The Chaos Pages

Web pages that exploit JavaScript to do animated explanations.

Quick start

Each page will have a link to the next one.

Start here with the Sierpiński Triangle, a classic fractal that emerges from chaos.

Compatibility

Some of the features of these pages (namely “canvas” and “slider”) are new in HTML 5. Your browser must be up-to-date. By slider, I mean a GUI control that you drag back and forth to set a number. If it is rendered like this

you are OK. (This one is just a picture, it won’t actually work.) If the blue knob is missing and there is just a text field to type in, a lot of the interactivity of the Web page will be missing.

Brief Explanation

These web pages illustrate graphically some of the interesting ideas in chaos and fractal theory. The basic idea is that they follow a plan, and you go through them from first to last, using the links at the top or bottom of the pages. This is a good idea if you haven’t seen them before.

But you can also visit them in arbitrary order. In the section below, there are links for each page, so you can jump to them directly.

Navigate yourself

• Sierpiński Triangle — A classic fractal drawn by a random process.
• Iteration and Orbits — What happens to the sequence x, f(x), f(f(x)), etc...?
• The Logistic Map 1 — Single-step through the orbits of a simple-looking function.
• The Logistic Map 2 — Graph the orbits of the logistic map.
• The Bifurcation Diagram — Another way of graphing the logistic orbits.
• Universality — Any function with the up-and-down shape of the logistic map will have similar chaotic behavior.
• Composition — When you look at f(f(f(x))), it turns out to be very complicated.
• Reflection — Iterating a function by reflecting it off the y=x diagonal.

Remarks

Although this set of eight pages comprises a linear narrative, at each page we could turn off toward a multitude of related topics. Perhaps this set of Web pages will eventually more resemble a tree than a line! Of course, eventually the large-scale connectivity would be web-like. This is just a start a getting the simple things illustrated. Each page is self-contained, and no code is hidden or obfuscated, so inquiring minds can inspect the programming and learn further.

See Also

• The Golden Ratio Pages — A similar series of pages exploring the famous ratio.
• Gear Ratios — These may not be golden, but they are important for people who ride and work with bicycles. Available as a mobile app and a Web page.
• Anagrams — A mobile app and a Web tool for finding anagrams.
• LL(1) Parsing — A Web tool for generating a parse table and using it to construct a leftmost derivation.
• Enumeration — A Web tool for finding permutations and combinations.
• Gaussian Elimination — A Web tool for reducing matrices to row echelon form.
• Combinatorial Music Theory — A lecture connecting graph theory with musical scales and chords.
• The 3D Pages — My JavaScript implementation of interactive 3D graphics .
• The DSP Pages — Explaining the Fourier transform in the discrete domains.
• Graph Clock — A good example of using JavaScript to make a self-modifying Web page, and a little puzzle about elementary connected graphs.
• Regular Expressions — Sometimes a non-match can hang the system.
• The Z-Board — A new kind of MIDI controller.