Conway criterion

A golygon which is a Prototile satisfying the Conway criterion. The four midpoints of centrosymmetric symmetry are indicated by black dots.

In the mathematical theory of tessellations, the Conway criterion, named for the English mathematician John Horton Conway, describes rules for when a prototile will tile the plane; it consists of the following requirements:^[1] The tile must be a closed topological disk with six consecutive points A, B, C, D, E, and F on the boundary such that:

• the boundary part from A to B is congruent by translation to the boundary part from E to D
• each of the boundary parts BC, CD, EF, and FA is centrosymmetric—that is, each one is congruent to itself when rotated by 180-degrees around its midpoint
• some of the six points may coincide but at least three of them must be distinct.^[2]

Any prototile satisfying Conway's criterion admits a periodic tiling of the plane—and does so using only translation and 180-degree rotations. The Conway criterion is a sufficient condition to prove that a prototile tiles the plane but not a necessary one; there are tiles that fail the criterion and still tile the plane.^[3]

Examples

A hexagonal tiling with centrosymmetric hexagons

The two nonominoes not satisfying the Conway criterion but able to tile the plane

In its simplest form the criterion states that any hexagon with a pair of opposite sides that are parallel and congruent will tessellate the plane by translation, called hexagonal parallelogons.^[4] But when some of the points coincide, the criterion can apply to other polygons and even to shapes with curved perimeters.^[5]

The Conway criterion discriminates many shapes, especially polyforms: except the two tiling nonominoes on the right, all tiling polyominos up to the nonominoes can form a patch of at least one tile which satisfies the criterion.^[3] These figures also show that the criterion is a sufficient but not necessary condition for a prototile to tile the plane.

References

External links

• History and introduction to polygon models, polyominoes and polyhedra, by Anthony J Guttmann
• G C Rhoads (2005) Planar tilings by polyominoes, polyhexes, and polyiamonds, Journal of Computational and Applied Mathematics, V 174 p 329-353

Tessellation   Periodic Pythagorean Rhombille Schwarz triangle Rectangle Domino Uniform tiling & honeycomb Coloring Convex Kisrhombille Wallpaper group Wythoff   Aperiodic Ammann–Beenker Aperiodic set of prototiles List Einstein problem Socolar–Taylor Gilbert Penrose Pentagonal Pinwheel Quaquaversal Rep-tile & Self-tiling Sphinx Truchet   Other Anisohedral & Isohedral Architectonic & catoptric Circle Limit III Computer graphics Honeycomb Isotoxal Packing Problems Domino Wang Heesch's Squaring Prototile Conway criterion Girih Regular Division of the Plane Regular grid Substitution Voronoi Voderberg   By vertex type Spherical 2n 33.n V33.n 42.n V42.n Regular 2∞ 36 44 63 Semi- regular 32.4.3.4 V32.4.3.4 33.42 33.∞ 34.6 V34.6 3.4.6.4 (3.6)2 3.122 42.∞ 4.6.12 4.82 Hyper- bolic 32.4.3.5 32.4.3.6 32.4.3.7 32.4.3.8 32.4.3.∞ 32.5.3.5 32.5.3.6 32.6.3.6 32.6.3.8 32.7.3.7 32.8.3.8 33.4.3.4 32.∞.3.∞ 34.7 34.8 34.∞ 35.4 37 38 3∞ (3.4)3 (3.4)4 3.4.62.4 3.4.7.4 3.4.8.4 3.4.∞.4 3.6.4.6 (3.7)2 (3.8)2 3.142 3.162 (3.∞)2 3.∞2 42.5.4 42.6.4 42.7.4 42.8.4 42.∞.4 45 46 47 48 4∞ (4.5)2 (4.6)2 4.6.12 4.6.14 V4.6.14 4.6.16 V4.6.16 4.6.∞ (4.7)2 (4.8)2 4.8.10 V4.8.10 4.8.12 4.8.14 4.8.16 4.8.∞ 4.102 4.10.12 4.122 4.12.16 4.142 4.162 4.∞2 (4.∞)2 54 55 56 5∞ 5.4.6.4 (5.6)2 5.82 5.102 5.122 (5.∞)2 64 65 66 68 6.4.8.4 (6.8)2 6.82 6.102 6.122 6.162 73 74 77 7.62 7.82 7.142 83 84 86 88 812 8.62 8.162 ∞3 ∞4 ∞5 ∞∞ ∞.62 ∞.82