Trifylfots

Here’s a simple fractal created by dividing an equilateral triangle into smaller equilateral triangles, then discarding (and rotating) some of those sub-triangles, then doing the same to the sub-triangles:

Fractangle (triangle-fractal) (stage 1)


Fractangle #2


Fractangle #3


Fractangle #4


Fractangle #5


Fractangle #6


Fractangle #7


Fractangle #8


Fractangle #9


Fractangle (animated)


I’ve used the same fractangle to create this shape, which is variously known as a swastika (from Sanskrit svasti, “good luck, well-being”), a gammadion (four Greek Γs arranged in a circle) or a fylfot (from the shape being used to “fill the foot” of a stained glass window in Christian churches):

Trifylfot


Because it’s a fylfot created ultimately from a triangle, I’m calling it a trifylfot (TRIFF-ill-fot). Here’s how you make it:

Trifylfot (stage 1)


Trifylfot #2


Trifylfot #3


Trifylfot #4


Trifylfot #5


Trifylfot #6


Trifylfot #7


Trifylfot #8


Trifylfot #9


Trifylfot (animated)


And here are more trifylfots created from various forms of fractangle:













































Elsewhere other-accessible

Fractangular Frolics — more on fractals from triangles

Fractangular Frolics

Here’s an interesting shape that looks like a distorted and dissected capital S:

A distorted and dissected capital S


If you look at it more closely, you can see that it’s a fractal, a shape that contains itself over and over on smaller and smaller scales. First of all, it can be divided completely into three copies of itself (each corresponding to a line of the fractangle seed, as shown below):

The shape contains three smaller versions of itself


The blue sub-fractal is slightly larger than the other two (1.154700538379251…x larger, to be more exact, or √(4/3)x to be exactly exact). And because each sub-fractal can be divided into three sub-sub-fractals, the shape contains smaller and smaller copies of itself:

Five more sub-fractals


But how do you create the shape? You start by selecting three lines from this divided equilateral triangle:

A divided equilateral triangle


These are the three lines you need to create the shape:

Fractangle seed (the three lines correspond to the three sub-fractals seen above)


Now replace each line with a half-sized set of the same three lines:

Fractangle stage #2


And do that again:

Fractangle stage #3


And again:

Fractangle stage #4


And carry on doing it as you create what I call a fractangle, i.e. a fractal derived from a triangle:

Fractangle stage #5


Fractangle stage #6


Fractangle stage #7


Fractangle stage #8


Fractangle stage #9


Fractangle stage #10


Fractangle stage #11


Here’s an animation of the process:

Creating the fractangle (animated)


And here are more fractangles created in a similar way from three lines of the divided equilateral triangle:

Fractangle #2


Fractangle #2 (anim)

(open in new window if distorted)


Fractangle #2 (seed)


Fractangle #3


Fractangle #3 (anim)


Fractangle #3 (seed)


Fractangle #4


Fractangle #4 (anim)


Fractangle #4 (seed)


You can also use a right triangle to create fractangles:

Divided right triangle for fractangles


Here are some fractangles created from three lines chosen of the divided right triangle:

Fractangle #5


Fractangle #5 (anim)


Fractangle #5 (seed)


Fractangle #6


Fractangle #6 (anim)


Fractangle #6 (seed)


Fractangle #7


Fractangle #7 (anim)


Fractangle #7 (seed)


Fractangle #8


Fractangle #8 (anim)


Fractangle #8 (seed)