General Inversion Sets

In my previous post I was looking into valid double inversions as a broader form of inversive limit sets. In this post I'm looking at adding general Mobius transformations in addition to sphere inversions, and what constitutes a valid configuration to generate a clean limit set. 

Clean here means non-chaotic, not inversion order dependent, and usually means the geometry is made of whole circles rather than haphazard sections joined together. A sufficient requirement is that it forms a Kleinian group, and this is governed by the Maskit Combination Theorems. But I don't think it is a necessary requirement; it appears to be a bit overly conservative. 

The Mobius transformations generalise translation, rotation, dilation, circulation, parabolic, hyperbolic and loxodromic transformations. All of these preserve orientation.

Single Mobius Transformation

This is where a single such transformation is applied each iteration before or after the sphere inversions.

For sphere inversions, the transformation reflects the generating set into the sphere in question. This generates an overlap between the original set and its transformation that must be at a 2pi/n angle. It so happens that the intersection is between the originally overlapping spheres and their own transformations.

For orientation-preserving transformations a similar thing happens. You transform the generating set and the result must overlap the original set by angles of 2pi/n. The only difference is that is may be a different sphere that the overlap happens with.

So the rules ought to be very similar to the original theory, if the generating set are disjoint then you get dist, if they kiss you get a void- fractal. If they form a filled polygon mesh then it'll generate a surface etc.

The complexity is really in choosing a division surface. This is ideally a mid-surface (bisecting) between the generating set and its transformation. As such it passes through the intersection circles of any sphere overlaps. It is a bit complex as it will be piecewise quadric surfaces.

The effect of Mobius transformation is to create a chain of the limit set associated with the original generating spheres. 

• If the Mobius transformation is a translation then it is a regular straight chain
• if it is a rotation then the angle must be a submultiple of 2pi and the chain is the corresponding finite ring
• if it is a circulation then it likewise needs to be 2pi/n and you get a ring around the circulation
• if it is a parabolic transformation then you get an infinite chain like the Real integers on the Riemann sphere
• if it is a hyperbolic transformation then you get an infinite chain following that 'surge' path
• if it is a loxodromic transformation then you get the corresponding helical infinite chain

Multiple Mobius Transformations

When there are more than one transformation we find the set of surfaces that bisect each transformation, and bisect with the generating set that applies the inverse of the transformation too.

The algorithm then compares the sample point to these surfaces to decide which transformation should be applied to bring it to the original set, and that transformation is applied. 

Unlike the single Mobius case, this doesn't generate chains of the original limit set but rather a mesh (of two transformations) or a 3D lattice if three are used. They follow something like wallpaper groups, you could use it to make an infinite fractal surface, or the 2D mesh can fold into a spherical (closed) shape. 

These shapes at their simplest are the polyhedral group, which is like the standard coxeter group but it is orientation preserving. The simplest example has bisection planes from the centre to the 12 edges, and rotate the volumes (for example) up to the top 'sextile'. 

General Thoughts

Mobius transformations add a (typically curved) chain or lattice of the original limit set, and obey the same classification rules with respect to how the transformations self-overlap. The chain or lattice is not just on the largest scale, the sphere inversions mean that it will be present throughout the fractal at all scales.

This part is conceptually easier than the double sphere inversions of the previous post, but practically it is quite hard to generate the bisection surfaces. Not massively hard, but a bit hard. 

On the other hand you don't necessarily need to generate these surfaces, just compute the two sets of spheres and find the shortest distance to each set in order to decide which transformation to apply. 

New Shapes

Are there any new sorts of shapes possible with these transformations? Well, chiral fractals is one thing. Using the pair of generating rotations for a particular polyhedral group. 

Here I use the cubic polyhedral group to make a chiral inversive limit set:

it is equivalent to the limit set of a set of inversive spheres on the vertices of an augmented snub cube. Their dihedral intersection angle is pi/3 since the spheres contact in threes. 

You can make the augmented part (the pyramid) concave instead to get a chiral sponge: