Planigon

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.{{cite book

| author1=Grünbaum, Branko

| author2=Shephard, G. C.

| title=Tilings and Patterns

| publisher=W. H. Freeman and Company

| year=1987

| isbn=0-7167-1193-1

| pages=[https://archive.org/details/isbn_0716711931/page/59 59, 96]

| url-access=registration

| url=https://archive.org/details/isbn_0716711931/page/59

}}{{cite book

| first1 = John H.

| last1 = Conway

| author-link = John Horton Conway

| first2 = Heidi

| last2 = Burgiel

| author-link2 =

| first3 = Chaim

| last3 = Goodman-Strauss

| author-link3 = Chaim Goodman-Strauss

| title = The Symmetries of Things

| chapter = Chapter 21, Naming the Archimedean and Catalan polyhedra and tilings, Euclidean Plane Tessellations

| date = April 18, 2008

| url = https://akpeters.com/product.asp?ProdCode=2205

| location =

| publisher = A K Peters / CRC Press

| page = 288

| isbn = 978-1-56881-220-5

| archive-url = https://web.archive.org/web/20100919143320/https://akpeters.com/product.asp?ProdCode=2205

| archive-date = 2010-09-19

}} They're also called Shubnikov–Laves tilings after Shubnikov, Alekseĭ Vasilʹevich.[https://books.google.com/books?id=5rPnCAAAQBAJ&dq=Shubnikov%E2%80%93Laves+tilings&pg=PA169 Encyclopaedia of Mathematics: Orbit - Rayleigh Equation], 1991 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.{{SpringerEOM|title=Planigon|id=Planigon&oldid=31578|first=A. B.|last=Ivanov}} 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 (or V4.8²) means isosceles triangle tiles with one corner with four triangles, and two corners containing eight triangles.

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-to-edge-midpoint line segments of all regular polygons in the k-uniform tilings.

All 14 uniform usable regular vertex planigons also hail{{Cite web|url=https://www.biglist-tilings.com/|title=THE BIG LIST SYSTEM OF TILINGS OF REGULAR POLYGONS|website=THE BIG LIST SYSTEM OF TILINGS OF REGULAR POLYGONS|language=en-US|access-date=2019-08-30}} from the 6-5 dodecagram (where each segment subtends $5\backslash pi/6$ radians, or 150 degrees).

The incircle of this dodecagram demonstrates that all the 14 VRPs are cocyclic, as alternatively shown by circle packings. The ratio of the incircle to the circumcircle is:

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

and the convex hull is precisely the regular dodecagons in the k-uniform tiling. The equilateral triangle, square, regular hexagon, and regular dodecagon; are shown above with the VRPs.

In fact, any group of planigons can be constructed from the edges of a $2k\backslash text\{-\}(k-1)$ polygram, where $k=\backslash gcd(n\_1,\backslash dots,n\_m)$ and $n\_i$ is the number of sides of sides in the RP adjacent to each involved vertex figure. This is because the circumradius $\backslash frac\{1\}\{2\}\backslash csc\backslash frac\{\backslash pi\}\{n\_i\}$ of any regular $n\_i$-gon (from the vertex to the centroid) is the same as the distance from the center of the polygram to its line segments which intersect at the angle $2\backslash pi/n\_i$, since all $2k\backslash text\{-\}(k-1)$ polygrams admit incircles of inradii $1/2$ tangent to all its sides.

In Tilings and Patterns, Grünbaum also constructed the Laves tilings using monohedral tiles with regular vertices. A vertex is regular if all angles emanating from it are equal. In other words:

1. All vertices are regular,
2. All Laves planigons are congruent.

In this way, all Laves tilings are unique except for the square tiling (1 degree of freedom), barn pentagonal tiling (1 degree of freedom), and hexagonal tiling (2 degrees of freedom):

When applied to higher dual co-uniform tilings, all dual coregular planigons can be distorted except for the triangles (AAA similarity), with examples below:

For edge-to-edge Euclidean tilings, the interior angles of the convex polygons meeting at a vertex must add to 360 degrees. A regular {{mvar|n}}-gon has internal angle $\backslash left(1-\backslash frac\{2\}\{n\}\backslash right)180^\backslash circ$ degrees. There are seventeen combinations of regular polygons whose internal angles add up to 360 degrees, each being referred to as a species of vertex; in four cases there are two distinct cyclic orders of the polygons, yielding twenty-one types of vertex.

In fact, with the vertex (interior) angles $60^\backslash circ,90^\backslash circ,108^\backslash circ,120^\backslash circ,128\backslash frac\{4\}\{7\}^\backslash circ,135^\backslash circ,140^\backslash circ,144^\backslash circ,$

147\frac{3}{11}^\circ,150^\circ,\dots, we can find all combinations of admissible corner angles according to the following rules:

1. Every vertex has at least degree 3 (a degree-2 vertex must have two straight angles or one reflex angle);
2. If the vertex has degree $d$, the smallest $d-1$ polygon vertex angles sum to over $180^\{\backslash circ\}$;
3. The vertex angles add to $360^\{\backslash circ\}$, and must be angles of regular polygons of positive integer sides (of the sequence $60^\backslash circ,90^\backslash circ,108^\backslash circ,120^\backslash circ,128\backslash frac\{4\}\{7\}^\backslash circ,135^\backslash circ,140^\backslash circ,144^\backslash circ,\; 147\backslash frac\{3\}\{11\}^\backslash circ,150^\backslash circ,\backslash dots$).

Using the rules generates the list below:

File:Clusters Of Not Usable Planigons.svg

*The $90^\{\backslash circ\}\backslash text\{-\}135^\{\backslash circ\}\backslash text\{-\}135^\{\backslash circ\}~(\backslash times\; 1)$ cannot coexist with any other vertex types.

The solution to Challenge Problem 9.46, Geometry (Rusczyk),{{Cite book|last=Rusczyk, Richard.|title=Introduction to geometry|date=2006|publisher=AoPS Inc|isbn=0977304523|location=Alpine, CA|oclc=68040014}} is in the Degree 3 Vertex column above. A triangle with a hendecagon (11-gon) yields a 13.2-gon, a square with a heptagon (7-gon) yields a 9.3333-gon, and a pentagon with a hexagon yields a 7.5-gon). Hence there are $1(1)+(1(2)+1)+(3(2)+1)+10=21$ combinations of regular polygons which meet at a vertex.

Only eleven of these angle combinations can occur in a Laves Tiling of planigons.

In particular, if three polygons meet at a vertex and one has an odd number of sides, the other two polygons must be the same. If they are not, they would have to alternate around the first polygon, which is impossible if its number of sides is odd. By that restriction these six cannot appear in any k-dual-uniform tiling:

File:Not Usable Planigons.svg

On the other hand, these four can be used in k-dual-uniform tilings:

File:Dualization_Deriving_Four_Semiplanigons.png dual for each planigon V3².4.12, V3.4.3.12, V3².6², V3.4².6. And all quadrilaterals can tile the plane.]]

Finally, assuming unit side length, all regular polygons and usable planigons have side-lengths and areas as shown below in the table:

Every dual uniform tiling is in a 1:1 correspondence with the corresponding uniform tiling, by construction of the planigons above and superimposition.

Such periodic tilings may be classified by the number of orbits of vertices, edges and tiles. If there are k orbits of planigons, a tiling is known as k-dual-uniform or k-isohedral; if there are t orbits of dual vertices, as t-isogonal; if there are e orbits of edges, as e-isotoxal.

k-dual-uniform tilings with the same vertex faces can be further identified by their wallpaper group symmetry, which is identical to that of the corresponding k-uniform tiling.

1-dual-uniform tilings include 3 regular tilings, and 8 Laves tilings, with 2 or more types of regular degree vertices. There are 20 2-dual-uniform tilings, 61 3-dual-uniform tilings, 151 4-dual-uniform tilings, 332 5-dual-uniform tilings and 673 6--dualuniform tilings. Each can be grouped by the number m of distinct vertex figures, which are also called m-Archimedean tilings.[http://www2.math.uu.se/research/pub/Lenngren1.pdf k-uniform tilings by regular polygons] {{webarchive|url=https://web.archive.org/web/20150630004622/http://www2.math.uu.se/research/pub/Lenngren1.pdf |date=2015-06-30 }} Nils Lenngren, 2009 {{verify source |date=September 2019 |reason=This ref was deleted Special:Diff/906953539 by a bug in VisualEditor and later restored by a bot from the original cite located at Special:Permalink/906952837 cite #2 - verify the cite is accurate and delete this template. User:GreenC bot/Job 18}}

Finally, if the number of types of planigons is the same as the uniformity (m = k below), then the tiling is said to be dual Krotenheerdt. In general, the uniformity is greater than or equal to the number of types of vertices (m ≥ k), as different types of planigons necessarily have different orbits, but not vice versa. Setting m = n = k, there are 11 such dual tilings for n = 1; 20 such dual tilings for n = 2; 39 such dual tilings for n = 3; 33 such dual tilings for n = 4; 15 such dual tilings for n = 5; 10 such dual tilings for n = 6; and 7 such dual tilings for n = 7.

The 3 regular and 8 semiregular Laves tilings are shown, with planigons colored according to area as in the construction:

File:1-Co-Uniform Tilings.png

File:Central Park Manhole Tiling.png

• A degree-six vertex can be replaced by a center regular hexagon and six edges emanating thereof;
• A degree-twelve vertex can be replaced by six deltoids (a center deltoidal hexagon) and twelve edges emanating thereof;
• A degree-twelve vertex can be replaced by six Cairo pentagons, a center hexagon, and twelve edges emanating thereof (by dissecting the degree-6 vertex in the center of the previous example).

This is done above for the dual of the 3-4-6-12 tiling. The corresponding uniform process is dissection, and is shown here.

There are 20 tilings made from 2 types of planigons, the dual of 2-uniform tilings (Krotenheerdt Duals):

File:2-Co-Uniform Tilings.png

There are 39 tilings made from 3 types of planigons (Krotenheerdt Duals):

File:3-Kroten Co-Uniform.png

There are 33 tilings made from 4 types of planigons (Krotenheerdt Duals):

File:Everything_Uniform_4_Krotenheerdt.png

There are 15 5-uniform dual tilings with 5 unique planigons:

File:5-Krotenheerdt Co-Uniform Tilings.png

There are 10 6-uniform dual tilings with 6 unique planigons:

File:6-Krotenheerdt Co-Uniform Tilings.png

There are 7 7-uniform dual tilings with 7 unique planigons:

File:7-Krotenheerdt Co-Uniform Tilings.png

The last two dual uniform-7 tilings have the same vertex types, even though they look nothing alike!

From $n\backslash ge\; 8$ onward, there are no uniform n tilings with n vertex types, or no uniform n duals with n distinct (semi)planigons.{{Cite web|url=https://oeis.org/search?q=11,20,39,33,15,10,7&sort=&language=&go=Search|title=11,20,39,33,15,10,7 - OEIS|website=oeis.org|access-date=2019-06-26}}

There are many ways of generating new k-dual-uniform tilings from other k-uniform tilings. Three ways is to scale by $1+\backslash sqrt\{3\},2+\backslash sqrt\{3\},3+\backslash sqrt\{3\}$ as seen below:

To enlarge the planigons V3².4.12 and V3.4.3.12 using the truncated trihexagonal method, a scale factor of $2(3+\backslash sqrt\{3\})$ must be applied:

alt=

By two 9-uniform tilings in a big fractalization is achieved by a scale factor of 3 in all planigons. In the case of s,C,B,H its own planigon is in the exact center:

alt=

The two 9-uniform tilings are shown below, fractalizations of the demiregulars DC and DB, and a general example on S²TC:

Dual co-uniform tilings (red) along with the originals (blue) of selected tilings.J. E. Soto Sánchez, [http://chequesoto.info/thesis.html On Periodic Tilings with Regular Polygons], PhD Thesis, IMPA, Aug 2020. Generated by centroid-edge midpoint construction by polygon-centroid-vertex detection, rounding the angle of each co-edge to the nearest 15 degrees. Since the unit size of tilings varies from 15 to 18 pixels and every regular polygon slightly differs, there is some overlap or breaks of dual edges (an 18-pixel size generator incorrectly generates co-edges from five 15-pixel size tilings, classifying some squares as triangles).

alt=

Other edge-edge construction comparisons. Rotates every 3 seconds.

Below are affine linear expansions of other uniform tilings, from the original to the dual and back:

The first 12-uniform tiling contains all planigons with three types of vertices, and the second 12-uniform tiling contains all types of edges.

File:104-Uniform 14VRP p4g (6px).gif. Such tilings can assume any wallpaper group except for p4m since p4m only admits planigons O, S, T, D, s, C, B, H.{{Cite web |title=Tessellation catalog |url=https://zenorogue.github.io/tes-catalog/?c=k-uniform/ |access-date=2022-03-21 |website=zenorogue.github.io}}]]

If $a$-$b$ tiling means $a$ dual uniform, $b$ Catalaves tiling, then there exists a 11-9 tiling, a 13-10 tiling, 15-11 tiling, a 19-12 tiling, two 22-13 tilings, and a 24-14 tiling. Also exists a 13-8 slab tiling and a 14-10 non-clock tiling. Finally, there are 7-5 tilings using all clock planigons:

{{See also|Circle packing#Other_packings}}

Each uniform tiling corresponds to a circle packing, in which circles of diameter 1 are placed at all vertex points, corresponding to the planigons. Below are the circle packings of the Optimized Tilings and all-edge tiling:

File:Circle_Packing_Optimal_Tilings.png

A slideshow of all 94 5-dual-uniform tilings with 4 distinct planigons. Changes every 6 seconds, cycles every 60 seconds.

alt=

All tilings with regular dodecagons in are shown below, alternating between uniform and dual co-uniform every 5 seconds:{{wide image|Clock Tilings (Uniform Tilings with Regular Dodecagons).gif|19631px|All tilings with regular dodecagons are shown below, alternating between uniform and dual co-uniform every 5 seconds.}}

A comparison of 65 k uniform tilings in uniform planar tilings and their dual uniform tilings. The two lower rows coincide and are to scale:{{wide image|Final Planar Tiling Project (30 Tiles).png|14916px|A comparison of 65 k uniform tilings in uniform planar tilings and their dual uniform tilings. The two lower rows coincide and are to scale.}}

{{reflist}}

• [https://onlinelibrary.wiley.com/doi/abs/10.1002/(SICI)1099-0526(199907/08)4:6%3C31::AID-CPLX7%3E3.0.CO;2-D Planigon tessellation cellular automata] Alexander Korobov, 30 September 1999
• [http://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=im&paperid=3749&option_lang=eng B. N. Delone, “Theory of planigons”], Izv. Akad. Nauk SSSR Ser. Mat., 23:3 (1959), 365–386

Category:Types of polygons

Category:Euclidean tilings