Hello all,

This is actually something I discovered (alert me if this is not the case and I will edit duly) a while ago and somehow overlooked. Recently though I figured that, if for no other reason, you guys might like it because it is pretty and something fairly novel to do with the M-Set.

One day I was wondering what the M-Set would look like on strange geometries. I went through the usual: polar coords projections, spheres and blah blah. Nothing seemed to really introduce new or interesting features until I considered the Mobius strip, or to be precise, the Mobius cylinder. The Mobius cylinder as I term it, is characterised by some length a on the real axis and is unbounded on the imaginary axis, and has the usual mobius transform at x=-a and x=a. That is, any point with |Re{z}|>a is wrapped around the cylinder horizontally until Re{z} in [-a,a) (or the complementary interval, or the closed one) and is vertically flipped (complex conjugate).

Code wise, we are looking at something akin to the following:

vec2 mobius(vec2 p) {

float dist=abs(p.x)-width;

if (p.x>width) {

p.x=-width+dist;

p.y*=-1;

}

else if (p.x<-width) {

p.x=width-dist;

p.y*=-1;

};

return p;

};

note here width=a and that this is technically the half-width. For some terminology, I call this the total Mobius Cylinder. A partial Mobius cylinder is thus where the wrapping is applied at only one boundary.

What does this have to do with the M-Set? Well, I found that if we adjust the iteration for points to z->MobiusWrap(z^2+c), that is, apply the coordinate transform after every iteration, beautiful new patterns emerge for all values of a ~ (0,2.7). Naturally, due to the fact that the only way for a point to escape is vertically (or in one horizontal direction and vertically), we must increase the bailout radius a tad to include all of the fractal. Values around six seem to work just fine. Most interestingly of all, despite what would seem like a non-conformal map, the fractal remains continuous.

Now for some pictures to pique some interest:

http://imgur.com/a/2Onxl#0The overall shape, for the lazy, with a=1 and total:

I would encourage all of you to play with this and discover the hidden beauties and let me know! (I have compiled a quick fragmentarium file to render the fractal, attached)

Hiato