Return to Home
|Sections: Counts |
Tetratiles, Hexatiles, Octatiles|
Decatiles | Dodecatiles | Tetradecatiles
| Hexadecatiles | Octadecatiles | Icosatiles
Triangles | Rhombi | Pentagons |
Strictly Convex Polytiles
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.
Polytiles of the form a.b^2 are rhombi, alternating 2 angles.
There are 7 convex decatiles
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 only equilateral triangle is regular. It can be constructed as 3k-tiles: k^3, shown for 6,9,12,15,18,21.
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.
There are a few near-misses, given a more generous tolerance for closing.
© 2020-2021 Created by Tom Ruen