The Sierpinski triangle, Wikipedia, and the Sierpinski square also in Wikipedia, are fractals with the overall shapes of either an equilateral triangle or a square, subdivided recursively into smaller equilateral triangles or squares. I have found that initially subdividing a square into four squares two of which are 90° triangles each having two equal sides and a diagonal in a ratio of 1:1:√2 and two regular squares, then subsequently performing recursion, can lead to a new type of mingled (intermixed) fractal.
To get the first iteration we start by taking a tilted square and bisecting each side four times to form four smaller squares, two of which are bisected by blue diagonals:
The four smaller squares of figure Square I are subsequently labeled A,B,C and D and B and D are colored gold to distinguish them from A and C but also to discern the squares as they shrink in size as the number of iterations increases. Alternatively, the four squares can be constructed as on the right (Square II):
Performing an iteration on Square I affords two types of subsquares: squares A and C follow a regular Sierpinki recursion, while B and D undergo a recursion involving a Cantor set where the number of golden squares double each time a recursion occurs, mingled with a second iterated Sierpinski.
The effect of subdividing Square I recursively by performing two, three and four iterations affords:
where the color of the diagonals at iteration three changes to orange to distinguish it from the blue and green diagonals.
And finally in a fifth iteration where only square B (enlarged) is shown, to afford:
Since B and D are equivalent, the fifth iteration would show a total of 32 golden subsquares. On the other hand, the fifth iteration of A or C are not shown since these are just the iterated blue 90° Sierpinski triangles.
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