Archive

A superposition is not a morph

For fifteen years the harmonic-oscillator demo on this site animated a superposition as cos t·ψ_A + sin t·ψ_B and called it time evolution. It was not — and the conviction doesn't even need the Rabi oscillation it happens to resemble: a real wavefunction carries identically zero probability current, so the morphing density was teleporting. The original animation, preserved as the bug it was, with the Born flow that convicts it.

The chaos game

Throw a point at random through the maps, plot where it lands, and a theorem guarantees the attractor draws itself. The renderer is trivial. The tone map is where the work is.

The derivative is nilpotent, and the shift is its exponential

Differentiate a quintic six times and it is gone. Exponentiate that same dying operator and you get the shift by h, invertible forever. The sentence connecting them is Taylor's theorem.

Products add, composition multiplies

Two ways to combine polynomials, two different laws for the degree. One of them proves in a single line that the only invertible polynomials are the affine ones.

Degree is a coordinate

Substituting ax + b into a polynomial is a linear map with a triangular matrix. The diagonal is the powers of a, which is the whole reason an affine change of variable can never change a degree.

What an affine IFS actually is

A handful of maps that shrink the plane, a theorem that says they have exactly one fixed shape, and a second theorem that tells you how to find the maps for a shape you already want.

What a function does to a space of functions

A polynomial is a point in a vector space. Shifting, scaling, differentiating, and substituting are maps of that space into itself, and the only question worth asking about any of them is what happens to the degree.

Five ways to draw the same fractal

The fractal designer now carries five renderers, an aspect-aware workspace, and a button that hands you a clean PNG. Three of the five were quietly drawing the wrong thing.

From exponential growth to the logistic curve

A single population grows without bound, predators can only cage it into a boom-and-bust cycle, and the honest fix is a population that limits itself: the S-curve.

Van der Pol and the birth of a limit cycle

An oscillator with friction that changes its mind. Small swings get pumped up, big ones get bled down, and everything meets on a single closed loop the whole plane falls onto.

The pendulum, the cat's eye, and a little friction

The pendulum is the spring with the small-angle lie removed. Putting the full sine back in bends its phase space into the cat's-eye pattern, and friction drains it.

The spring equation and the shape of simple motion

Where the spring equation comes from, and why its phase space is a family of nested ellipses that damping turns into a spiral.

Drawing differential equations as wind

An old widget of mine, rebuilt in plain javascript, that draws the phase space of a differential equation the way those wind maps draw the wind.

Evil Cauliflower

Clearing some stuff out

New Feature - Download demo image

Symmetric - almost

Hexagonal snowflake

Hexagonal grid fractal

Replicated triangles

Faded paper

Fractal Grid2

Fractal Ruler

Grid deconstructed

Another image, busy busy

Fractal Friday - wisp

Fractal Friday - alignment

Fractal Friday - darkness

More crossing lines images.

Fractal Friday

Playing around with overlapping lines.

Some simple geometric fractals

Just post something, Jeeezz

Varying circle grid and color

Grid of overlapping circles

A bunch of circles

Roiling sparks

Experimenting with pixel sizes

Geometric spiral fractal

Hard to believe there is a pattern to this one.

Generative Convergence #2 - Ghosts

Generative convergence #1 - Shadows

More one dimensional generative experiments

New Technique - Iterated function system interpolation functions

This is what additivity gets you

All filled in - Animated Generative circles #2

Technique work

Another generative geometric fractal experiment

Ring around

Current project

My First canvas experiment

Twist again - more geometric generative op art

Askew Op Art Disk - Where'd the symmetry go?

Circling back to a dead end.

Spider web fractal

Simple Op Art disk

Twist #3

Bouncing Circle Grid

Twist #2

Simple twist

A single twist and spray

Supercharged performance

Grids of circles

Experimental generative images

All things must pass

Circled out

Circular Cellular Automata

Infinite Flow

Bouncing circles

Connected components in cellular automata

More animated Circles

A Pencil of Circles

Happy Accident #2

Circle Test

A Random Walk Through Some Cellular Automatons

Cellular automaton #1

Parameterized Prey Peaks

Disintegration

Chaos #1

Distortion

A happy accident

A minor update - circles

Canvas Line Based Image Renderer

HTML5 Canvas Image Effects

Making sense of space-time

HTML5 color eyedropper

Visual studio 11 and javascript

Messing around with the canvas mask api

Random dynamics on a triangular grid

Fractal Drawing Tool - Users Guide

Flitter and flutter

Nothing fancy: Winter bubbles

Forgot to upload this image

Something different about this one

Gravitational slingshots

A less idealized central force

Central force images #2

These aren't all that pretty

Islands of stability

Sensitive dependence on initial conditions

Ensemble #2

Ensemble #1

More envelopes

sine waves and envelopes

I hadn't thought of stacking multiple iterations on top of each other

WebGL buffer fractal renders

Playing around with straight lines

A little free time

Got the math right with this one

Big day tomorrow.

The other side of the world

Another one I just like

Complexification

Another image

Rough and Random page

Quality of Movement

Spiral

What makes the Mandelbrot set so special?

A Mandelbrot Image I've been working on.

Odds and ends

Something I'm playing around with.

I never knew it looked like that

Release often

Static images

Mondrian meets javascript

Electron sausages?

Fuzzy and indistinct

Atomic Orbitals now a chrome experiment

I think I've decided

I'm thinking about how best to handle this

The best surprises come from unexpected places

Just a quick update

three.js particle system

This is one of the coolest things I've seen in quite a while

I'm just playing around with particles for this one

New gallery - keep it simple

The most basic IFS Fractal of all - the cantor set

square + triangle = ???

Now something completely different

Things are slightly askew

more square

nearly square

Speed of sound

Now we're getting somewhere

Lorenz attractor v1

Closer to a good stopping point

I've got to hand it to IE 9.0 beta

I think this is the last moving fractal for a while

faster, denser

Visual complexity

New streaking and pause

Improvements and effects

Animate2

Now with Movement

Gritty Block Header

chrome fractals