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 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 such tilings, and (additional) non-Krötenheerdt tilings with clocks. They are dual uniform tilings to those uniform tilings with regular dodecagons. Finally, there are up to 394[1] distinct clocks.
- n-Gonal Duals with an emphasis on coloring by vertex regular planigon (VRP), up to 5-uniform. There are such tilings and (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[1], both consisting of 14 distinct vertex regular planigons (VRPs), the largest number of VRPs which can exist in a dual uniform tiling.[2]
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):
- Isosceles obtuse triangle (V3.122): O.
- 30-60-90 right triangle (V4.6.12): 3.
- Skew quadrilateral (V32.4.12): S.
- Tie kite (V3.4.3.12): T.
- Equilateral triangle (V63): E.
- Isosceles trapezoid (V32.62): I.
- Rhombus (V(3.6)2): R.
- Right trapezoid (V3.42.6): r.
- Deltoid (V3.4.6.4): D.
- Floret pentagon (V34.6): F.
- Square (V44): s.
- Cairo pentagon (V32.4.3.4): C.
- Barn pentagon (V33.42): B.
- Hexagon (V36): H.
- Isosceles right triangle (V4.82): i.
or O3STEIRrDFsCBHi for short. For isomeric tilings, the subscripts 1,2,3,... will be used.
Edge-Lattice Duality
editRegular and 1-Uniform Tilings
edit
Select Tilings
editKrötenheerdt Tilings
editRegular Tilings
edit
1-Uniform Tilings
edit
2-Uniform Tilings
edit
3-Uniform Tilings
edit
4-Uniform Tilings
edit
5-Uniform Tilings
edit
6-Uniform Tilings
edit
7-Uniform Tilings
editClock Tilings
editAll tilings with regular dodecagons in [3] are shown below, alternating between uniform and dual co-uniform every 5 seconds:
n-Gonal Duals
editSelect Tilings
editBonus
edit92-Uniform Tiling (Poster Size)
editBelow is a 92-dual-uniform tiling with all 14 arbitrary uniform vertex regular planigons (VRPs), and its fundamental unit, to scale at . 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.
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)
editBelow is a 174-dual-uniform tiling with all 14 arbitrary uniform vertex regular planigons (VRPs), and its fundamental unit, to scale at . This is courtesy of Paul Hofmann[1]. 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).
Fundamental Unit | Whole Tiling |
---|---|
- ^ a b c "THE BIG LIST SYSTEM OF TILINGS OF REGULAR POLYGONS". THE BIG LIST SYSTEM OF TILINGS OF REGULAR POLYGONS. Retrieved 2019-08-31.
- ^ Grünbaum, Branko, author. Tilings and patterns. ISBN 9780486469812. OCLC 962406815.
{{cite book}}
:|last=
has generic name (help)CS1 maint: multiple names: authors list (link) - ^ "n-Uniform Tilings". probabilitysports.com. Retrieved 2024-01-14.