Making sense of space-time

minkAdding Minkowski options in my Voronoi┬ápage made me realize that I really wasn’t sure how to interpret the results, and that I’d always been puzzled thinking about the zero distance between points of the light cone means.

Reading: Where Is Now? The Paradox Of the Present┬ámade a few things snap into place for me. We are small slow moving creatures and the natural speed of space-time is way too fast for us to grasp. It’s only when we look at things that are far, far away like the night sky that we have to come to grips with being denizens of a Minkowski space.

I’ve often thought that one of the biggest hurdles with trying to understand relativity is that most introductions tackle multiple mathematical abstractions all at the same time. Space-time is a complicated concept that is still useful in non-relativistic mechanics. Thinking about points in time is a skill which takes time to learn. Thinking about 4 dimensional and higher spaces is another skill that can be tackled on it’s own. There are many useful and interesting abstract non-Euclidean geometries that don’t require quite the same leap of understanding as relativity.


This is one of the sites that got me back into blogging.


I’ve been trying to put my finger on what I really enjoy about this collection. To start with, the drawing algorithm used throughout the site is extremely pleasing and has a real organic feel to it. The peaceful and relaxing palette doesn’t hurt either.

The first few images really stand out and look meticulously hand drawn. As I go back and look through the site, I am continually amazed at the differences in the different series and how well they are drawn together through the artist’s brush.

While the pictures themselves are beautiful, I love that the algorithms are public as well, and that these images are all algorithmically generated. It’s neat to see things like sorting algorithms visually represented, but even more beautiful when the code expresses it’s inner beauty through a canvas.

What makes the Mandelbrot set so special?

The classic Mandelbrot set

We’ve all seen galleries or videos of zooming into the Mandelbrot set, but not as many for julia sets. This video sums it up, while the boundary of the Mandelbrot set is a map of a number of different julia sets, the julia sets are almost exactly self similar. If you pick 2 areas on a julia set, there is a function that will map one part exactly onto the other. This is true for the linear IFS fractals I’ve been writing apps for.

These maps don’t exist the same way for the Mandelbrot set, which seems strange when the algorithms for generating the images seem so similar. There is a fundamental difference between the two types of objects though. A julia set is defined by one single function and the different points in the image capture the behavior of a starting point. Each point of the Mandelbrot set corresponds to a function, and that to a Julia set. To do the standard rendering each Julia set is reduced to a single numeric value.

Now, this value hasn’t been chosen at random, but happens to answer a particular question. Is the Julia set of that function connected or is it a dust of disconnected specs? If we don’t care about that question, we can milk the rendering to get some more interesting fractal images. If we move the starting point, then the boundary of our new set moves accordingly. Instead of showing just barely touching Julias, it moves off center and some of the edges stray into the solid territory while others move into the dust realm.

I’ve put together some images using modifications I made to an HTML5 canvas renderer. (Link to my changes here) They show various alternate Mandelbrot sets generated from changing the starting position of the rendering. In further installments we can look at computing other ways to reduce the Julia fractals down to a single value for computing still other Mandelbrots. I’ll also cover generating Mandelbrot sets for the IFS fractals as well. That uses a more complicated algorithm.

These images show what happens when you go off script with the Mandelbrot. There are now pieces floating off the main island and large chunks are cut off at rough but boring boundaries. The transitional areas between the vast boring solid areas and the ethereal dusty areas contain a little burst of activity.

One of the early casualties is the symmetry and regularity of the Mandelbrot set.

Mondrian meets javascript

The first poster my wife got for me after we were married was a Mondrian print. I love how he created such pleasing images from such simple forms. Now there is a web app for creating your own Mondrian style creations.

What really struck me was how quickly things go off the rails. It only takes a few clicks before a balanced harmonious composition gets too busy or just plain ugly.

I’m thinking about how best to handle this

I like the idea of putting demos up directly on the blog like the last one. However, they don’t show up on feed readers so you either get the text without the image and then click through. I can throw down some javascript to include a place holder link to the demo in the feed, or do excerpts, which I really don’t like, or go back to the old format of talking about something and linking to it at the end, which I don’t really like either.

Does anyone have any ideas or suggestions on how they handle this issue?

Visual complexity

Visual complexity is a bad ass site. It introduced me to the term generative art and reawakened my interest in exploring it myself. I fell in love with the mathematics and images of fractals that exploded in the eighties. One of my favorite books in high school was chaos by James Gleick. The pictures were beautiful, and the math to compute them could be grasped by someone with just a high school education.

I eventually tired of the images and wrote the whole thing off as I realized that most of the slick pictures were little more than thomas kincaid paintings for geeks. They looked pretty, but lacked message or meaning. I still played around with this stuff from time to time, but it was just a personal hobby of mine.

When I stumbled onto visual complexity things changed. Here were images and programs that just dripped and oozed with meaning and message. Here were artists that distilled meaning out of data and poured it onto the page, so that every inch of the screen illustrated their point. They pull the narrative out of the noise and instead of telling it with words and numbers, they distill it down to raw pixels so it leaps out at you with a glance. That is what I aspire to.

So poke around the site a little bit, there is a ton there and I plan on linking to some of the ones that have really inspired me, and others that are just plain cool.