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Sections: Counts | Tetratiles, Hexatiles, Octatiles| Decatiles | Dodecatiles | Tetradecatiles | Hexadecatiles | Octadecatiles | Icosatiles
Triangles | Rhombi | Pentagons |

Strictly Convex Polytiles

Counts

This table shows the enumeration of strictly-convex p-tiles, by sides with a brute-force search. Chiral pairs are counted as 1. Digon are excluded.

The highest-sided strictly convex p-tile is a regular p-gon. Identical polygons can repeat between p-tile families as multiplies. For instance, a square is the tetratile 1^4 and octatile 2^4, and dodecatile 3^4, etc.

You can see the counts generally increase exponentially, but not always consecutively. For instance there are more 24-tiles than 26-tiles, and more 30-tiles than 32 tiles.

Counts of convex p-tiles by sides
p Total  Sides
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 31 32
4 1   1
6 3 1 1   1
8 4   2   1 1
10 7   2 1 2   1   1
12 16 1 3 1 4 1 3 1 1   1
14 17 3 4 1 4 3   1 1
16 28 4 5 8 5   4 1 1
18 70 1 4 1 9 4 12 6 12 4 9 1 4 1 1 1
20 85 5 1 8 1 16 2 17 2 16 1 8 1 5 1 1
22 125 5 10 20 26 1 26 20 10 5 1 1
24 392 1 6 2 15 8 33 20 50 27 66 27 50 20 33 8 15 2 6 1 1 1
26 379 6 14 35 57 76 1 76 57 35 14 6 1 1
28 704 7 16 1 47 1 79 3 126 4 134 4 126 3 79 1 47 1 16 7 1 1
30 3359 1 7 3 24 17 71 60 173 145 329 249 442 315 442 249 329 145 173 60 71 17 24 3 7 1 1 1
32 2248 8 21 72 147 280 375 440 375 280 147 72 21 8 1 1


Tetratiles, Hexatiles and Octatiles

  • There only one tetratile, a square, 1^4.
  • There are 3 hexatiles, a regular hexagon, 1^6, rhombus, 1.2^2, and an equilateral triangle, 2^3.
  • There are 4 octatiles: a regular octagon, 1^8, a flat hexagon 1.1.2^2, a square, 2^4, and a rhombus, 1.3^2.

Polytiles of the form a.b^2 are rhombi, alternating 2 angles.


Sets of convex Tetratiles, Hexatiles, and Octatiles


Decatiles

There are 7 convex decatiles

There is no special names for the irregular hexagons. The form a.(1^c)^2 , like 3.1.1^2 might be called a polygonal-lens, approximating 2 attached circular arcs.


Set of convex Decatiles


Dodecatiles

There are 16 or 17 convex dodecatiles (depending whether chiral copies are counted as 1 or 2):

Polytiles of the form a.b^n are called isotoxal, having one edge type withing its symmetry, and 2 vertex types which alternate.

The decagon  2.1.1.1.1^2 , octagon 3.1.1.1^2, and hexagon 4.1.1^2 can be called polygonal-lens.


Set of convex Dodecatiles


Triangles

The only equilateral triangle is regular. It can be constructed as 3k-tiles: k^3, shown for 6,9,12,15,18,21.


Pentagons

There are a relatively few number of strictly-convex equilateral pentagons (searched p=4 to 100). All of them besides a regular pentagon can be dissected into an equilateral triangle of a rhombus, so only occur for p-tiles with p as a multiple of 3.


Set of convex equilateral pentagons, p=4..100

There are a few near-misses, given a more generous tolerance for closing.

Near-miss pentagonal polytiles


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