wikipedia.org/wiki/File:CDW_dot.pnghttp://en.wikipedia.org/wiki/File:CDW_infin.pnghttp://en.wikipedia.org/wiki/File:CDW_dot.pnghttp://en.wikipedia.org/wiki/File:CDW_2.pnghttp://en.wikipedia.org/wiki/File:CDW_dot.pnghttp://en.wikipedia.org/wiki/File:CDW_p.pnghttp://en.wikipedia.org/wiki/File:CDW_dot.pnghttp://en.wikipedia.org/wiki/File:CDW_dot.pnghttp://en.wikipedia.org/wiki/File:CDW_4.pnghttp://en.wikipedia.org/wiki/File:CDW_dot.pnghttp://en.wikipedia.org/wiki/File:CDW_2.pnghttp://en.wikipedia.org/wiki/File:CDW_dot.pnghttp://en.wikipedia.org/wiki/File:CDW_infin.pnghttp://en.wikipedia.org/wiki/File:CDW_dot.pnghttp://en.wikipedia.org/wiki/File:CDW_dot.pnghttp://en.wikipedia.org/wiki/File:CDW_2.pnghttp://en.wikipedia.org/wiki/File:CDW_dot.pnghttp://en.wikipedia.org/wiki/File:CDW_2.pnghttp://en.wikipedia.org/wiki/File:CDW_dot.pnghttp://en.wikipedia.org/wiki/File:CDW_infin.pnghttp://en.wikipedia.org/wiki/File:CDW_dot.pnghttp://en.wikipedia.org/wiki/File:CD_dot.pnghttp://en.wikipedia.org/wiki/File:CD_2.pnghttp://en.wikipedia.org/wiki/File:CD_righttriangle-000.pnghttp://en.wikipedia.org/wiki/File:CDW_dot.pnghttp://en.wikipedia.org/wiki/File:CDW_2.pnghttp://en.wikipedia.org/wiki/File:CDW_dot.pnghttp://en.wikipedia.org/wiki/File:CDW_3.pnghttp://en.wikipedia.org/wiki/File:CDW_dot.pnghttp://en.wikipedia.org/wiki/File:CDW_6.pnghttp://en.wikipedia.org/wiki/File:CDW_dot.pnghttp://en.wikipedia.org/wiki/File:CDW_dot.pnghttp://en.wikipedia.org/wiki/File:CDW_2.pnghttp://en.wikipedia.org/wiki/File:CDW_dot.pnghttp://en.wikipedia.org/wiki/File:CDW_4.pnghttp://en.wikipedia.org/wiki/File:CDW_dot.pnghttp://en.wikipedia.org/wiki/File:CDW_4.pnghttp://en.wikipedia.org/wiki/File:CDW_dot.pnghttp://en.wikipedia.org/wiki/File:Tile_33344.svghttp://en.wikipedia.org/wiki/File:Elongated_triangular_prismatic_honeycomb.pnghttp://en.wikipedia.org/wiki/File:Uniform_tiling_63-snub.pnghttp://en.wikipedia.org/wiki/File:Snub_triangular-hexagonal_prismatic_honeycomb.pnghttp://en.wikipedia.org/wiki/File:CDW_dot.pnghttp://en.wikipedia.org/wiki/File:CDW_infin.pnghttp://en.wikipedia.org/wiki/File:CDW_ring.pnghttp://en.wikipedia.org/wiki/File:CDW_2.pnghttp://en.wikipedia.org/wiki/File:CDW_hole.pnghttp://en.wikipedia.org/wiki/File:CDW_3.pnghttp://en.wikipedia.org/wiki/File:CDW_hole.pnghttp://en.wikipedia.org/wiki/File:CDW_6.pnghttp://en.wikipedia.org/wiki/File:CDW_hole.pnghttp://en.wikipedia.org/wiki/File:Uniform_tiling_63-t012.pnghttp://en.wikipedia.org/wiki/File:Omnitruncated_triangular-hexagonal_prismatic_honeycomb.pnghttp://en.wikipedia.org/wiki/File:CDW_dot.pnghttp://en.wikipedia.org/wiki/File:CDW_infin.pnghttp://en.wikipedia.org/wiki/File:CDW_ring.pnghttp://en.wikipedia.org/wiki/File:CDW_2.pnghttp://en.wikipedia.org/wiki/File:CDW_ring.pnghttp://en.wikipedia.org/wiki/File:CDW_3.pnghttp://en.wikipedia.org/wiki/File:CDW_ring.pnghttp://en.wikipedia.org/wiki/File:CDW_6.pnghttp://en.wikipedia.org/wiki/File:CDW_ring.pnghttp://en.wikipedia.org/wiki/Omnitruncated_triangular-hexagonal_prismatic_honeycombhttp://en.wikipedia.org/wiki/Omnitruncated_triangular-hexagonal_prismatic_honeycombhttp://en.wikipedia.org/wiki/Great_rhombitrihexagonal_tilinghttp://en.wikipedia.org/wiki/Snub_triangular-hexagonal_prismatic_honeycombhttp://en.wikipedia.org/wiki/Snub_hexagonal_tilinghttp://en.wikipedia.org/wiki/Elongated_triangular_prismatic_honeycombhttp://en.wikipedia.org/wiki/Elongated_triangular_tilinghttp://en.wikipedia.org/w/index.php?title=Convex_uniform_honeycomb&action=edit§ion=11http://en.wikipedia.org/wiki/Crystalhttp://en.wikipedia.org/wiki/Wikipedia:Citation_neededhttp://en.wikipedia.org/wiki/Tetrahedral-octahedral_honeycombhttp://en.wikipedia.org/wiki/Close-packinghttp://en.wikipedia.org/wiki/Trusshttp://en.wikipedia.org/wiki/Alexander_Graham_Bellhttp://en.wikipedia.org/wiki/Buckminster_Fullerhttp://en.wikipedia.org/wiki/Octet_trusshttp://tabletoptelephone.com/~hopspage/Fuller.htmlhttp://members.cruzio.com/~devarco/energy.htmhttp://www.n55.dk/MANUALS/DISCUSSIONS/OTHER_TEXTS/CM_TEXT.htmlhttp://www.cjfearnley.com/fuller-faq-2.htmlhttp://en.wikipedia.org/w/index.php?title=Convex_uniform_honeycomb&action=edit§ion=12http://en.wikipedia.org/wiki/Cell_(mathematics)http://en.wikipedia.org/wiki/Uniform_tilinghttp://en.wikipedia.org/w/index.php?title=Cubic_prismatic_slab_honeycomb&action=edit&redlink=1http://en.wikipedia.org/w/index.php?title=Tri-hexagonal_prismatic_slab_honeycomb&action=edit&redlink=1http://en.wikipedia.org/w/index.php?title=Triangular_prismatic_slab&action=edit&redlink=1http://en.wikipedia.org/w/index.php?title=Cubic_column_honeycomb&action=edit&redlink=1http://en.wikipedia.org/w/index.php?title=Prismatic_column_honeycomb&action=edit&redlink=1 -
8/9/2019 PANALES CONVEXOS
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Examples (partially drawn): Cubic slab honeycomb and Alternated hexagonal slabhoneycomb.
[edit] Hyperbolic forms
There are 9 Coxeter group families of compact uniform honeycombs in hyperbolic 3-space,
generated as Wythoff constructions, and represented by ring permutations of theCoxeter-Dynkin diagrams for each family.
From these 9 families, there are a total of 76 unique honeycombs generated:
1. [3,5,3] :
2. [5,3,4] :
3. [5,3,5] :
4. [5,31,1] :
5. [4,3,3,3:] :
6. [4,3,4,3:] :
7. [5,3,3,3:] :
8. [5,3,4,3:] :
9. [5,3,5,3:] :
The full list of hyperbolic uniform honeycombs has not been proven and an unknown
number of non-wythoffian exist. One known example is given with the {3,5,3} family
below.
[edit] [3,5,3] family
There are 9 forms, generated by ring permutations of the Coxeter group: [3,5,3] or
One related non-wythoffian form is constructed from the {3,5,3} vertex figure with 4
(tetrahedrally arranged) vertices removed, creating pentagonal antiprisms and dodecahedra
filling in the gaps.[1]
# Honeycomb Cell counts/vertex
http://en.wikipedia.org/w/index.php?title=Convex_uniform_honeycomb&action=edit§ion=13http://en.wikipedia.org/wiki/Coxeter_grouphttp://en.wikipedia.org/wiki/Hyperbolic_spacehttp://en.wikipedia.org/wiki/Wythoff_constructionhttp://en.wikipedia.org/wiki/Coxeter-Dynkin_diagramhttp://en.wikipedia.org/wiki/Coxeter-Dynkin_diagramhttp://en.wikipedia.org/wiki/