Monday, March 2, 2015

Conformal transforms in Minkowski space

I'm writing this mainly for my record, the question is what does a conformal transformation of Minkowski space look like?
The reason it is an interesting question is that relativistic physics is based on Minkowski space (or distorted versions) and the most fundamental symmetry of this is conformal symmetry.

The answer is a combination of:

  • translations
  • scales
  • rotations
  • boosts
  • inversions (which invert the distance metric (positive or negative), so flip around the double hyperboloid in the time-like region and around the hyperboloid of revolution in the space-like region).
However, this isn't totally useful as it doesn't give a clear image of your average conformal transformation. For comparison, we can define any arbitrary conformal transformation of 3d Euclidean space as:

An arbitrary rotation around an arbitrary circle in space. 

This covers all cases, sometimes in degenerate form, for example an ordinary rotation is when the circle has infinite radius, an ordinary translation is the limit as the radius increases, the circle becomes farther away and the rotation around it decreases.

The equivalent definition for 3+1 Minkowski space can be described in two ways, in terms of linear operations (e.g. rotations, boosts) or in terms of curvature. In both cases the transform is described in the time-like region when it involves a time flip, and in the space-like region when it involves a space flip:
An arbitrary rotation around an arbitrary circle in space, with equal and opposite arbitrary boosts through the circle at two times around an arbitrary point in time. 
An arbitrary boost through an arbitrary sphere in space at an arbitrary point in time. 
An arbitrary cupping (positive curvature) of space together with an arbitrary (boost) acceleration through the axis of the bulge. 
An arbitrary radial expansion/contraction at an arbitrary point in space and time.
I'm still not sure if this covers all combinations of transforms but it seems close at least. It is a nice and simple generalisation of 'rotation around a circle' for space-like regions and 'boost through a plane' for time-like regions. 

Sunday, February 15, 2015

Tetrahedral wrapping function

A useful result of mapping complex numbers to the Riemann sphere is that elementary functions like z^2 and z^3 can be interpreted geometrically. 
z^2 wraps the sphere around onto itself, take the one faced spherical polyhedron shown above, cut it along the shown arc and stretch it around itself, as if doubling the longitude coordinates, until you can sew it back up at the original cut. The space naturally pinches toward the two poles such that the transform remains conformal. This is z^2.
Similary, z^3 takes the two-faced spherical polyhedron and wraps it around onto itself, as if stretching the complex plane by tripling the longitude around for example the Greenwich meridian.
z^4 wraps a three-faced spherical polyhedron onto itself, and z^5 wraps a four-faced spherical polyhedron onto itself. These shapes are all called hosohedra
Given this interpretation, it is interesting to consider mappings that wrap other spherical polyhedra onto themselves. Afterall, while the first two example hosohedra are regular, the three and four faced ones are not. The four faced hosohedra is also an unstable partition of the sphere in some senses, the stable and regular four faced spherical polyhedron is a spherical tetrahedron:

So it is interesting that you can indeed wrap a spherical tetrahedron onto itself, and it does remain conformal. The function is this:
If the complex plane origin is point D in the above image then the constants in the numerator of the product are the vertices A,B and C. The constants in the denominator are the poles which map to the centre of the spherical triangle ABC, they are located at the centre of the three remaining spherical triangles. 
The mapping can be visualised on the complex plane using David Bau's complex function viewer:
The grey circles are the unit disk in the complex plane, so you can see that this function can equally be seen to map a two sided spherical polyhedra (the the edge being the unit disk) to a spherical octahedron.
This animation I made shows the fold as applied to a non-spherical tetrahedron:

This tetrahedral wrapping function, for which I'll use the letter tau, can be simplified slightly if we allow it to wrap onto its dual tetrahedron (so removing the minus sign) and rearrange slightly to:

The mandelbrot set of this function requires care to visualise, unlike the standard mandelbrot set it doesn't bailout on large values. The only parts not in the set are those that hit a pole and so remain at infinity, you can do this by iterating the radius of the pixel, not just the point, and bailing out when either of the three poles are within this radius. Here I view it using Fractal Lab:

I find the look of this fractal to be quite different to others, even though you can clearly see mandelbrot elements within it. Here are two more close ups of the holes:

and here is part of the negative tau version:

You can also apply this function as a form of Mandelbulb fractal. In this case the transform operates on a real cartesian sphere around the origin, and the radius or each point squares each iteration, the mandelbrot version gives:
which was rendered using Fragmentarium.

Dihedral version

While the above function is I think quite unique as a regular polyhedron that wraps onto itself, it is also possible to do so with a two-sided spherical polyhedron called a Dihedron {4,2} (which means four vertices and two faces):
The wrap is similar to taking a square (with two sides, like an envelope) and folding in half then in half again to form a smaller square, but in our case it is done on the inflated/spherical surface.
The function is:
Which can be visualised here. You can visualise the mandelbrot set of it by using its reciprocal:

Wow, these are great pics.. and the negative version:

It also produces a mandelbulb which has some similarities to the tetrahedral version: