Planigon

s, eight planigons, four demiregular planigons, and six not usable planigon triangles which cannot take part in dual uniform tilings; all to scale. 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). In the Euclidean plane there are 3 regular planigons; equilateral triangle, squares, and regular hexagons; and 8 semiregular planigons; and 4 demiregular planigons which can tile the plane only with other planigons. All angles of a planigon are whole divisors of 360°. 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). Tilings made from planigons can be seen as dual tilings to the regular, semiregular, and demiregular tilings of the plane by regular polygons.

s, eight planigons, four demiregular planigons, and six not usable planigon triangles which cannot take part in dual uniform tilings; all to scale. 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). In the Euclidean plane there are 3 regular planigons; equilateral triangle, squares, and regular hexagons; and 8 semiregular planigons; and 4 demiregular planigons which can tile the plane only with other planigons. All angles of a planigon are whole divisors of 360°. 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). Tilings made from planigons can be seen as dual tilings 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 (or V4.8 2 ) means isosceles triangle tiles with one corner with four triangles, and two corners containing eight triangles. == 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-to-edge-midpoint line segments of all regular polygons in the k-uniform tilings. \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 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\text{-}(k-1) polygram, where k=\gcd(n_1,\dots,n_m) and n_i is the number of sides in the RP adjacent to each involved vertex figure. This is because the circumradius \frac{1}{2}\csc\frac{\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\pi/n_i , since all 2k\text{-}(k-1) polygrams admit incircles of inradii 1/2 tangent to all its sides. The solution to Challenge Problem 9.46, Geometry (Rusczyk), 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. 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. B. N. Delone, “Theory of planigons”, Izv. Akad. Nauk SSSR Ser. Mat., 23:3 (1959), 365–386

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