Triheptagonal tiling
In geometry , the triheptagonal tiling is a semiregular tiling of the hyperbolic plane, representing a rectified Order-3 heptagonal tiling . There are two triangles and two heptagons alternating on each vertex . It has Schläfli symbol of r{7,3}.
Compare to trihexagonal tiling with vertex configuration 3.6.3.6 .
Images
Klein disk model of this tiling preserves straight lines, but distorts angles
The dual tiling is called an Order-7-3 rhombille tiling , made from rhombic faces, alternating 3 and 7 per vertex.
7-3 Rhombille
In geometry , the 7-3 rhombille tiling is a tessellation of identical rhombi on the hyperbolic plane . Sets of three and seven rhombi meet two classes of vertices.
7-3 rhombile tiling in band model
The triheptagonal tiling can be seen in a sequence of quasiregular polyhedrons and tilings:
Quasiregular tilings: (3.n)2
Sym. *n32 [n,3]
Spherical
Euclid.
Compact hyperb.
Paraco.
Noncompact hyperbolic
*332 [3,3] Td
*432 [4,3] Oh
*532 [5,3] Ih
*632 [6,3] p6m
*732 [7,3]
*832 [8,3]...
*∞32 [∞,3]
[12i,3]
[9i,3]
[6i,3]
Figure
Figure
Vertex
(3.3)2
(3.4)2
(3.5)2
(3.6)2
(3.7)2
(3.8)2
(3.∞)2
(3.12i)2
(3.9i)2
(3.6i)2
Schläfli
r{3,3}
r{3,4}
r{3,5}
r{3,6}
r{3,7}
r{3,8}
r{3,∞}
r{3,12i}
r{3,9i}
r{3,6i}
Coxeter
Dual uniform figures
Dualconf.
V(3.3)2
V(3.4)2
V(3.5)2
V(3.6)2
V(3.7)2
V(3.8)2
V(3.∞)2
From a Wythoff construction there are eight hyperbolic uniform tilings that can be based from the regular heptagonal tiling.
Drawing the tiles colored as red on the original faces, yellow at the original vertices, and blue along the original edges, there are 8 forms.
Uniform heptagonal/triangular tilings
Symmetry: [7,3], (*732)
[7,3]+ , (732)
{7,3}
t{7,3}
r{7,3}
t{3,7}
{3,7}
rr{7,3}
tr{7,3}
sr{7,3}
Uniform duals
V73
V3.14.14
V3.7.3.7
V6.6.7
V37
V3.4.7.4
V4.6.14
V3.3.3.3.7
See also
References
External links