Saturday, September 27, 2014

Black rings

Assuming black holes exist, it is true that every one of them must spin, since the chance of one forming with exactly 0 angular velocity is 0. Black holes are thought to create a singularity in their centre, which is hidden behind their Schwartzchild radius, or event horizon. It seems to me (without thinking too hard about the maths) that a spinning black hole should produce an infinitely dense ring rather than a single point. A ring spinning around its axis should hold itself open.

What is interesting about this is that a singularity in the form of a ring is an allowable kind of analytic function of 3d space with two or more layers of volumes. By this I mean that you could pass through the ring and into a different version of 3d space without I think violating general relativity. This is a bit like ideas about space-time wormholes but it is different, it doesn't connect two distance areas of space by a shortcut, it is just a particular geometry of space where everywhere is continuous (apart from the ring singularity) but the world through the ring is different from the world if you pass by the ring without going through it. Moreover, there are potentially an unbounded number of layers each time you circle around the ring and through it.

Would make a nice concept for a sci-fi movie.

I wonder if such a physical setup could be simulated... the ring doesn't have to be massive, it could be the size of a door, so long as the ring is infinitely dense (which doesn't mean massive).

Friday, September 12, 2014

Reaction Diffusion Fractals

Previous work with fractal automata has the disadvantage that it lacks continuous rotational symmetry (it also lacks continuous translational and scale symmetry). An interesting idea is to instead work with reaction-diffusion systems which are already continuous in rotation and translation symmetry, and try to add scale symmetry. The normal formula is as follows (and is explained in the link above):

Scaling Du and Dv scales the size of the patterns formed, it seems that the pattern size is proportional to the square root of this scale. Therefore we choose to scale by t2 where t is our scaling factor.

Next we change all of our variables to be vectors rather than scalars. This represents a list of independent reaction diffusion systems. We then choose vector t to be (1,t,t2,....), in other words the difference between each reaction diffusion system is a geometric increase in scale. If we view the first three components of v in the red,green and blue channels respectively, then the result is simply a superimposed set of three reaction diffusion systems (using Du = 1, Dv = 0.5):
Here the red scale is the smallest, the green is noticeably twice the scale, and blue twice again.
The final ingredient is to have these separate scales interact with eachother, we do that in the reaction part of the formula, which is the uv2...
each u is reacting with two vs, we replace the v vector (v for each scale) with a weighted average of all the vs where the weighting is an exponential dropoff s|x| around each component. Effectively the vector v has been convolved with an exponential dropoff function, or low pass filtered. This gives us an extra parameter s, when s=0 we have no interaction like the image above, s=1 gives equal interaction which prevents any separation of the different components. Values in between are interesting.

Here we choose a small value of s = 0.05. t = 0.5 so we double the scale of each component (red,green,blue). I plot the reaction diffusion system with varying parameters F and k on the horizontal and vertical axes respectively (increasing right and up). 
zoomed out

s=0.05. zoomed in slightly. Varying the parameters in x,y shows the different patterns possible.
Next are with s = 0.1. The three components are more correlated. Notice that the thin bridges are more red (small scale) and the large areas more blue. 

zoom around 0.027, 0.058, range 0.08

zoom around 0.026, 0.057, range 0.04

zoom around 0.023, 0.055, range 0.02

zoom around 0.023, 0.051, range 0. (unvarying in x,y)

similar area, unvarying parameters. Notice the similar worm shapes at each scale.

The system does not require the scales are doubled for each component, here we show a zoomed out image for t=0.7, so green is roughly 1.4 times the width of red.   
In fact the results are an approximation of a continuous scale symmetry as t1 from below. Make sure to change the dropoff to s|x|log2(1/t)