User:Harry Princeton/Planigons and Dual Uniform Tilings

In geometry, a planigon is a polygon that can fill the plane with only copies of itself. In the Euclidean plane there are 3 regular forms equilateral triangle, squares, and regular hexagons, and 8 semiregular forms, and 4-demiregular forms which can tile the plane with other planigons.

All angles of a planigon are whole divisors of 360°. Tilings are made by edge-to-edge connections.

Tilings made from planigons can be seen as duals to the regular, semiregular, and demiregular tilings of the plane by regular polygons.

History
In the 1987 book, Tilings and Patterns, Branko Grünbaum calls the vertex-uniform tilings Archimedean in parallel to the Archimedean solids. Their dual tilings are called Laves tilings in honor of crystallographer Fritz Laves. They're also called Shubnikov–Laves tilings after Shubnikov, Alekseĭ Vasilʹevich. John Conway calls the uniform duals Catalan tilings, in parallel to the Catalan solid polyhedra.

The Laves tilings have vertices at the centers of the regular polygons, and edges connecting centers of regular polygons that share an edge. The tiles of the Laves tilings are called planigons. This includes the 3 regular tiles (triangle, square and hexagon) and 8 irregular ones. Each vertex has edges evenly spaced around it. Three dimensional analogues of the planigons are called stereohedrons.

These tilings are listed by their face configuration, the number of faces at each vertex of a face. For example V4.8.8 means isosceles triangle tiles with one corner with four triangles, and two corners containing eight triangles.

Three Regular Polygons, Eight Uniform Planigons, Four Demiregular Planigons (15 Usable Polygons)
Archimedean (monohedral) tilings are also 1-uniform as well. These are the convex regular polygons and vertex regular planigons which can tile the plane alone. The original regular tilings and 1-uniform tilings are faintly outlined. All tilings are to simultaneous scale, and all vertex regular planigons are to simultaneous scale at a different size.

Finally, a scaled comparison vertex regular polygons in dual uniform tilings. From here, any dual uniform tiling can be created using flips and 90-degree rotations of these polygons. The edges of the original uniform tiling are faintly shown (dual compound).

A similar treatment is done for usable regular polygons, using the exact same scale (200 pixels per unit side). In fact, only the colors and importance of each lattice (uniform and dual uniform) change, if the uniform tiling and dual uniform tiling are compared, using the respective set of polygons. Moreover, both images occupy-to-scale on this page.

Construction
The Conway operation of dual interchanges faces and vertices. In Archimedean solids and k-uniform tilings alike, the new vertex coincides with the center of each regular face, or the centroid. In the Euclidean (plane) case; in order to make new faces around each original vertex, the centroids must be connected by new edges, each of which must intersect exactly one of the original edges. Since regular polygons have dihedral symmetry, we see that these new centroid-centroid edges must be perpendicular bisectors of the common original edges (e.g. the centroid lies on all edge perpendicular bisectors of a regular polygon). Thus, the edges of k-dual uniform tilings coincide with centroid-edge midpoint line segments of all regular polygons in the k-uniform tilings.

So, we can alternatively construct k-dual uniform tilings (and all 21 planigons) equivalently by forming new centroid-edge midpoint line segments of the original regular polygons (dissecting the regular n-gons into n congruent deltoids, ortho), and then removing the original edges (dual). Closed planigons will form around interior vertices, and line segments of (many possible) planigons will form around boundary vertices, giving a faithful k-dual uniform lattice (ortho-superimposable and to scale). On the other hand, centroid-centroid connecting only yields interior planigons (with variable translation and scale), but this construction is nonetheless equivalent in the interior. If the original k-uniform tiling fills the entire frame, then so will the k-dual uniform lattice by the first construction, and the boundary line segments can be ignored (equivalent to second construction). Below, rendered in Asymptote: Some types are different from their mirror images and listed twice. A triangle a.b.c and c.b.a are mirror images if a,b,c are all unique. In these images, the regular polygons are listed counter clockwise from the right, with Planigons in various colors: translucent green for regular polygons, translucent blue/cyan for planigons, translucent yellow for demiregular planigons, and translucent orange/red for unusable polygons. The 4.82 cyan planigon cannot coexist with any other planigon in any k-uniform tiling, and the red unusable polygon has one axis of symmetry unlike the others. The below image constructs all vertex regular polygons to scale, with the top image inserted as reference:

An Alternate Construction
All 14 arbitrary uniform usable vertex regular planigons (VRPs) also hail from the 6-5 dodecagram (where each segment subtends $$5\pi/6$$ radians, or $$150$$ degrees).

The incircle of this dodecagram demonstrates that all the 14 VRPs are cocyclic, as alternatively shown by circle ambo packings section below. As a refresher, the ratio of the incircle to the circumcircle is


 * $$\sin\frac{\pi}{12}=\sin 15^{\circ}=\frac{\sqrt{6}-\sqrt{2}}{4}\approx 0.258819$$

and the convex hull is precisely the regular dodecagons in the arbitrary uniform tilings! In fact, the equilateral triangle, square, regular hexagon, and regular dodecagon; are shown below along with the VRPs:

Circle Ambo Packings
We can show that the vertex regular planigons (VRPs) are the polar reciprocals of the configuration regular vertex polygons (CVRPs) about circles of radii $$1/\sqrt{2}$$ centered about the vertex. This is done using similar right triangles when taking the perpendicular diagonals of the $$n$$ congruent deltoids of a regular $$n$$-gon. Or, taking the convex hull of the ortho-intersection points of the VRPs with the edges of regular $$n$$-gons, we construct half-size configuration regular vertex polygons (hCVRPs). As the side length is common to all regular $$n$$-gons around a vertex, we can inscribe a circle of radius $$1/2$$ in each VRP (see circle packings). Then each VRP is explicitly the polar reciprocal of the hCVRP about the circle of radius $$1/2$$. The VRP makes the dual uniform tiling; the $$1/2$$-radius circle makes the uniform circle packing, and the hCVRPs outline ambo uniform tilings in checkerboard fashion, with checker gaps made by rectified (shrunk as well as polar reciprocated) regular polygons.

In other words, the hCVRP is the rectification of the VRP, and both form a one-dimensional dual compound about the common circle of radius $$1/2$$ (e.g. the dual uniform tiling is a 2-dimensional dual) It works for the regular polygons of uniform tilings too, but with circles of different radii.

Construction of a Dual Tiling (Grunbaum, et. al. 174)
If the tiles of an arbitrary graph allow circles of positive radii inside their interiors, then we can construct dual graph tilings by drawing the circles about dual vertices (anywhere within the interior of the tiles of the original graph). Then, if the tile of the original graph has $$n$$ edges, then the circle is split into $$n$$ equiangular sectors by $$n$$ rays (forming a equiangular dual vertex of degree $$n$$). Finally, connect the endpoints of the $$n$$ rays to points along the edges of the original graph tiling (preferably orthogonally). Note that there may not be clearly defined 'midpoints' because these are arbitrary graphs.

This is seen below, where the original lattice is in black, the dual lattice is in blue, and the circles are in thick blue.

Relationship to Ambo Uniform Circle Packing
The interior circles was a way to construct dual graph tilings.

The exact opposite happens if we try to recover k-uniform tilings from k-dual uniform tilings. It has been shown that the vertex regular planigons are cocyclic in Ambo Circle Packings. This means we can inscribe actual incircles of radius $$1/2$$ in vertex regular planigons. The inparameter is maximal at $$1/2$$, so there is no room to change the orientation with respect to an edge; each incircle is tangent to at least three others (giving circle packings). We could draw equiangular rays, but they certainly would not yield the original k-uniform tiling.


 * 1) Instead, we need to draw rays from the incenters to the points of tangency (yielding the corresponding half-size configuration vertex regular polygon as the convex hull), which are not equiangular.
 * 2) On the other hand, this guarantees that the new k-uniform edges are perpendicularly bisected by those of the k-dual uniform tiling, so there is a well-defined notion of midpoint.

This is shown as a ambo circle packing below (there are eight distinct VRPs and nine planigons):

Ways to Color Dual Uniform Tilings
The vertex regular planigons (VRPs) can be colored in one of five ways:


 * By number of sides (green for hexagon, cyan for pentagons, red for quadrilaterals, and yellow for triangles). This generally does not permit enough variety or distinction between tiles.
 * By vertex regular planigon (with frequency of color negatively correlated with size). This is generally the best adaptable way, especially for high-dual-uniform tilings. This is also called Archimedianess.
 * By uniformity (so two of the same VRPs could be colored differently if they correspond to different orbits). For large tilings, uniformity tends to increase quadratically; a screen-size fundamental unit usually has uniformity in the hundreds.
 * As the dual lattice of a uniform tiling. This helps with construction, but the dual tile outlines are hard to see.
 * As an ambo circle packing with explicit polar reciprocation between the VRP, a circle of radius $$1/2$$, and a hCVRP. This illustrates a uniform circle packing and an ambo uniform tiling, along with faint original and dual lattices.

Below, the 4-dual-uniform V[3.4.3.12; (3.4.6.4)2; 32.4.3.4] tiling is colored in all five respective ways. The scale and translation parameters coincide.

Below, the 2nd 4-dual-uniform V[32.4.12; 3.122; 32.4.3.4; 36] Krötenheerdt tiling (a gem tiling) is colored in all five respective ways. The scale and translation parameters coincide. In this case, the first three coloring methods are essentially redundant (but need to be illustrated):

Finally, the 2nd seven-dual-uniform V[4.6.12; 32.4.12; 32.4.3.4; 3.42.6; 34.6; 33.42; 36] Krötenheerdt tiling is colored in all five respective ways. The scale and translation parameters coincide. (Not done yet)

Regular and Laves tilings
The 3 regular and 8 semiregular Laves tilings are shown, with cyan polygons (the vertex polygons are magenta and cyan, respectively). Faint dissections/insets are shown whenever possible.

Krötenheerdt Duals with 2 planigons (Colored by Vertex Regular Planigon)
Note that Archimedean (monohedral) tilings are also 1-uniform as well; therefore, all 2-uniform tilings must be Krötenheerdt (every orbit must have a different vertex regular planigon tile). The vertex regular planigons in each tiling are listed by priority (percent of area of whole tiling), with vertex regular planigons in that order below. The original 2-uniform tilings are faintly outlined. All tilings are to simultaneous scale, and all vertex regular planigons are to simultaneous scale at a different size.

Fidget Spinners, Crosses, Hexagrams, and Clocks (Clusters or Semi-Complete Rings )
While we use regular polygons to construct vertex regular planigons, we can do the reverse; recover/construct regular polygons (e.g. the k-uniform tilings) by connecting the incenters of the cocyclic vertex regular planigons in these structures. Nonetheless, polar reciprocation about configuration vertices polygons of these clusters only yield equiangular polygons whose edges may not coincide, unlike edges of ambo circle packings.


 * Semi-complete rings of vertex regular planigons surrounding 3-equiangular vertices are called fidget spinners.
 * Semi-complete rings of vertex regular planigons surrounding 4-equiangular vertices are called crosses.
 * Semi-complete rings of vertex regular planigons surrounding 6-equiangular vertices are called hexagrams.
 * Semi-complete rings of vertex regular planigons surrounding 12-equiangular vertices are called clocks.

There is one single semi-complete ring of vertex regular planigons surrounding an 8-equiangular vertex (the pinwheel), which is found in the tetrakis square tiling. And there is one corresponding single semi-complete ring of vertex regular planigons surrounding the 4-equiangular vertex (the kisquadrille). So far, it is known that:

In fact, there are exactly 648 toys to play with (650 toys including the toys of the tetrakis square tiling). This implies an upper bound of $$648-77-67-110=394$$ (394) clocks!
 * There are at least 67 non-isometric fidget spinners.
 * There are at least 77 non-isometric crosses.
 * There are at least 110 non-isometric hexagrams.
 * There are more than 100 non-isometric clocks.

Also, seen below:


 * Smallest fidget spinner: triple hexagon.
 * Smallest cross: rectangular barn pentagons.
 * Smallest hexagram: sextuplet Floret pentagon.
 * The pinwheel: quadrille isosceles right triangles.
 * Smallest clock: dodectuplet tie kite

Finally, seen below:


 * Largest fidget spinner: triple obtuse triangle (dual fractalizing triangle).
 * Largest cross: rectangular 30-60-90 triangles (the kisrhombille).
 * Largest hexagram: triangular 30-60-90 triangles (the meta-triangle).
 * Largest clock: hexagramal obtuse triangle (in triakis triangular tiling).



Extending Convex Uniform Patches (Dual Uniform Version)
If we construct dual uniform patches made from isoperimetric and same-area planigons, then these are most of the results (Grunbaum, et. al., Tilings and Patterns, Section 2.8):

=== Constructing Unbalanced Hexagonal Rings of Triangles and Hexagons (Dual Uniform Version - Using Floret pentagon and Isosceles Trapezoid Transitions) === Non-uniform tilings (or arbitrary uniformity tilings) can be constructed by alternating hexagonal triangular rings around hexagons. The dual uniform version involves hexagonal hexagonal rings, transitioned by Floret pentagons and isosceles trapezoids, enclosing equilateral triangles.

Select Dual Krötenheerdt Tilings (Grunbaum, et al. 68)
A 3-, 4-, 5-, 6-, and 7-uniform Krötenheerdt tiling is featured on p. 68. Here are the duals, colored by vertex regular planigon:

Krotenheerdt Duals with 2 planigons
There are 20 tilings made from 2 unique planigons, the dual of 2-uniform tilings (Krotenheerdt Duals). Cyan polygons occupy more area than light pink polygons. The vertex polygons below each tiling are ordered in importance.

Krotenheerdt Duals with 3 planigons
There are 39 tilings made from 2 unique planigons, the dual of 3-uniform tilings (Krotenheerdt Duals). Cyan polygons occupy more area than light pink polygons, which occupy more area than green polygons (occasionally, the role of cyan and light pink switch). The duals are respective to the table below.

Krotenheerdt Duals with 4 planigons
There are 33 4-uniform dual tilings with 4 unique planigons. Cyan polygons occupy more area than light pink polygons, which occupy more area than green polygons, then turquoise polygons (occasionally, the role of cyan, light pink, and turquoise switch). The duals are respective to the table below.

Krotenheerdt Duals with 5 planigons
There are 15 5-uniform dual tilings with 5 unique planigons.

Krotenheerdt Duals with 6 planigons
There are 10 6-uniform dual tilings with 6 unique planigons.

Krotenheerdt Duals with 7 planigons
There are 7 7-uniform dual tilings with 7 unique planigons. Strangely enough, the 5th and 7th Krotenheerdt dual uniform-7 tilings have the same vertex types, even though they look nothing alike!

From $$n\ge 8$$ onward, there are no uniform n tilings with n vertex types, or no uniform n duals with n distinct (semi)planigons.

Fractalizing Dual k-Uniform Tilings
There are many ways of generating new k-dual-uniform tilings from old k-uniform tilings. For example, notice that the 2-uniform V[3.12.12; 3.4.3.12] tiling has a square lattice, the 4(3-1)-uniform V[343.12; (3.122)3] tiling has a snub square lattice, and the 5(3-1-1)-uniform V[334.12; 343.12; (3.12.12)3] tiling has an elongated triangular lattice. These higher-order uniform tilings use the same lattice but possess greater complexity. The dual fractalizing basis for theses tilings is as follows: The side lengths are dilated by a factor of $$2+\sqrt{3}$$:


 * Each triangle is replace by three V[3.122] polygons (the unit of the 1-dual-uniform V[3.122] tiling);
 * Each square is replaced by four V[3.122] and four V[3.4.3.12] polygons (the unit of the 2-dual-uniform V[3.122; V3.4.3.12] tiling);
 * Each hexagon is replaced by six deltoids V[3.4.6.4], six tie kites V[3.4.3.12], and six V[3.122] polygons (the unit of that 3-dual-uniform tiling)
 * Each dodecagon is dissected into six big triangles, six big squares, and one central hexagon, all which consist of above.

This can similarly be done with the truncated trihexagonal tiling as a basis, with corresponding dilation of $$3+\sqrt{3}$$.


 * Each triangle is replace by three V[4.6.12] polygons (the unit of the 1-dual-uniform V[4.6.12] tiling);
 * Each square is replaced by one square, four V[33.42] polygons, four V[3.4.3.12] polygons, and four V[32.4.12] polygons (the unit of that Krotenheerdt 4-dual-uniform tiling);
 * Each hexagon is replaced by six deltoids V[3.4.6.4] and thirty-six V[4.6.12] polygons (the unit of that 5-dual-uniform tiling)
 * Each dodecagon is dissected into six big triangles, six big squares, and one central hexagon, all which consist of above.

92-Uniform Tiling
Below is the fundamental unit of a 92-dual uniform tiling consisting of 14 vertex regular polygons, the maximum allowed in an dual uniform tiling; the V4.82 planigon cannot exist with any other planigon. This uses.

Miscellaneous
Listed on top are 65 of k-uniform tilings originally taken from Euclidean tilings by convex regular polygons. Another version is compared with the dual uniform tilings, coinciding and to the same scale. The dual uniform tilings are adequately colored''. Faint dissections/insets are shown whenever applicable. Scroll to see all such tilings.