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Definition  Compounds  Chirality  Partitions  Alternated  Equiangular  Polychains and partials  Fractional  Extended 
Polytile NotationDefinitionA polytile or ptile 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 ngon tile is represented by n 1s. The sum of these numbers equal n. For example 111111 or 1.1.1.1.1.1 or 1^{6} is a regular hexagon, with 6 60° angles. Oddsided regular polygons can only be generated by even, 2ntiles. For example a triangle is 222 or 2.2.2 or 2^{3}, with 3 120° angles. Zero indices can be given for colinear edges, and negative indices allow for concave, selfcontacting and selfintersecting tiles. For example 1111 or 11^{2} or 1^{4} is a square, while a 2:1 rectangle is 110110 or 110^{2}. 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 ngon 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, a hexagon is both 1^{6} as a cycle of 6 hexagons, and 1^{6} as a cycle of 6 triangles in alternating orientations. Covers and compound notationA 4th parameter can be extracted, ccover. If p:a_{1}.a_{2}…a_{m}^n is a valid polytile, then p:a_{1}.a_{2}…a_{m}^nc is a degenerate ccover of it, repeating the same vertices and edges c times. Multicovered polygons are degenerate and can’t be seen, but have a topological existence. One reinterpretation of a ccover regular polygon {ca/cb} factors out as c{a/b}, and draws it as a ccompound or (cpart), adding rotated copies, giving acfold cyclic symmetry. This can be generalized for any polytile. A ccover polytile, '''p:a_{1}.a_{2}…a_{m}^nc''', is written as a ccompound c*p:a_{1}.a_{2}…a_{m}^n, interpreted as c rotated copies of '''p:a_{1}.a_{2}…a_{m}^n'''. A ccompound madic ncgram'' has mnc vertices. For example, as a dodecatile, a square is 12:3^4, while a doublecover square is 12:3^8, triplecover 12:3^12, and quadruplecover 12:3^16, with a 2compound square 2*12:3^4, and 3compound square 3*12:3^4. The doublecover and quadruple covers are unfilled due to evendensities being unfilled. Chiral pairsNegative exponents repeat a sequence backwards, including ^1. This allows chiral pairs to be expressed by the same sequence. If a polytile has reflection symmetry, this will have no effect.
For example, 12:1.2.3^2 is a chiral dodecatile, and 12:3.2.1^2 is its chiral copy, but you can also describe as 12:1.2.3^2. Partition notationReflective polytiles have the form p:a.(b_{1}.b_{2}…b_{m}).c.(b_{m}.b_{m1}…b_{1})^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, and B can be sequence length zero if no vertices off the reflection lines. This can be expressed more compactly with 4 pipe operator , as 3 partitions, aBc^n, allowing the reverse polychain B^1 or b_{m}.b_{m1}…b_{1} to be suppressed.
For example, these convex octadecatile (18tiles) are shown in partition notation if they have reflections, and otherwise ordinary polytile notation. Alternated powersSome polychain sequences have the form (a_{1}.a_{2}…a_{m}).(a_{1}.a_{2}…a_{m}), with a second set repeating the first, but turning in the opposite direction. This has an notation a_{1}.a_{2}…a_{m}^~2, signifying 2 copies, the second opposite turns. Such expressions can exist in crosspolygons, and also in frieze groups, so 3:1^~10 = 3:1.1^5, 5 cycle zipzag pattern. And 4:1.1^~10 = 4:1.1.1.1^5, 5 square wave patterns. a_{1}.a_{2}…a_{m}^~n makes glide symmetry, frieze group p11g, and doubles to p2mg for a_{1}.a_{2}…a_{m}^~n. For example, 12:1.2.3.1.2.3^~5 can be written more compactly as 12:1.2.3^~10. Similarly a_{1}.a_{2}…a_{m}^~2 = (a_{1}.a_{2}…a_{m}).(a_{m}.…a_{2}.a_{1}), reversing both direction and signs of alternate sets. And a_{1}.a_{2}…a_{m}^~2n = (a_{1}.a_{2}…a_{m}).(a_{m}…a_{2}.a_{1})^n makes 2fold rotations, frieze group p2, doubles to p2mg for a_{1}.a_{2}…a_{m}^~n. For example, 12:1.2.3.3.2.1^~5 can be written more compactly as 12:1.2.3^~10.
Equiangular notationPolytile notation allows the construction of equiangular polygons with integer edge lengths, expressed as (0^(a1)) for an alength 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>:e_{1}.e_{2}…e_{m}^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^(e11))....±q.(0^(em1))^n. 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. Polychains and Partial notationA polytile expression that does not close is called a polychain. A polychain expression is not a polygon, but has uses. A polychain can be closed by adding a final edge between the first and last vertices.
Fractional notationA fractional polytile is a central dissection a polytile and adding 2 radial edges and the central point. Polytiles notation is p:a_{1}…a_{m}^n/f, with f as a divisor of n, starting at vertex a_{1}. This allows symmetric fractional polytiles like isosceles triangles, kite, dart, and other common and uncommon shapes of interest.
Extended notationA polytile, p:a_{1}.a_{2}…a_{m}^n, can be extended by operator ! by a polychain p:b_{1}.b_{2}…b_{m}^k with a special notation, and optionally repeated recursively r times..
There are 2 forms:
For example, the hexagonal concave star 12:2.4^6 has a simple extend of 0,1,1, or alternating +1,1.
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