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User:Tomruen/Polytile

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This heptagonal dodecatile, 1312131, has bilateral symmetry, i2

In geometry, a polytile is an equilateral polygon defines with specific angles. An n-tile is a polygon defined by an edge-to-edge cycle of regular n-gons. The vertices of the n-tile are positioned at the center of the n-gons.

A polytiling is a tessellation made from one or more polytiles from the set defined by the angles of a regular n-gon.

Polytile definition

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There is one equilateral concave pentagonal decatile, 24-242. It exists in the endo-dodecahedron

A 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 p-tile notational is: p:a1.a2…am^n. (A square is 4:1^4 as four quarter turns, 90°.)
  • Each turn angle ai' index is an integer less than p/2. (Indices ±p/2 are half turns, 180°.)
  • A ^n exponent repeats sequence n times. A chiral pair is mirrored as ^-n, including ^-1.
  • The turning number is computed by: t=n(a1+a2+…+am)/p. If |t|=1, it is a simple polytile (convex or concave), otherwise, a star polytile.
  • If the “p:” is not given, we assume simple (t=1), and compute p=n(a1+a2+…+am).
  • Turn angles ai may be allowed outside (-p/2,p/2), but to correctly compute the turning number, they need be given within the signed range modulo p.
  • A p-tile can be up-scaled: p:a1.a2..am^n to kp:ka1.ka2..kam^n, for any whole number k. In reverse, any kp-tile, with all indices as multiples of k, can be down-scale to a p-tile.
  • A polytile expression that does not close as cycle is called a polychain.

A regular n-gon tile is represented by n 1's. 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 collinear 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, and applies to the full string of indices. 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, a hexagon is both 16 as a cycle of 6 hexagons, and 16 as a cycle of 6 triangles in alternating orientations.

Geometric interpretation

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This heptagonal decatile, -124-2223, has no symmetry, a1. It can be seen as the union of a concave pentagon and a rhombus

Polytiles have a helpful geometric interpretation where each p-tile can be described by a cyclic path of regular p-gons connected edge-to-edge. The p-tile is constructed with its vertices centered on each regular p-gon of the cycle, with equal edges defined between adjacent p-gons.

Classifying and naming

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A p-tile can be called a t-turn m-adic n-gram, with 3 parameters n, m, t:

  1. A polygram or n-gram, like monogram, digram, trigram, tetragram, pentagram, hexagram, etc. is any equilateral polygon with n-fold rotational symmetry.
  2. A polyad or m-ad, like monad, dyad, triad, tetrad, pentad, hexad, etc. as the number of vertices with its symmetry of a polygram.
  3. t-turn is the turning number. Crossed polygons, like a regular polygon with two vertices flipping places, can be a 0-turn which may be called counterturn. If we are interested in clockwise polygons, we can call them contra-t-turn.

An m-adic n-gram is an equilateral nm-gon. If m is 1, that qualifier can be suppressed.

Covers and compounds

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A 4th parameter can be extracted, c-cover. If 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.

Multicovered polygons are degenerate and can’t be seen, but have a topological existence.

An ordinary interpretation of a c-cover regular polygon {ca/cb} factors out as c{a/b}, and draws it as a c-compound, adding rotated copies, giving ac-fold cyclic symmetry. This 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.

Shorter names for regular polygons (-are suffix)

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Regular polygons (monadic, m=1) have no general specific names, except for the square. I propose we use the square as a guide, and use -are suffix for all regulars. An equilateral triangle becomes a trigram-are or triare for short. Regular polygons become: triare, tetrare, pentare, hexare, heptare, octare, etc.

Regular star polygons apply as well. A pentagram {5/2}, 2-turn monadic pentagram, becomes a 2-turn pentare. Generally, a regular {p/q} star polygon, p:q^p becomes a q-turn p-are.

Shorter names for isotoxal polygons (-us suffix)

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An isotoxal or edge-transitive polygon (dyadic, m=2) has one edge type, 2 vertices within its symmetry. Polytile notation is: p:a.b^n. The smallest class are rhombi: p:a.b^2. Using the name rhombus as a standard, I propose a -us suffix. Then a rhombus is a digramus, or di-us for short. And in sequence: dius, trius, tetrus, pentus, etc.

Here we also find a new quality, convexity. The concave form will have one negative index, like p:-a.b^n (a, b positive). I differentiate as stella- prefix if concave and (optional) arch- prefix if convex, and lineo- if colinear edges. A concave isotoxal decagon can be called a stellapentus.

Shorter names for m-adic polygons

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An m-adic n-gram can also be written as one word as n-gram-m-ad. An m-gon with no symmetry, a m-adic monogram, can be called a monogram-m-ad and mono-m-ad for short. An equilateral pentagon with no cyclic symmetry, a pentadic monogram, can be called a monogram-pentad and monopentad.

Number of convex tiles

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A full set of convex polytiles represent all solutions for a given polygon. The numbers of forms are given below, chiral column counting chiral pairs. The number of tiles by sides are also given, with bold 1 given for regular polygon solutions.

Odd family polytiles, (2k+1)-tiles, are given for completeness. They are a subset of 2(2k-1)-tiles, identified with all odd-turn angle multipliers. The smallest odd solution is one triatile, with 6 triangles around a point, making a regular hexagon, and same as 6 hexagons for the first hexatile.

Number of convex tiles
2n Unique
tiles
With
chirals
Counts
3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
4 1 1 0 1+3
6 3 3 1 1 0 1+1
3 1 1 0 1
8 4 4 0 1+1 0 1 0 1
10 7 7 0 2 1 2 0 1 0 1
5 2 2 0 1 0 1
12 16 17 1 1+2 1 1+3 1 3 1 1 0 1
14 17 19 0 3 0 4 1 4 0 3 0 1 0 1
7 4 4 0 2 0 1 0 1
16 28 34 0 1+3 0 5 0 1+7 0 5 0 4 0 1 0 1
18 70 92 1 4 1 1+8 4 12 1+5 12 4 9 1 4 1 1 0 1
9 10 11 0 1+3 0 3 1 1 0 1
20 85 115 0 1+4 1 8 1 16 2 1+16 2 16 1 8 1 5 0 1 0 1
22 125 187 0 5 0 10 0 20 0 26 1 26 0 20 0 10 0 5 0 1 0 1
11 15 18 0 4 0 5 0 4 0 1 0 1
24 392 616 1 6 2 15 8 33 20 50 27 1+65 27 50 20 33 8 15 2 6 1 1 0 1
26 379 631 0 6 0 14 0 35 0 57 0 76 1 76 0 57 0 35 0 14 0 6 0 1 0 1
13 30 40 0 5 0 10 0 8 0 5 0 1 0 1
28 704 1201 0 7 0 16 1 47 1 79 3 126 4 1+133 4 126 3 79 1 47 1 16 0 7 0 1 0 1
30 3359 3359 1 7 1+2 1+23 17 71 60 1+172 145 329 249 442 1+314 442 249 329 145 173 60 71 17 24 3 7 1 1 0 1
15 82 116 0 9 0 1+17 5 21 3 15 2 6 1 1 0 1

Tetratiles

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Square
1111=14, r8

A tetratile only has turn angles of 90°. There is only one convex tetratile, the square itself. The only tiling is the square tiling.

Hexatiles

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Convex Hexatiles

"Hexare"
16, r12

Rhombus
212, d4

"Triare"
23, r6

A hexatile uses turn angles of 60° and 120°. With 3 tiles, it is the smallest nontrivial set. The 3 tiles are: the regular hexagon 111111 or 16, the equilateral triangle 222 or 23, and 60 degree rhombus 1212 or (12)2. These are related to the rhombille tiling, and trihexagonal tiling.

Examples

Triatiles

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Triatiles

Hexagon
16, r12

There is only 1 convex triatile, repeated as odd-only-indices from the hexatiles.

Convex Triatiles

Octatiles

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Convex Octatiles

"Octare"
18, r16

Hexagon
1212, i4

Square
24, r8

Rhombus
312, d4

An octatile has turn angles of 45°, 90° and 135°. There are 4 convex octatiles: the "octare" (11111111) or (18), Flatted hexagon (311211) or (311)3, and square (2222) or 24, and rhombus (4131) or (31)3.

The tiles are related to the Ammann-Beenker tiling which includes the square and rhombus. The chamfered square tiling is a simple tiling with the right hexagon.

Examples

Square tiling

Truncated square tiling

Ammann-Beenker tiling

Chamfered square tiling

Decatiles

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Convex Decatiles

"Decare"
110, r20

Octagon
21112, d4

Tall hexagon
2122, i4

"Pentare"
25, r10

Flat hexagon
1312, i4

Rhombus
322, d4

Rhombus
412, d4

A decatile has turn angles of 36°, 72°, 108°, and 144°. There are 7 convex decatiles: the regular decagon 11111111 or 110, octagon 21112111 or 21112, tall hexagon 221221 or 2212, and regular pentagon 22222 or 25, flat hexagon 113113 or 1132, wide rhombus 3232, or 322, and thin rhombus, 4141 or 412.

The penrose tiling contains the wide and narrow rhombus.


Penrose tiling

Rhombille tiling

decagon-octagon-rhomb

pentagon-octagon-rhomb

Pentatiles

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Convex Pentatiles

Decagon
110, r20

Flat hexagon
1312, i4

There are 2 convex pentatiles, repeated as odd-only-indices from the dodecatiles.

Dodecatiles

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Convex Dodecatiles

"Dodecare"
112, r24

Decagon
112112, i4

Enneagon
1213, i6

Tall octagon
12212, p4

Octagon
214, d4

"Hexare"
26, r12

Octagon
31112, d4

Heptagon
2131131, i2

Hexagon
3212 (a), g2

Hexagon
3122 (b), g2

"Trius"
313, d6

Square
34, r8

Flat hexagon
1412, i4

Right pentagon
41331, i2

Wide rhombus
422, d4

"Triare"
43, r6

Thin rhombus
512, d4

A dodecatile has turn angles of 30°, 60°, 120°, and 150°.

There are 16 convex dodecatiles. The dodecatiles are used in pattern blocks, triangle, wide and narrow rhombs, square, and hexagon.


Pattern blocks

Dissections of convex dodecatiles
Examples

2222

1114 and 444

2222 and 444

444

1412
Truncated square tiling variations

34 and 214

34 and 11222

34 and 11222

Tetradecatiles

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Convex Tetradecatiles

Tetradecagon
114, r28

Dodecagon
2111112, d4

Decagon
211122, i4

Decagon
212122, i4

Octagon
22212, d4

Heptagon
27, r14

Decagon
113112, i4

Decagon
32112 (a), g2

Octagon
31122 (b), g2

Octagon
31212, d4

Hexagon
3132, i4

Hexagon
2322, i4

Octagon
41112, d4

Hexagon
4122 (a), g2

Hexagon
4212 (b), g2

Rhombus
432, d4

Hexagon
5112, i4

Rhombus
522, d4

Rhombus
612, d4

There are 17 convex tetradecatiles, 19 with chiral pairs.

Heptatiles

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Convex Heptatiles

Tetradecagon
114, r28

Decagon
113112, i4

Hexagon
3132, i4

Hexagon
1512, i4

There are 4 convex heptatiles, repeated as odd-only-indices from the tetradecatiles.

Hexadecatiles

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Convex Hexadecatiles

Hexadecagon, r32
1^16

Tetradecagon, i4
2111111^2

Dodecagon, i4
221111^2

Dodecagon, p4
211121^2

Dodecagon, i8
211^4

Decagon, i4
22112^2

Decagon, g2
21212^2

Octagon, r16
2^8

Dodecagon, d4
113111^2

Decagon, g2
32111^2 (a)

Decagon, g2
32111^2 (b)

Decagon, g2
31121^2 (a)

Decagon, g2
31211^2 (b)

Octagon, p4
1331^2

Octagon, g2
1223^2

Octagon, d8
13^4

Octagon, d4
1232^2

Octagon, d4
1322^2

Hexagon, i4
323^2

Decagon, i4
11411^2

Octagon, g2
1124^2 (a)

Octagon, g2
1142^2 (b)

Octagon, d4
1214^2

Hexagon, g2
143^2 (a)

Hexagon, g2
134^2 (b)

Hexagon, g2
242^2

Square, r8
4444

Octagon, d4
5111^2

Hexagon, g2
125^2 (a)

Hexagon, g2
152^2 (b)

Rhombus, d4
53^2

Hexagon, i4
161^2

Rhombus, d4
62^2

Rhombus, d4
71^2

There are 28 convex hexadecatiles, 34 with chiral pairs.

Octadecatiles

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Convex Octadecatiles

Octadecagon, r36
1^18

Hexadecagon, d4
11111112^2

Tetradecagon, i4
1113111^2

Tetradecagon, i4
1111122^2

Dodecagon, d4
111114^2

Pentadecagon, i6
11211^3

Tetradecagon, i4
1121211^2

Tridecagon, i2
1111221121113 (a)

Tridecagon
3111211221111 (b)

Dodecagon
112311^2 (a)

Dodecagon
113211^2 (b)

Hendecagon
11114112114

Decagon
11511^2

Tetradecagon
2111211^2

Dodecagon
312111^2 (a)

Dodecagon
312111^2 (b)

Dodecagon
222111^2

Hendecagon
11122221114

Decagon
42111^2 (a)

Decagon
42111^2 (b)

Dodecagon
3111^3

Hendecagon
11131221123 (a)

Hendecagon
32112213111 (b)

Decagon
33111^2

Decagon
1114113114

Enneagon
111422115 (a)

Enneagon
511224111 (b)

Octagon
1116^2

Dodecagon
112113^2

Dodecagon
112122^2 (a)

Dodecagon
221211^2 (b)

Decagon
11214^2 (a)

Decagon
41211^2 (b)

Dodecagon
1122^3

Hendecagon
11222113113

Decagon
1122222114

Decagon
11223^2 (a)

Decagon
32211^2 (b)

Decagon
11232^2

Enneagon
112411314 (a)

Enneagon
421141311 (b)

Octagon
1125^2 (a)

Octagon
5211^2 (b)

Decagon
11313^2

Enneagon
113222214 (a)

Enneagon
412222311 (b)

Octagon
1134^2 (a)

Octagon
4311^2 (b)

Enneagon
411^3

Octagon
11414115

Octagon
11422224

Heptagon
1144116

Heptagon
1152225

Hexagon
711^2

Dodecagon
21^6

Decagon
31212^2 (a)

Decagon
32121^2 (b)

Octagon
5121^2

Decagon
31221^2

Decagon
22221^2

Octagon
4122^2 (a)

Octagon
4221^2 (b)

Enneagon
312^3 (a)

Enneagon
321^3 (b)

Octagon
3312^2 (a)

Octagon
3321^2 (b)

Octagon
4212^2

Heptagon
1242315 (a)

Heptagon
5132421 (b)

Hexagon
612^2 (a)

Hexagon
621^2 (b)

Octagon
4131^2

Octagon
3132^2

Hexagon
513^2 (a)

Hexagon
531^2 (b)

Heptagon
1414224

Hexagon
441^2

"Trius"
51^3

Pentagon
15426 (a)

Pentagon
62451 (b)

Rhombic
81^2

Enneagon
2^9

Octagon
3222^2

Hexagon
522^2

Hexagon
423^2 (a)

Hexagon
432^2 (b)

"Trius"
42^3

Rhombic
72^2

Hexagon
3^6

Rhombic
63^2

Rhombic
54^2

"Triare"
6^3

There are 70 convex octadecatiles, 92 with chiral pairs.

Enneatiles

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Convex Enneatiles

Octadecagon, r16
1^18

Tetradecagon, i4
3111111^2

Decagon, i4
51111^2

Dodecagon, d6
3111^3

Decagon, i4
33111^2

Decagon, i4
31131^2

Hexagon, i4
711^2

Hexagon, g2
513^2 (a)

Hexagon, g2
531^2 (b)

"Trius", d6
51^3

Hexagon, r12
3^6

There are 10 convex enneatiles, 11 with chiral pairs, identical to the subset of octadecatile with only odd turn angle multiplers.

Icosatiles

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Convex Icosatiles

Icosagon
1^20

Octadecagon
211111111^2

Hexadecagon
31111111^2

Hexadecagon
22111111^2

Tetradecagon
4111111^2

Hexadecagon
21211111^2

Tetradecagon
2311111^2 (a)

Tetradecagon
3211111^2 (b)

Dodecagon
511111^2

Hexadecagon
21111211^2

Tetradecagon
2131111^2 (a)

Tetradecagon
3121111^2 (b)

Tetradecagon
2221111^2

Dodecagon
241111^2 (a)

Dodecagon
421111^2 (b)

Dodecagon
331111^2

Decagon
61111^2

Hexadecagon
2111^4

Tetradecagon
3111211^2 (a)

Tetradecagon
3112111^2 (b)

Tetradecagon
2211121^2 (a)

Tetradecagon
2212111^2 (b)

Dodecagon
411121^2 (a)

Dodecagon
412111^2 (b)

Dodecagon
311122^2 (a)

Dodecagon
322111^2 (b)

Dodecagon
321112^2

Decagon
25111^2 (a)

Decagon
52111^2 (b)

Dodecagon
311131^2

Decagon
41113^2 (a)

Decagon
43111^2 (b)

Octagon
7111^2

Pentadecagon
211^5

Tetradecagon
2211211^2

Tridecagon
1121131211213

Dodecagon
411211^2

Tetradecagon
2112121^2

Dodecagon
311212^2 (a)

Dodecagon
321211^2 (b)

Dodecagon
321121^2 (a)

Dodecagon
312112^2 (b)

Hendecagon
11214121133

Decagon
51121^2 (a)

Decagon
51211^2 (b)

Dodecagon
311221^2 (a)

Dodecagon
312211^2 (b)

Dodecagon
222211^2

Decagon
41122^2 (a)

Decagon
42211^2 (b)

Decagon
33112^2 (a)

Decagon
33211^2 (b)

Decagon
42112^2

Octagon
6112^2 (a)

Octagon
6211^2 (b)

Dodecagon
311^4

Hendecagon
11313121313

Decagon
41131^2 (a)

Decagon
41311^2 (b)

Decagon
31132^2

Enneagon
113412143

Octagon
5113^2 (a)

Octagon
5311^2 (b)

Octagon
4411^2

Hexagon
811^2

Dodecagon
312121^2

Dodecagon
222121^2

Decagon
41212^2 (a)

Decagon
42121^2 (b)

Decagon
33121^2

Octagon
6121^2

Dodecagon
221^4

Decagon
41221^2

Decagon
31222^2 (a)

Decagon
32221^2 (b)

Decagon
32122^2 (a)

Decagon
32212^2 (b)

Octagon
5122^2 (a)

Octagon
5221^2 (b)

Decagon
31231^2 (a)

Decagon
32131^2 (b)

Octagon
4123^2 (a)

Octagon
4321^2 (b)

Octagon
4312^2 (a)

Octagon
4213^2 (b)

Octagon
5212^2

Hexagon
712^2 (a)

Hexagon
721^2 (b)

Decagon
31^5

Enneagon
131331314

Octagon
5131^2

Octagon
4132^2 (a)

Octagon
4231^2 (b)

Octagon
3331^2

Heptagon
1343144

Hexagon
613^2 (a)

Hexagon
631^2 (b)

Octagon
41^4

Hexagon
514^2 (a)

Hexagon
541^2 (b)

Tetragon
91^2

Decagon
2^10

Octagon
4222^2

Octagon
3322^2

Hexagon
622^2

Octagon
32^4

Hexagon
523^2 (a)

Hexagon
532^2 (b)

Hexagon
442^2

Tetragon
82^2

Hexagon
433^2

Tetragon
73^2

Pentagon
4^5

Tetragon
64^2

Tetragon
5^4

There are 85 icosatiles, 115 including chiral pairs.

Icosiditiles

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There are 125 convex icosiditiles, 187 with chiral pairs.

Hendecatiles

[edit]
Hendecatiles

Hendecagon
122

Octadecagon
1111311112

Tetradecagon
11151112

Tetradecagon
11111332

Tetradecagon
11113132

Decagon
111172

Tetradecagon
11131132

Decagon
111352 (a)

Decagon
153112 (b)

Decagon
131512 (a)

Decagon
151312 (b)

Decagon
133312

Hexagon
1912

Decagon
313132

Hexagon
1372 (a)

Hexagon
7312 (b)

Hexagon
5152

Hexagon
3352

Icosihexatiles

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There are 379 convex icosihexatiles, 631 with chiral pairs.

Tridecatiles

[edit]
Hendecatiles

Icosihexagon
113

Icosidigon
311111111112

Octadecagon
5111111112

Octadecagon
3311111112

Octadecagon
3131111112

Tetradecagon
71111112

Octadecagon
3113111112

Tetradecagon
35111112 (a)

Tetradecagon
53111112 (b)

Octadecagon
3111131112

Tetradecagon
51111312 (a)

Tetradecagon
51311112 (b)

Tetradecagon
33311112

Decagon
911112

Tetradecagon
51113112 (a)

Tetradecagon
51131112 (b)

Tetradecagon
33111312 (a)

Tetradecagon
33131112 (b)

Decagon
711132 (a)

Decagon
731112 (b)

Decagon
551112

Tetradecagon
33113112

Tetradecagon
31131312

Decagon
711312 (a)

Decagon
713112 (b)

Decagon
511332 (a)

Decagon
533112 (b)

Decagon
531132

Decagon
515112

Hexagon
1.1.112

Decagon
513132 (a)

Decagon
531312 (b)

Decagon
513312

Decagon
333312

Hexagon
9132 (a)

Hexagon
9312 (b)

Hexagon
7152 (a)

Hexagon
7512 (b)

Hexagon
73372

Hexagon
5532

There are 30 tridecatiles, 40 with chiral pairs. They represent a subset of icosihexatiles with only odd turn angle multipliers.

Triacontatiles

[edit]

There are 3359 convex triacontatiles, 3359 with chiral pairs.

Pentadecatiles

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Pentadecatiles

"Triacontare"
130

Icosihexagon
11111111111132

Icosidigon
111111111152

Icosidigon
111111111332

Icosidigon
111111113132

Octadecagon
1111111172

Icositetragon
111111133

Icosidigon
111111131132

Icosagon
11111113311113111115 (a)

Icosagon
51111131111331111111 (b)

Octadecagon
1111111352 (a)

Octadecagon
5311111112 (b)

Icosidigon
111111311132

Octadecagon
1111113152 (a)

Octadecagon
5131111112 (b)

Octadecagon
1111113332

Tetradecagon
11111192

Icosidigon
1111134

Octadecagon
1111131152 (a)

Octadecagon
5113111112 (b)

Octadecagon
1111131332 (a)

Octadecagon
3313111112 (b)

Hexadecagon
1111133133111117

Tetradecagon
11111372 (a)

Tetradecagon
73111112 (b)

Octadecagon
1111153

Hexadecagon
1111151133111135 (a)

Hexadecagon
5311113311511111 (b)

Tetradecagon
11111552

Octadecagon
1111311152 (a)

Octadecagon
5111311112 (b)

Octadecagon
1111311332

Octadecagon
3311311112 (b)

Octadecagon
1111313132

Tetradecagon
11113172 (a)

Tetradecagon
71311112 (b)

Octadecagon
1111333

Hexadecagon
1111333111151115

Tetradecagon
11113352 (a)

Tetradecagon
53311112 (b)

Tetradecagon
11113532

Tetradecagon
11115152

Decagon
1.1.1.1.112

Icosagon
11135

Octadecagon
1113111332

Octadecagon
1113113132 (a)

Octadecagon
3131131112 (b)

Tetradecagon
11131172 (a)

Tetradecagon
71131112 (b)

Octadecagon
1113133

Tetradecagon
11131333131117

Tetradecagon
11131352 (a)

Tetradecagon
53131112 (b)

Tetradecagon
11131532

Tetradecagon
51311132 (b)

Tetradecagon
11133152 (a)

Tetradecagon
51331112 (b)

Tetradecagon
11133332

Decagon
111392 (a)

Decagon
931112 (b)

Tetradecagon
11151152

Dodecagon
111531333117 (a)

Dodecagon
711333135111 (b)

Decagon
111572 (a)

Decagon
751112 (b)

Dodecagon
11173

Decagon
1117331337

Octadecagon
1136

Tetradecagon
11311352 (a)

Tetradecagon
53113112 (b)

Tetradecagon
11313152 (a)

Tetradecagon
51313112 (b)

Tetradecagon
11313332 (a)

Tetradecagon
33313112 (b)

Tetradecagon
11315132

Decagon
113192 (a)

Decagon
913112 (b)

Tetradecagon
1133112

Tetradecagon
11331332

Decagon
113372 (a)

Decagon
733112 (b)

Dodecagon
11353 (a)

Dodecagon
53113 (b)

Decagon
113552 (a)

Decagon
553112 (b)

Decagon
113732

Decagon
115172 (a)

Decagon
715112 (b)

Decagon
115352

Hexagon
1.1.132

Tetradecagon
13131332

Decagon
131372 (a)

Decagon
731312 (b)

Dodecagon
13153

Decagon
131552

Decagon
133172

Dodecagon
13333

Decagon
133352 (a)

Decagon
533312 (b)

Decagon
133532 (a)

Decagon
5331353313 (b)

Decagon
135152 (a)

Decagon
531512 (b)

Hexagon
1.3.112 (a)

Hexagon
11.3.12 (b)

Decagon
155

Hexagon
1592 (a)

Hexagon
9512 (b)

Hexagon
1772

"Trius"
193

Decagon
310

Hexagon
3392

Hexagon
3572 (a)

Hexagon
7532 (b)

"Trius"
373

Hexagon
56

There are 82 convex pentadecatiles, 116 including chiral pairs. They represent a subset of tricontatiles with only odd turn angle multipliers.

Examples

[edit]

Regular tiles

[edit]

Each set of n-tiles has regular p-gon tiles for whole number divisors of n, 3 or larger.

Example regular tiles
ab Tetratile
4
Hexatile
3,6
Octatile
4,8
Decatiles
5,10
Dodecatiles
3,4,6,12
Tetradecatiles
7,14
Hexadecatiles
4,8,16
Octadecatiles
3,6,9,18
Icosatiles
4,5,10,20
1b
14

16

18

110

112

114

116

118

120
2b
23

24

25

26

27

28

29

210
3b
34

36
4b
43

44

45
5b
54
6b
63

Rhombic tiles

[edit]

Rhombi are named as ab2, where a and b are positive integers, making a 2(a+b)-tile. A rhombus has d4 symmetry.

Example rhombi
ab2 Tetratile Hexatile Octatile Decatiles Dodecatiles Tetradecatiles Hexadecatiles Octadecatiles Icosatiles
a12
112

212

312

412

512

612

712

812

912
a22
222

322

422

522

622

722

822
a32
332

432

532

632

732
a42
442

542

642

Elongated rhombic tiles

[edit]

An elongated rhombus (hexagon) is named by aabaab or aba2, where a and b are positive integers, making a 2(2a+b)-tile. If a>b, it makes a short hexagon, and b>a makes a tall hexagon. An elongated rhombus has i4 symmetry.

Example elongated rhombi
aba Tetratile Hexatile Octatile Decatiles Dodecatile Tetradecatiles Hexadecatiles Octadecatiles Icosatiles
1b1
1012

1112

1212

1312

1412

1512

1612

1712

1812
2b2
2022

2122

2222

2322

2422

2522

2622
3b3
3032

3132

3232

3332

3432
4b4
4142

4242

Chiral zonogonal polytiles

[edit]
The smallest chiral zonogon is a dodecatile, 1232 and 1322

A chiral zongonal has 2-fold rotational symmetry, like a hexagonal one is abc2 where a,b,c are unique.