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Sections: Convex polytile | Concave polytilesConvex star polytiles | Concave star polytiles

Isotoxal (convex) Polytiles

An isotoxal polygon alternates between two vertex angles. A simple isotoxal polytile has the form a.b^n, with p=(a+b)n. Vertices exist at 2 radii.

If both a and b are positive, it will be convex. If they are equal it will be regular, a.a^n = a^2n. If n=2, it will be a rhombus.

The symmetry of a nonregular isotoxal polytile is d_2n. The number of rhombi for p-tiles is p/4-1, including a square solution.

d6 symmetry isotoxals include 12:13^3, 18:2.4^3, and 18:1.5^3.

d8 symmetry isotoxals include: 12:1.2^4, 16:1.3^4, 20:2.3^4, 20:1.4^4.

d12 symmetry isotoxals include: 18:1.2^6.


Set of convex isotoxal polytiles, p=4..12


Isotoxal (concave) Polytiles

Concave isotoxal polytiles have the form p:-a.b^n, with p=(b-a)n. So to exist, b-a must be a divisor of p, including 1. Also a,b are limited to values 1..p/2-1, makes for a


Set of concave isotoxal polytiles, p=4..16

 

Set of concave isotoxal stars


Isotoxal (convex stars) Polytiles

 Isotoxal convex star polygons are p:a.q^p, with gcd(a+q,p)=1, a<q, having (a+q) turns. Coinciding vertices and edges cause the appearance of a {p/(q+a)} regular star.


First set of convex isotoxal stars, p=4..12


Second set of convex isotoxal stars

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