cross Trianguloids

In addition to the fractals generated by combining the 0- (P) and 60-isotrianguloids (Z), or by combining the 0- (T) and 60-allotrianguloids (W), further fractals can be generated by combining isotranguloids with allotrianguloids. There are 8 such combinations of 2 fractals. (This number increases rapidly as the number combined increases.)

The trianguloids described here and on the preceding pages are not all the d3-symmetric fractals.

For example the arrays of 9 or 16 elements that make up the 2nd and 3rd order equilateral triangles can be considered as being made up of 2 or 5 rings of elements around a central point. The orientation of each ring can be independently varied. In the case of the second order equilateral triangle none of the resulting fractals has a uniform measure, but in the case of the third order equilateral triangle several of the 32 possible fractals have a uniform measure, including some not described here and on the preceding pages.

I have reason to believe that all these tile the plane, but do not have constructions for all of these.

Source: All are independently discovered.

References:

© 2000 Stewart R. Hinsley