It is labeled V4.6.14 because each right triangle face has three types of vertices: one with 4 triangles, one with 6 triangles, and one with 14 triangles. It is the dual tessellation of the truncated triheptagonal tiling which has one square and one heptagon and one tetrakaidecagon at each vertex.
The name 3-7 kisrhombille is given by Conway, seeing it as a 3-7 rhombic tiling, divided by a kis operator, adding a center point to each rhombus, and dividing into four triangles.
Symmetryedit
There are no mirror removal subgroups of [7,3]. The only small index subgroup is the alternation, [7,3]+, (732).
It is topologically related to a polyhedra sequence; see discussion. This group is special for having all even number of edges per vertex and form bisecting planes through the polyhedra and infinite lines in the plane, and are the reflection domains for the (2,3,n) triangle groups – for the heptagonal tiling, the important (2,3,7) triangle group.
The kisrhombille tilings can be seen as from the sequence of rhombille tilings, starting with the cube, with faces divided or kissed at the corners by a face central point.
*n32 symmetry mutation of omnitruncated tilings: 4.6.2n
Visualization of the map (2,3,∞) → (2,3,7) by morphing the associated tilings.[1]
Just as the (2,3,7) triangle group is a quotient of the modular group (2,3,∞), the associated tiling is the quotient of the modular tiling, as depicted in the video at right.
Referencesedit
^Platonic tilings of Riemann surfaces: The Modular Group, Gerard Westendorp
John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008, ISBN978-1-56881-220-5 (Chapter 19, The Hyperbolic Archimedean Tessellations)
kisrhombille, geometry, tiling, semiregular, dual, tiling, hyperbolic, plane, constructed, congruent, right, triangles, with, triangles, meeting, each, vertex, typedual, semiregular, hyperbolic, tilingfacesright, triangleedgesinfiniteverticesinfinitecoxeter, d. In geometry the 3 7 kisrhombille tiling is a semiregular dual tiling of the hyperbolic plane It is constructed by congruent right triangles with 4 6 and 14 triangles meeting at each vertex 3 7 kisrhombilleTypeDual semiregular hyperbolic tilingFacesRight triangleEdgesInfiniteVerticesInfiniteCoxeter diagramSymmetry group 7 3 732 Rotation group 7 3 732 Dual polyhedronTruncated triheptagonal tilingFace configurationV4 6 14Propertiesface transitiveWikimedia Commons has media related to Uniform dual tiling V 4 6 14 The image shows a Poincare disk model projection of the hyperbolic plane It is labeled V4 6 14 because each right triangle face has three types of vertices one with 4 triangles one with 6 triangles and one with 14 triangles It is the dual tessellation of the truncated triheptagonal tiling which has one square and one heptagon and one tetrakaidecagon at each vertex Contents 1 Naming 2 Symmetry 3 Related polyhedra and tilings 4 References 5 See alsoNaming editThe name 3 7 kisrhombille is given by Conway seeing it as a 3 7 rhombic tiling divided by a kis operator adding a center point to each rhombus and dividing into four triangles Symmetry editThere are no mirror removal subgroups of 7 3 The only small index subgroup is the alternation 7 3 732 Small index subgroups of 7 3 732 Type Reflectional Rotationalindex 1 2Diagram nbsp nbsp Coxeter orbifold 7 3 nbsp nbsp nbsp nbsp nbsp 732 7 3 nbsp nbsp nbsp nbsp nbsp 732 Related polyhedra and tilings editThree isohedral regular or quasiregular tilings can be constructed from this tiling by combining triangles Projections centered on different triangle points Poincarediskmodel nbsp nbsp nbsp Center Heptagon Triangle RhombicKleindiskmodel nbsp nbsp nbsp Relatedtiling nbsp nbsp nbsp Heptagonal tiling Triangular tiling Rhombic tilingUniform heptagonal triangular tilings vteSymmetry 7 3 732 7 3 732 nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp 7 3 t 7 3 r 7 3 t 3 7 3 7 rr 7 3 tr 7 3 sr 7 3 Uniform duals nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp V73 V3 14 14 V3 7 3 7 V6 6 7 V37 V3 4 7 4 V4 6 14 V3 3 3 3 7It is topologically related to a polyhedra sequence see discussion This group is special for having all even number of edges per vertex and form bisecting planes through the polyhedra and infinite lines in the plane and are the reflection domains for the 2 3 n triangle groups for the heptagonal tiling the important 2 3 7 triangle group See also the uniform tilings of the hyperbolic plane with 2 3 7 symmetry The kisrhombille tilings can be seen as from the sequence of rhombille tilings starting with the cube with faces divided or kissed at the corners by a face central point n32 symmetry mutation of omnitruncated tilings 4 6 2n vteSym n32 n 3 Spherical Euclid Compact hyperb Paraco Noncompact hyperbolic 232 2 3 332 3 3 432 4 3 532 5 3 632 6 3 732 7 3 832 8 3 32 3 12i 3 9i 3 6i 3 3i 3 Figures nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp Config 4 6 4 4 6 6 4 6 8 4 6 10 4 6 12 4 6 14 4 6 16 4 6 4 6 24i 4 6 18i 4 6 12i 4 6 6iDuals nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp Config V4 6 4 V4 6 6 V4 6 8 V4 6 10 V4 6 12 V4 6 14 V4 6 16 V4 6 V4 6 24i V4 6 18i V4 6 12i V4 6 6i nbsp Visualization of the map 2 3 2 3 7 by morphing the associated tilings 1 Just as the 2 3 7 triangle group is a quotient of the modular group 2 3 the associated tiling is the quotient of the modular tiling as depicted in the video at right References edit Platonic tilings of Riemann surfaces The Modular Group Gerard Westendorp 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 See also editHexakis triangular tiling Tilings of regular polygons List of uniform tilings Uniform tilings in hyperbolic plane Retrieved from https en wikipedia org w index php title 3 7 kisrhombille amp oldid 999394021, wikipedia, wiki, book, books, library,
