Truncated tetrapentagonal tiling

In geometry, the truncated tetrapentagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of t0,1,2{4,5} or tr{4,5}.

Truncated tetrapentagonal tiling
Truncated tetrapentagonal tiling

Poincaré disk model of the hyperbolic plane
Type Hyperbolic uniform tiling
Vertex configuration 4.8.10
Schläfli symbol tr{5,4} or
Wythoff symbol 2 5 4 |
Coxeter diagram CDel node 1.pngCDel 5.pngCDel node 1.pngCDel 4.pngCDel node 1.png or CDel node 1.pngCDel split1-54.pngCDel nodes 11.png
Symmetry group [5,4], (*542)
Dual Order-4-5 kisrhombille tiling
Properties Vertex-transitive

Symmetry

Truncated tetrapentagonal tiling with mirrors
Truncated tetrapentagonal tiling with mirror lines. CDel node c1.pngCDel 5.pngCDel node c1.pngCDel 4.pngCDel node c2.png

There are four small index subgroup constructed from [5,4] by mirror removal and alternation. In these images fundamental domains are alternately colored black and white, and mirrors exist on the boundaries between colors.

A radical subgroup is constructed [5*,4], index 10, as [5+,4], (5*2) with gyration points removed, becoming orbifold (*22222), and its direct subgroup [5*,4]+, index 20, becomes orbifold (22222).

See also

References

  • John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008, ISBN 978-1-56881-220-5 (Chapter 19, The Hyperbolic Archimedean Tessellations)
  • Coxeter, H. S. M. (1999). "Chapter 10: Regular honeycombs in hyperbolic space". The Beauty of Geometry: Twelve Essays. Dover Publications. ISBN 0-486-40919-8. LCCN 99035678.

External links

4-5 kisrhombille

In geometry, the 4-5 kisrhombille or order-4 bisected pentagonal tiling is a semiregular dual tiling of the hyperbolic plane. It is constructed by congruent right triangles with 4, 8, and 10 triangles meeting at each vertex.

The name 4-5 kisrhombille is by Conway, seeing it as a 4-5 rhombic tiling, divided by a kis operator, adding a center point to each rhombus, and dividing into four triangles.

The image shows a Poincaré disk model projection of the hyperbolic plane.

It is labeled V4.8.10 because each right triangle face has three types of vertices: one with 4 triangles, one with 8 triangles, and one with 10 triangles.

Small index subgroups of [5,4]
Index 1 2 10
Diagram 542 symmetry 000 542 symmetry 00a 542 symmetry bb0 542 symmetry zz0
Coxeter
(orbifold)
[5,4] = CDel node c1.pngCDel 5.pngCDel node c1.pngCDel 4.pngCDel node c2.png
(*542)
[5,4,1+] = CDel node c1.pngCDel 5.pngCDel node c1.pngCDel 4.pngCDel node h0.png = CDel node c1.pngCDel split1-55.pngCDel nodeab c1.png
(*552)
[5+,4] = CDel node h2.pngCDel 5.pngCDel node h2.pngCDel 4.pngCDel node c2.png
(5*2)
[5*,4] = CDel node g.pngCDel 5g.pngCDel 3sg.pngCDel node g.pngCDel 4.pngCDel node c2.png
(*22222)
Direct subgroups
Index 2 4 20
Diagram 542 symmetry aaa 542 symmetry bba 542 symmetry zza
Coxeter
(orbifold)
[5,4]+ = CDel node h2.pngCDel 5.pngCDel node h2.pngCDel 4.pngCDel node h2.png
(542)
[5+,4]+ = CDel node h2.pngCDel 5.pngCDel node h2.pngCDel 4.pngCDel node h0.png = CDel node h2.pngCDel split1-55.pngCDel branch h2h2.pngCDel label2.png
(552)
[5*,4]+ = CDel node g.pngCDel 5g.pngCDel 3sg.pngCDel node g.pngCDel 4.pngCDel node h0.png
(22222)
*n42 symmetry mutation of omnitruncated tilings: 4.8.2n
Symmetry
*n42
[n,4]
Spherical Euclidean Compact hyperbolic Paracomp.
*242
[2,4]
*342
[3,4]
*442
[4,4]
*542
[5,4]
*642
[6,4]
*742
[7,4]
*842
[8,4]...
*∞42
[∞,4]
Omnitruncated
figure
Spherical octagonal prism2
4.8.4
Uniform tiling 432-t012
4.8.6
Uniform tiling 44-t012
4.8.8
H2 tiling 245-7
4.8.10
H2 tiling 246-7
4.8.12
H2 tiling 247-7
4.8.14
H2 tiling 248-7
4.8.16
H2 tiling 24i-7
4.8.∞
Omnitruncated
duals
Spherical octagonal bipyramid2
V4.8.4
Spherical disdyakis dodecahedron
V4.8.6
1-uniform 2 dual
V4.8.8
Order-4 bisected pentagonal tiling
V4.8.10
Hyperbolic domains 642
V4.8.12
Hyperbolic domains 742
V4.8.14
Hyperbolic domains 842
V4.8.16
H2checkers 24i
V4.8.∞
*nn2 symmetry mutations of omnitruncated tilings: 4.2n.2n
Symmetry
*nn2
[n,n]
Spherical Euclidean Compact hyperbolic Paracomp.
*222
[2,2]
*332
[3,3]
*442
[4,4]
*552
[5,5]
*662
[6,6]
*772
[7,7]
*882
[8,8]...
*∞∞2
[∞,∞]
Figure Spherical square prism Uniform tiling 332-t012 Uniform tiling 44-t012 H2 tiling 255-7 H2 tiling 266-7 H2 tiling 277-7 H2 tiling 288-7 H2 tiling 2ii-7
Config. 4.4.4 4.6.6 4.8.8 4.10.10 4.12.12 4.14.14 4.16.16 4.∞.∞
Dual Spherical square bipyramid Spherical tetrakis hexahedron 1-uniform 2 dual H2checkers 245 H2checkers 246 H2checkers 247 H2checkers 248 H2checkers 24i
Config. V4.4.4 V4.6.6 V4.8.8 V4.10.10 V4.12.12 V4.14.14 V4.16.16 V4.∞.∞
Uniform pentagonal/square tilings
Symmetry: [5,4], (*542) [5,4]+, (542) [5+,4], (5*2) [5,4,1+], (*552)
CDel node 1.pngCDel 5.pngCDel node.pngCDel 4.pngCDel node.png CDel node 1.pngCDel 5.pngCDel node 1.pngCDel 4.pngCDel node.png CDel node.pngCDel 5.pngCDel node 1.pngCDel 4.pngCDel node.png CDel node.pngCDel 5.pngCDel node 1.pngCDel 4.pngCDel node 1.png CDel node.pngCDel 5.pngCDel node.pngCDel 4.pngCDel node 1.png CDel node 1.pngCDel 5.pngCDel node.pngCDel 4.pngCDel node 1.png CDel node 1.pngCDel 5.pngCDel node 1.pngCDel 4.pngCDel node 1.png CDel node h.pngCDel 5.pngCDel node h.pngCDel 4.pngCDel node h.png CDel node h.pngCDel 5.pngCDel node h.pngCDel 4.pngCDel node.png CDel node.pngCDel 5.pngCDel node.pngCDel 4.pngCDel node h.png
Uniform tiling 54-t0 Uniform tiling 54-t01 Uniform tiling 54-t1 Uniform tiling 54-t12 Uniform tiling 54-t2 Uniform tiling 54-t02 Uniform tiling 54-t012 Uniform tiling 54-snub Uniform tiling 542-h01 Uniform tiling 552-t0
{5,4} t{5,4} r{5,4} 2t{5,4}=t{4,5} 2r{5,4}={4,5} rr{5,4} tr{5,4} sr{5,4} s{5,4} h{4,5}
Uniform duals
CDel node f1.pngCDel 5.pngCDel node.pngCDel 4.pngCDel node.png CDel node f1.pngCDel 5.pngCDel node f1.pngCDel 4.pngCDel node.png CDel node.pngCDel 5.pngCDel node f1.pngCDel 4.pngCDel node.png CDel node.pngCDel 5.pngCDel node f1.pngCDel 4.pngCDel node f1.png CDel node.pngCDel 5.pngCDel node.pngCDel 4.pngCDel node f1.png CDel node f1.pngCDel 5.pngCDel node.pngCDel 4.pngCDel node f1.png CDel node f1.pngCDel 5.pngCDel node f1.pngCDel 4.pngCDel node f1.png CDel node fh.pngCDel 5.pngCDel node fh.pngCDel 4.pngCDel node fh.png CDel node fh.pngCDel 5.pngCDel node fh.pngCDel 4.pngCDel node.png CDel node.pngCDel 5.pngCDel node.pngCDel 4.pngCDel node fh.png
Uniform tiling 54-t2 Order-5 tetrakis square tiling Order-5-4 quasiregular rhombic tiling Order-4 pentakis pentagonal tiling Uniform tiling 54-t0 Deltoidal tetrapentagonal tiling Order-4 bisected pentagonal tiling Order-5-4 floret pentagonal tiling Uniform tiling 552-t2
V54 V4.10.10 V4.5.4.5 V5.8.8 V45 V4.4.5.4 V4.8.10 V3.3.4.3.5 V3.3.5.3.5 V55

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