For a while now, I’ve ocasionally used the MobiusN variation in a final transform to create multiple copies of the fractal and tile them – clearly, the technique is best suited to flames with sharp borders, for example triangles:
and sierpinski tiles:
With the perfect square of the Sierpinski tile, it dawned on me that I might use different values of the Power variable (other than the obvious 4) with a corresponding distortion of the final transform’s triangle. The brainwave to switch to polar coordinates on the Transform tab really opened up the method – I was soon able to deduce the theta values for the Y and O coordinates based on the Power (n) value as 360/n and 360/n/2 respectively. Thus far, though, I’ve been unable to deduce a formula for the r-value of the O coordinate, resorting to trial-and-error. The following table gives the theta and approximate r values for the O coordinate for 3 <= n <= 12.. Additionally, I've also provided the length l of the XY line of the triangle as I've a hunch this might somehow be involved. Any mathematical geniuses out there who know, or can deduce, the formula for r?
As a bonus, here’s a special effect from using a basic Sierpinski tile with MobiusN final transform with Power = 4, but NOT shifted, so that all 4 copies overlay. As each is at 90 degrees from the its neighbours, the result is a misting of the colours to give an unusual lighting effect.