Ying Tong Iddle I Po

Well after a bit of hitting myself over the head with quite how simple it was I’ve managed to generate some hyperbolic Julia sets:

The problem I was having is that I was getting hung up over the distance function rather than letting the algebra take care of it for me. The big advantage of our approach is that one can just ‘twist a knob’ at the start and the method works the same in Euclidean geometry as Hyperbolic. I was trying to re-scale my point twice for no good reason effectively collapsing me down to the origin or flying off to the Poincaré disc rim at each iteration.

Both pictures show the set from the same starting point (I think) although they differ quite a bit because (I imagine) the hyperbolic Mandelbrot differs. In the hyperbolic case I have mapped the Poincaré disc back onto the plane (i.e. the rim has been pushed to inifinty) to more clearly show the structure of the fractal although one can plot it on the disc. The colours are assigned using the normal ‘minimal iterations to test for inclusion’ principle (I have some shortcuts for the axes which is why there are some artifacts there).

It looks like the hyperbolic set is far more ’swirly’ than their Euclidean counterparts with a fine internal structure. Should be fun playing with the 3d version now — although I need to re-cast a load of complex analysis to deduce a suitable distance function.

Update: There now exists a high resolution version showing the internal stucture nicely. This one is on the Poincaré disc so the blue part around the outside should be considered not part of the space.
Update 2: A hyperbolic Mandelbrot has also been generated. If the Euclidean one looks like a squashed bug I think this one looks like a roast turkey.

I’ve decided to start posting recipies for anything I make up that turns out particularly good so that I can recreate them for my darling girlfriend upon request. Read more for tonight’s creation — Culture Fusion Curry (veggie)

Ingredients

• 2 sweet peppers (red and green for æsthetic reasons).
• 1 medium onion.
• 1 clove of garlic.
• 2 tsp olive oil.
• 1 tsp sugar.
• 3 or 4 smallish mushrooms.
• 2 tbsp Thai yellow curry paste.
• 4 tsp tumeric.
• 2 tsp ground corriander.
• 4 tsp ground cumin.
• Chilli powder or crushed dried chillies to taste.
• 2 or 3 medium potatoes.
• 1 tin of coconut milk (400ml).
• 1 tin of chopped tomatoes (400ml).
• 1 stock cube.
• 3 tsp cornflower.

Method

1. Coarsely chop the onions and peppers — ideally 1 inch square pieces of pepper and half-rings of onion. Heat the oil in a pan and crush the garlic into it. Add the peppers and onions and lightly sweat off for 2-3 minutes. Sprinkle the sugar into the pan and leave until the onions begin to turn golden brown. Add the mushrooms, again coarsely chopped and stir for 2-3 minutes until the mushrooms change colour.
2. Add the tumeric, corriander, cumin and chilli powder then stir for 30 seconds until the spices become aromatic. Stir in curry paste until well coated.
3. Add coconut milk, tomatoes and potatoes peeled and cut into 1 cm slices. Crumble stock cube over mixture, stir and bring to boil.
4. Leave to simmer for 15-20 minutes until potatoes offer little resistance when poked with a sharp knife.
5. Mix cornflower in a little water to form a liquid with the consistency of single cream and add to the mixture. Stir and continue simmering for 5 minutes.
6. Serve with (in this case) home-baked naan-bread and rice.

As promised there now exists a DVD resolution movie of the ray-traced Julia sets. If anyone wants it the source is available but remember it is a prototype and I am quite ashamed at the coding to be honest.

4 parameter personality test results:

Your brain: 0% interpersonal, 5% visual, 25% verbal, and 70% mathematical!

Congratulations on being 100% smart! The above score breaks down what kind of thinking you most enjoy doing. It says nothing about how good you are at any one, just how interested you are in each, relatively. A substantial difference in scores between two people means, conclusively, that they are different kinds of thinkers. Matching Summary: Each of us has different tastes. Still, I offer the following advice, which I think is obvious: Don’t date someone if your interpersonal percentages differ by more than 50% . Don’t be friends with someone if your verbal percentages differ by more than 60% . Don’t have sex with someone if their math percentage is over 50% .

I’ve got my ray-tracer working. I’ve been generating pretty pictures of 3d Julia sets for a while now. Stay tuned for some animations. P.S. If you couldn’t tell it is an orthographic projection.

I’m pretty much posting crap at the moment in protest to claims that I’m using the bandwidth from the Xbox Live LAN party downstairs. Even with my fastest web-browsing I’m only managing around 5-6kbps. I’m pretty far away from registering on their sustained 128kpbs up/downstream usage. I’ve redoubled my bandwidth efforts and we’ll see if they come up and moan again.

Update: no moans yet so I’m going to sit here in a smug and annoyed fashion :).

Odd BBC article. Quote:

The prosecution’s case is built on the theory that Michael Jackson could be a serial pædophile.

Now call me odd but, just as one is homosexual or not, one is a pædophile or not methinks. I find it unlikely that Jackon was attracted to kids, then not, then was, then wasn’t. I suspect that the BBC meant ’serial child abuser’. One could quite easily be a closet pædophile and do nothing about it I imagine, just as one could be a closet homosexual. The BBC has been getting sloppier with its use of language recently. The number of times I’ve heard the word ‘literally’ used incorrectly in news reports makes me terribly sad.

Well it was pretty good I though. I laughed out loud at some bits (in a good way) and would have been truly frightened by others when I was 8 years old (the Bertie-Basset monster during the McCoy era scared the hell out of me when I was 8 - I think I’m probably the youngest you can be and still remember the Doctor being broadcast on BBC 1). I’m actually really looking forward to the next episode and was really quite upset when the geek got shot :(.

Let us hope they continue the good work - and I love the new(-ish) theme tune.

Some of my PhD involved implementing Geometric Algebra algorithms on the GPU. Since we’re starting to investigate fractals on said here is a 3d julia set animation which was rendered in real time with 0% CPU utilisation.

Now if only I could get a ray-traced fractal to do the same…