Planigon
In geometry, a planigon is a convex polygon that can fill the plane with only copies of itself (isotopic to the fundamental units of monohedral tessellations).
Tilings are made by edge-to-edge connections by perpendicular bisectors of the edges of the original uniform lattice, or centroids along common edges (they coincide).
In the 1987 book, Tilings and patterns, Branko Grünbaum calls the vertex-uniform tilings Archimedean in parallel to the Archimedean solids.
The Laves tilings have vertices at the centers of the regular polygons, and edges connecting centers of regular polygons that share an edge.
This includes the 3 regular tiles (triangle, square and hexagon) and 8 irregular ones.
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.
All 14 uniform usable regular vertex planigons also hail[5] from the 6-5 dodecagram (where each segment subtends
The incircle of this dodecagram demonstrates that all the 14 VRPs are cocyclic, as alternatively shown by circle packings.
and the convex hull is precisely the regular dodecagons in the k-uniform tiling.
-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
In other words:[1] 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.
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.
The solution to Challenge Problem 9.46, Geometry (Rusczyk),[6] is in the Degree 3 Vertex column above.
Only eleven of these angle combinations can occur in a Laves Tiling of planigons.
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 tiling of regular polygons: On the other hand, these four can be used in k-dual-uniform tilings: Finally, assuming unit side length, all regular polygons and usable planigons have side-lengths and areas as shown below in the table: Side Lengths: 1 Side Lengths:
Each can be grouped by the number m of distinct vertex figures, which are also called m-Archimedean tilings.
[8] 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.
The 3 regular and 8 semiregular Laves tilings are shown, with planigons colored according to area as in the construction:
There are 39 tilings made from 3 types of planigons (Krotenheerdt Duals):
There are 33 tilings made from 4 types of planigons (Krotenheerdt Duals):
The last two dual uniform-7 tilings have the same vertex types, even though they look nothing alike!
as seen below: To enlarge the planigons V32.4.12 and V3.4.3.12 using the truncated trihexagonal method, a scale factor of
By two 9-uniform tilings in [10] a big fractalization is achieved by a scale factor of 3 in all planigons.
[7][11] 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,[7] 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).
Finally, there are 7-5 tilings using all clock planigons:[10] Each uniform tiling corresponds to a circle packing, in which circles of diameter 1 are placed at all vertex points, corresponding to the planigons.
