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Dodecahedral honeycombs of order 4

Dodecahedral honeycombs of order 4
H3 534 CC center.png
Type ofHyperbolic regular cells
Schläfli Symbol{5.3,4}
{5.3 1.1 }
Charts
Coxeter - Dynkin		CDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
CDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node h0.png ↔ CDel node 1.pngCDel 5.pngCDel node.pngCDel split1.pngCDel nodes.png
Cells{5.3} Uniform polyhedron-53-t0.png
Verge ofPentagons {5}
Rib figuresquares {4}
Vertex figureOrder-4 dodecahedral honeycomb verf.png
Octahedron
Dual Honeycombs
Coxeter groupBH 3 , [5,3,4]
DH 3 , [5.3 1.1 ]
PropertiesRegular, quasi-right honeycomb

In a hyperbolic three-dimensional space, dodecahedral cells of order 4 are one of the four compact regular space-filling mosaics (or cells ). Having a Schlefli {5,3,4} symbol , the cells have four dodecahedrons around each edge and 8 dodecahedrons around each vertex in an octahedral arrangement. The tops of the cells are built on 3 orthogonal axes. The dual body of honeycombs are .

Geometric cells are multi - faceted cells that fill the space in such a way that there are no free spaces left. Honeycombs are an example of a more general mathematical concept of paving in spaces of any dimension.

Honeycombs are usually built in the usual Euclidean ("flat") space like . They can also be constructed in non-Euclidean spaces , such as . Any finite can be projected onto its described sphere in order to form homogeneous cells on a spherical space.

Content

Description

The dihedral angle of the dodecahedron is ~ 116.6 °, so it is impossible to place 4 dodecahedrons on an edge in a Euclidean 3-dimensional space. However, in a hyperbolic space for a dodecahedron, one can choose the size so that its dihedral angles are reduced to 90 degrees, and then four dodecahedrons precisely fill the space around each edge.

Symmetry

Honeycombs are constructed with half symmetry, {5.3 1.1 }, with two types (colors) of hexagonal mosaics in the Witthoff structure .         ↔       .

Drawings

 
Honeycombs contain a two-dimensional hyperbolic , {5,4}

 
Beltrami - Klein Model

Related polyhedra and honeycombs

There are four types of regular compact cells in a hyperbolic 3D space: Template: Regular compact H3 cells

There are in the family [5, 3, 4] of the Coxeter groups , including these regular forms.

Cell family [5,3,4]
{5.3,4}
       		r {5,3,4}
       		t {5,3,4}
       		rr {5.3,4}
       		t 0,3 {5,3,4}
       		tr {5.3,4}
       		t 0,1,3 {5,3,4}
       		t 0,1,2,3 {5,3,4}
       
        
       
{4.3,5}
       		r {4,3,5}
       		t {4,3,5}
       		rr {4,3,5}
       		2t {4,3,5}
       		tr {4,3,5}
       		t 0,1,3 {4,3,5}
       		t 0,1,2,3 {4,3,5}
       

There are in an extensive family [5,3 1,1 ] of Coxeter groups, including cells in alternating form. This construction can be represented by alternation (as on a chessboard) with two colors of dodecahedral cells.

These cells are also connected to the 16-cell , cubic cells and , all have octahedral vertex shapes: Template: Tiles with octahedral vertex shapes

These cells are part of a sequence of four-dimensional polyhedra and cells with dodecahedral cells:

{5.3, p}
SpaceS 3H 3
ViewFiniteCompactParacompactNon-compact
Title{5.3,3}
       		{5.3,4}
       
     		{5.3,5}
       		{5.3,6}
       
     		{5.3,7}
       		{5,3,8}
       
      		... {5,3, ∞}
       
      
Picture       
Vertex
figure
     		 
{3.3}
     		 
{3,4}
     		 
{3,5}
     		 
{3.6}
     		 
{3.7}
     		 
{3.8}
     		 
{3, ∞}
     

Full-length dodecahedral cells of order 4

Full-length dodecahedral cells of order 4
Type of
Schläfli Symbolr {5,3,4}
r {5.3 1.1 }
Charts
Coxeter - Dynkin		       
        ↔      
Cellsr {5.3}  
{3,4}  
Verge ofTriangles {3}
pentagons {5}
Vertex figure 
cube
Coxeter groupBH 3 , [5,3,4]
DH 3 , [5.3 1.1 ]
PropertiesVertex transitive, edge transitive

Full-length dodecahedral cells of order 4 ' ,               , have alternating octahedral and icosododecahedral cells with a cube as a vertex figure .

  
 
Honeycombs can be considered as an analogue of a two-dimensional hyperbolic , r {5,4}

Associated Honeycomb

There are four types of full-cut compact regular cells:

Four full-cut regular compact honeycombs in H 3
Picture    
Designationr {5,3,4}
       		r {4,3,5}
       		r {3,5,3}
       		r {5,3,5}
       
Vertex
figure		    

Truncated dodecahedral cells of order 4

Truncated dodecahedral cells of order 4
Type of
Schläfli Symbolt {5,3,4}
t {5.3 1.1 }
Charts
Coxeter - Dynkin		       
        ↔      
Cellst {5.3}  
{3,4}  
Verge ofTriangles {3}
Decagons {10}
Vertex figure 
Square pyramid
Coxeter groupBH 3 , [5,3,4]
DH 3 , [5.3 1.1 ]
PropertiesVertex transitive

Truncated dodecahedral cells of order 4 ,               , have octahedral and truncated dodecahedral cells with a cube as a vertex shape .

 

Honeycombs can be considered as an analogue of two-dimensional hyperbolic t {5,4} with faces in the form of truncated pentagons and squares:

 

Associated Honeycomb

Four types of truncated regular regular compact cells in H 3
Picture    
Designationt {5,3,4}
       		t {4,3,5}
       		t {3,5,3}
       		t {5,3,5}
       
Vertex
figure		    

Shapeless dodecahedral cells of order 4

Bi-sectional dodecahedral cells of order 4
Chiselled cubic honeycombs of order 5
Type of
Schläfli Symbol2t {5.3,4}
2t {5.3 1.1 }
Charts
Coxeter - Dynkin		       
        ↔      
Cellst {3,5}  
t {3,4}  
Verge ofTriangles {3}
squares {4}
hexagons {6}
Vertex figure 
Tetrahedron
Coxeter groupBH 3 , [5,3,4]
DH 3 , [5.3 1.1 ]
PropertiesVertex transitive

Bi-sectioned dodecahedral cells of order 4 or bi-cut cubic cells of order 5 ,               , have truncated octahedra and truncated icosahedra as cells and a tetrahedron as a vertex figure .

 

Associated Honeycomb

Three types of bi-truncated regular compact cells in H 3
Picture   
Designation2t {4,3,5}
       		2t {3,5,3}
       		2t {5,3,5}
       
Vertex
figure		   

Oblique dodecahedral cells of order 4

Oblique dodecahedral honeycombs of order 4
Type of
Schläfli Symbolrr {5.3,4}
rr {5.3 1.1 }
Charts
Coxeter - Dynkin		       
        ↔      
Cellsrr {3,5}  
r {3,4}  
{} x {4} cube  
Verge ofTriangles {3}
squares {4}
pentagons {5}
Vertex figure 
Triangular prism
Coxeter groupBH 3 , [5,3,4]
DH 3 , [5.3 1.1 ]
PropertiesVertex transitive

Oblique dodecahedral cells of order 4 ,               , have rhomboicosododecahedral , cubooctahedral and cubic cells and a triangular prism as a vertex figure .

 

Associated Honeycomb

Four kinds of canted regular compact honeycombs in H 3
Picture    
Designationrr {5.3,4}
       		rr {4,3,5}
       		rr {3,5,3}
       		rr {5,3,5}
       
Vertex
figure		    

Bevel-truncated dodecahedral cells of order 4

Skewed-truncated dodecahedral cells of order 4
Type of
Schläfli Symboltr {5.3,4}
tr {5.3 1.1 }
Charts
Coxeter - Dynkin		       
        ↔      
Cellstr {3,5}  
t {3,4}  
{} x {4} Cubes  
Verge ofsquares {4}
hexagons {6}
Decagons {10}
Vertex figure 
mirror sphenoid
Coxeter groupBH 3 , [5,3,4]
DH 3 , [5.3 1.1 ]
PropertiesVertex transitive

Bevel-truncated dodecahedral cells of order 4 are homogeneous cells with a Coxeter-Dynkin diagram               and having a mirror sphenoid as a vertex figure .

 

Associated Honeycomb

Four kinds of beveled-truncated regular compact cells in H 3
Picture    
Designationtr {5.3,4}
       		tr {4,3,5}
       		tr {3,5,3}
       		tr {5,3,5}
       
Vertex
figure		    

Strug-truncated dodecahedral cells of order 4

Strug-truncated dodecahedral cells of order 4
Type of
Schläfli Symbolt 0,1,3 {5,3,4}
Charts
Coxeter - Dynkin		       
Cellst {5.3}  
rr {3,4}  
{} x {10}  
{} x {4}  
Verge ofTriangles {3}
squares {4}
Decagons {10}
Vertex figure 
quad pyramid
Coxeter groupBH 3 , [5,3,4]
PropertiesVertex transitive

Structural-truncated dodecahedral cells of order 4 are homogeneous cells with a Coxeter-Dynkin diagram               and a quadrilateral pyramid as a vertex shape .

 

Associated Honeycomb

Four types of strug-truncated regular compact cells in H 3
Picture    
Designationt 0,1,3 {5,3,4}
       		t 0,1,3 {4,3,5}
       		t 0,1,3 {3,5,3}
       		t 0,1,3 {5,3,5}
       
Vertex
figure		    

See also

  • Poincaré homology sphere Poincaré dodecahedral space
  • Seifert-Weber space Seifert – Weber dodecahedral space
  • List of valid multidimensional polyhedra and compounds

Notes

Literature

  • Coxeter . Tables I and II: Regular polytopes and honeycombs // . - 3rd. ed .. - Dover Publications, 1973. - p. 294–296. - ISBN 0-486-61480-8 .
  • Coxeter . Chapter 10: Regular Honeycombs in Hyperbolic Space; Summary Tables II, III, IV, V // The Beauty of Geometry: Twelve Essays. - Dover Publications, 1999. - p. 212-213. - ISBN 0-486-40919-8 .
  • Jeffrey R. Weeks. Chapter 16-17: Geometries on Three-manifolds I, II // The Shape of Space. - 2nd. - ISBN 0-8247-0709-5 .
  • Uniform Polytopes. - 1991. - (Manuscript).
    • The Theory of Uniform Polytopes and Honeycombs. - University of Toronto, 1966. - (Ph.D. Dissertation).
    • Chapter 13: Hyperbolic Coxeter groups // Geometries and Transformations. - 2015.
Source - https://ru.wikipedia.org/w/index.php?title=Dodecahedral_the orders_4_old_4&oldid = 99141583


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Clever Geek | 2019