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Strictly Convex Polytiles

The number of strictly convex p-tiles can be catalogued by brute force for any finite even whole number p. The highest-sided strictly convex p-tile is a regular p-gon.

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.

Decatiles

There are 7 convex decatiles, a regular decagon, 1^10, an octagon, 2.1.1.1^2, two hexagons: 2.2.1^2 and 3.1.1^2, a regular pentagon, 2^5, and two rhombi: 3.2^2, and 4.1^2.

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.

Dodecatiles

There are 16 or 17 convex dodecatiles (depending whether chiral copies are counted as 1 or 2): a regular dodecagon, 1^12, decagon 2.1.1.1.1^2, enneagon 2.1.1^3, 3 octagons 2.1^4, 2.2.1.1^2, and 3.1.1.1^2, a heptagon 1.3.1.2.1.3.1, 4 or 5 hexagons: 2^6, 3.1^3, 4.1.1^2, and chiral pair 1.2.3^2 and 3.2.1^2, one pentagon: 4.1.3.3.1, three quadrilaterals (square) 3^4,  (and rhombi) 4.2^2, and 5.1^2, and an equilateral triangle 4^3.

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.


© 2020 Created by Tom Ruen