User:Harry Princeton/Dual Uniform Tilings: Krötenheerdt Tilings, Clock Tilings, Edge Lattice Duality, and n-Gonal Duals

See User:Harry Princeton/Planigons and Dual Uniform Tilings for main results.

This page consists of high resolution dual-superimposed images and results of:

• Edge-Lattice Duality is the most faithful representation of a dual uniform tiling as there is a combinatorial isomorphism between the edges of the uniform and dual uniform lattices. There are 10 edges which exist in arbitrary uniform tilings, and 2 edges exclusive to the kisquadrille tiling. All Euclidean Catalaves tilings and select k-uniform dual tilings will be shown, along with a 25-uniform dual tiling which contains all 10 edges.

• Krötenheerdt Tilings from regular to 7-uniform. There are $3+8+20+39+33+15+10+7=135$ such tilings. These were found by Otto Krötenheerdt, and they have the same Archimedean (n-hedral) order as uniformity (n-isohedral).
• Clock Tilings from regular to 6-uniform. There are $2+3+8+(9+7+1+1)+(2+14+5+3+2+2)+48=107$ such tilings, and $9+26+45=80$ (additional) non-Krötenheerdt tilings with clocks. They are dual uniform tilings to those uniform tilings with regular dodecagons. Finally, there are up to 394 distinct clocks.
• n-Gonal Duals with an emphasis on coloring by vertex regular planigon (VRP), up to 5-uniform. There are $4+(1+2+4+12+30+54)+(3+2+3+3+7)+1=126$ such tilings and $94+12=106$ (additional) non-Krötenheerdt tilings (no k ≥ 2-uniform n-Gonal duals have clocks!), but we may only investigate select n-gonal duals.
• Bonus a 92-uniform tiling (pmm), and a 179-uniform tiling (pmg) by Paul Hofmann, both consisting of 14 distinct vertex regular planigons (VRPs), the largest number of VRPs which can exist in a dual uniform tiling.

There will be approximately 250 k-dual uniform tilings on this page.

Finally, the tilings will be labeled by initials, according to the 15 usable vertex regular planigons (VRPs):

1. Isosceles obtuse triangle (V3.12²): O.
2. 30-60-90 right triangle (V4.6.12): 3.
3. Skew quadrilateral (V3².4.12): S.
4. Tie kite (V3.4.3.12): T.
5. Equilateral triangle (V6³): E.
6. Isosceles trapezoid (V3².6²): I.
7. Rhombus (V(3.6)²): R.
8. Right trapezoid (V3.4².6): r.
9. Deltoid (V3.4.6.4): D.
10. Floret pentagon (V3⁴.6): F.
11. Square (V4⁴): s.
12. Cairo pentagon (V3².4.3.4): C.
13. Barn pentagon (V3³.4²): B.
14. Hexagon (V3⁶): H.
15. Isosceles right triangle (V4.8²): i.

or O3STEIRrDFsCBHi for short. For isomeric tilings, the subscripts _1,2,3,... will be used.

Edge-Lattice Duality

Krötenheerdt Tilings

Clock Tilings

All tilings with regular dodecagons in are shown below, alternating between uniform and dual co-uniform every 5 seconds:

All tilings with regular dodecagons are shown below, alternating between uniform and dual co-uniform every 5 seconds.

n-Gonal Duals

Bonus

92-Uniform Tiling (Poster Size)

Below is a 92-dual-uniform tiling with all 14 arbitrary uniform vertex regular planigons (VRPs), and its fundamental unit, to scale at $1:10$. This is the exact same tiling used in Special Tilings (Expand and Ortho), and k-Uniform Circle Packing Examples. Again, the VRPs are colored with frequency inverse to area.

92-Uniform Tiling: the Fundamental Unit and the Whole Tiling

Fundamental Unit	Whole Tiling

Empirically, there is a 3px horizontal discrepancy in the fundamental unit due to anti-aliased boundaries, per 3975px of height. This does not occur if 104-pixel uniformized VRPs in MS Paint are used instead.

174-Uniform Tiling (Poster Size)

Below is a 174-dual-uniform tiling with all 14 arbitrary uniform vertex regular planigons (VRPs), and its fundamental unit, to scale at $1:10$. This is courtesy of Paul Hofmann. Again, the VRPs are colored with frequency inverse to area.

Of note is that if the equilateral triangle/isosceles trapezoid (EI) parts are eliminated, the uniformity is essentially divided by 4 (around 50-dual-uniform). Hence on average, the tiling is 92-dual-uniform. So the previous tiling is as efficient, and it has better colors as well (this tiling below has too much Cairo pentagonal blue).

179-Uniform Tiling: the Fundamental Unit and the Whole Tiling

Fundamental Unit	Whole Tiling