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| Sections: Definition | Odds | Compounds | Chirality | Partitions | Alternated | Fractional | Sectional | Polychains and partials | Extended | Equiangular | Specification |
Polytile NotationDefinitionA polytile or p-tile as an equilateral polygon notated by a set of integers representing vertex angles. The p is an even whole number 4 or greater. The polytile angles representing multiples of a 360°/p degree turns. Angles are measured as turn angles, zero for straight (colinear edges), positive for counterclockwise turns, and negative for clockwise turns. For example: Tetratiles, Hexatiles, Octatiles, Decatiles, and Dodecatiles have turn angles that are integer multiples of 90°, 60°, 45°, 36° and 30° turns respectively.
A regular n-gon tile is represented by n 1s. The sum of these numbers equal n. For example 111111 or 1.1.1.1.1.1 or 16 is a regular hexagon, with 6 60° angles. Odd-sided regular polygons can only be generated by even, 2n-tiles. For example a triangle is 222 or 2.2.2 or 23, with 3 120° angles. Zero indices can be given for colinear edges, and negative indices allow for concave, self-contacting and self-intersecting tiles. For example 1111 or 112 or 14 is a square, while a 2:1 rectangle is 110110 or 1102. An exponent is always at the end of an expression, and applies to the full sequence. A negative exponent can imply a reverse order for a chiral pair. For example 4321 = 1234^-1. A regular n-gon has r2n symmetry. For example a square is r8 symmetry. Odd (2n+1)-tiles are only a subset of 2(2n+1)-tiles and are geometrically and notionally identical. For example, there are 3 strictly convex hexatiles, 6:1^6, 6:1.2^2, and 6:2^3, with one concave star shown 6:-1.2^6.
Odd polytilesPolytile expressions with odd p, can be defined, but fail to have a simple graphical representation since the regular p-gons must alternate in two orientations to connect edge-to-edge. Such angles are expressed more easily as a 2p-tiles, converted by doubling p, and doubling all the angles ai: p:a1.a2…am^n --> 2p:2a1.2a2…2am^n. For example a regular pentagon, represented as 5:1^5, can be represented by a doubling as decatile 10:2^5, and a regular pentagram 5:2^5, as 10:4^5. Actual construction of equilateral polygons from edge-to-edge odd regular p-gons exist as the subset of 2p-tiles which have only odd integer turns. For example we could say there are 2 strictly-convex pentatiles, represented notationally as decatiles, 10:1^10, and 10:1.1.3^2. On the pentagons, turns have to be counted as opposite vertex (an illegal connection 0), and first edge turn as 1, skipping illegal vertex at 2, and then second turn-edge as 3...
Covers and compound notationIf p:a1.a2…am^n is a valid polytile, then p:a1.a2…am^nc is a degenerate c-cover of it, repeating the same vertices and edges c times. A multicovered, or c-cover polygon is degenerate and can’t be seen, but have a topological existence. A standard reinterpretation of a c-cover is a c-compound or (c-part). This is expressed in regular polygons and stars {p/q} --> c{a/b}, where c=gcd(p,q), and a=p/c, b=q/c. Compounds add rotated copies, giving ac-fold cyclic symmetry. For example {6/2} is a double-covered triangle, can be reinterpreted as a compound of 2 triangles 2{3}, also called a hexagram. This compounding can be generalized for any polytile. A c-cover polytile, '''p:a1.a2…am^nc''', is written as a c-compound c*p:a1.a2…am^n, interpreted as c rotated copies of '''p:a1.a2…am^n'''. A c-compound m-adic nc-gram'' has mnc vertices. For example, as a dodecatile, a square is 12:3^4, while a double-cover square is 12:3^8, triple-cover 12:3^12, and quadruple-cover 12:3^16, with a 2-compound square 2*12:3^4, and 3-compound square 3*12:3^4. The double-cover and quadruple covers are unfilled due to even-densities being unfilled.
Chiral pairsNegative exponents repeat a sequence backwards, including ^-1. This allows chiral pairs to be expressed by the same sequence and is helpful in larger polytiles to avoid a need to relist the vertices backwards.
If a polytile has reflective symmetry, this will generate the same polytile. This equivalence is a definition of a reflective polytile. For example, 12:1.2.3^2 is a chiral dodecatile, and 12:1.2.3^-2 is its chiral copy, 12:3.2.1^2.
Reflective symmetry notationReflective polytiles have the form p:a.(b1.b2…bm).c.(bm.bm-1…b1)^n, or p:a.B.c.(B^-1)^n, where a and c are optional, exist if a vertex passes through the lines of reflection. B can be sequence length zero if there are no vertices off the reflection lines. This can be expressed more compactly with 4 pipe operators | , defining 3 partitions, |a|B|c|^n = p:a.B.c.(B^-1)^n, allowing the reverse polychain B^-1 or bm.bm-1…b1 to be suppressed.
For example, these convex octadecatile (18-tiles) are shown in partition notation if they have reflections, and otherwise ordinary polytile notation.
Alternated powersSome polychain sequences have the form (a1.a2…am).(-a1.-a2…-am), with a second set repeating the first, but turning in the opposite direction. This has an notation a1.a2…am^~2, signifying 2 copies, the second opposite turns. Such expressions can exist in cross-polygons, and also in frieze groups, so 3:1^~10 = 3:1.-1^5, 5 cycle zip-zag pattern. And 4:1.1^~10 = 4:1.1.-1.-1^5, 5 square wave patterns. a1.a2…am^~n makes glide symmetry, frieze group p11g, and doubles to p2mg for ||a1.a2…am||^~n. For example, 12:1.2.3.-1.-2.-3^~5 can be written more compactly as 12:1.2.3^~10. Similarly a1.a2…am^~-2 = (a1.a2…am).(-am.…-a2.-a1), reversing both direction and signs of alternate sets. And a1.a2…am^~-2n = (a1.a2…am).(-am…-a2.-a1)^n makes 2-fold rotations, frieze group p2, doubles to p2mg for ||a1.a2…am||^~-n. For example, 12:1.2.3.-3.-2.-1^~5 can be written more compactly as 12:1.2.3^~-10.
Fractional notationA fractional polytile is a central dissection a polytile and adding 2 radial edges and the central point. Polytiles notation is p:a1…am^n/f, with f as a divisor of nm, starting at vertex a1. This allows symmetric fractional polytiles like isosceles triangles, kite, dart, and other common and uncommon shapes of interest.
Extended notationA polytile, p:a1.a2…am^n, can be extended by operator ! by a polychain p:b1.b2…bl^k with a special notation, and optionally repeated recursively r times.
There are 2 forms:
If the polychain has sum s=(b1+b2…+bl)k nonzero, that sum must be subtracted from every element of a1.a2…am^n. a'i = (ai -s) for all i. For example, the hexagonal concave star 12:-2.4^6 has a simple extend of 0,1,-1, or alternating +1,-1.
Equiangular notationPolytile notation allows the construction of equiangular polygons with integer edge lengths, expressed as (0^(a-1)) for an a-length edge. The colinear edges (with turn 0) are interpreted as vertices for equilateral and creating integer length edges for equiangular polytiles. Equiangular notation allows this in a more compact format <p/q>:e1.e2…em^n, using turn angles of a regular {p/q} polygon, sequential edge integer lengths e1..em, repeated m times. If edge lengths are negative, the turn will go cw instead of ccw. This is translated into p:±q.(0^(e1-1))....±q.(0^(em-1))^n, each (0^a) terms extends as equilateral colinear edges. Partitions can also be applied for equiangular notation to express reflective symmetry. For example, these equiangular hexagons have 1, 2, or 3 different edge lengths. <6>:1.2.3^2 is the same as equilateral notation: 6:1.0.1.0.0.1^2.
Polytile specification
© 2020-2022 Created by Tom Ruen
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