I'm a professor of mathematics at the University of North Carolina Asheville. I've also done a fair amount of consulting work over the years focusing on scientific and data visualization.
I'm happy to be part of the first Observable Ambassador's cohort.
Published
Edited
Aug 1, 2022
9 stars
A Julia set on the Riemann sphereThe Z-CurveBarnsley's fernA stochastic digraph IFS algorithmSelf-affine tilesThe TwindragonThe Eisenstein fractionsA self-affine tile with holesSelf-affine tiles via polygon mergeGolden rectangle fractalsBifurcation diagram with critical curvesThe tame twindragonIllustrations for the proof of Green's theoremNon-orientability of a Mobius stripExamples of parametric surfacesPenrose tilingThe extended unit circlePenrose three coloringNewtons's method on the Riemann sphereConic sectionsDivisor graphsThe dance of Earth and VenusIterating multiples of the sine functionBorderline fractalsSelf-similar intersectionsBox-counting dimension examplesMandelbrot by dimensionInverse iteration for quadratic Julia setsInteger Apollonian PackingsIllustrations of two-dimensonal heat flowThe logistic bifurcation locusThe eleven unfoldings of the cubeA unimodal function with fractal level curvesGreen's theorem and polygonal areaThe geometry and numerics of first order ODEsThe xxx^xxx-spindleAnimated beatsRauzy FractalsHilbert's coordinate functionsPluckNot PiDrum strikeThe Koch snowflakeFractalized squareA Taylor series about π/4\pi/4π/4PlotX3D HyperboloidA PlotX3D animationModular arithmetic in 5th grade artSimple S-I-R ModelThe Poisson KernelPoly-gasketsClassification of 2D linear systems via trace and determinantJulia sets and the Mandelbrot setWater wavesFourier SeriesDisks for a solid of revolutionTracing a path on a spherePlot for mathematiciansFunctions of two variablesPartial derivativesDijkstra's algorithm on an RGGGradient ascentUnfolding polyhedraTangent plane to a level surfaceA strange discontinuityExamples of level surfacesMcMullen carpetsHills and valleysThe definition of ⇒Double and iterated integralsMST in an RGGTrees are bipartiteFractal typesettingd3.hierarchy and d3.treeK23 is PlanarPolar CoordinatesParametric region generatorParametric Plot 2DContour plotsGreedy graph coloringGraph6A few hundred interesting graphsThe Kings ProblemFirst order, autonomous systems of ODEsRunge-Kutta for systems of ODEs
Orbit detection for the Mandelbrot set
Also listed in…
Complex Dynamics
function mandelbrot_period_cnt(c, opts = {}) {
let { bail = 1000, max_orbit_length = 50, tolerance = 10 ** -15 } = opts;
let zi = new Complex(0, 0);
let i = 0;
let iterates = [zi];
while (i++ < bail) {
zi = zi.squared().plus(c);
if (zi.abs2() > 4) {
return [-1, i];
}
let idx = zi.indexOf(iterates, tolerance);
if (idx > -1) {
idx = zi.indexOf(iterates, tolerance * 10 ** 4);
return [iterates.length - idx, i];
} else {
iterates.push(zi);
}
if (iterates.length == max_orbit_length) {
iterates.shift();
}
}
return [0, i];
}
function draw(S, opts = {}) {
let { bail = 1000, max_orbit_length = 50, tolerance = 10 ** -15 } = opts;
let xmin = -2;
let xmax = 0.6;
let ymin = -1.3;
let ymax = 1.3;
let xScale = d3.scaleLinear().range([xmin, xmax]).domain([0, S]);
let yScale = d3.scaleLinear().range([ymin, ymax]).domain([S, 0]);
let canvas = d3.create("canvas").attr("width", S).attr("height", S);
let context = canvas.node().getContext("2d");
for (let i = 0; i < S; i++) {
Promises.delay().then(() => draw_row(i));
}
return canvas.node();
function draw_row(i) {
for (let j = 0; j < S; j++) {
let c = new Complex(xScale(j), yScale(i));
let [p, it] = mandelbrot_period_cnt(c, { bail, max_orbit_length, tolerance });
context.fillStyle = d3
.color(d3.schemeCategory10[(p + 1) % 10])
.darker(it / 100)
.toString();
context.fillRect(j, i, 1, 1);
}
}
}
//draw(500)
Complex = {
let complex = class Complex {
constructor(a, b) {
if (b) {
this.re = a;
this.im = b;
} else if (a) {
this.re = a;
this.im = 0;
} else {
this.re = 0;
this.im = 0;
}
}
// Magnitude squared, magnitude, and squared.
abs2() {
return this.re ** 2 + this.im ** 2;
}
abs() {
return Math.sqrt(this.abs2());
}
squared() {
return new Complex(this.re ** 2 - this.im ** 2, 2 * this.re * this.im);
}
// Algebra
plus(z) {
if (z instanceof Complex) {
return new Complex(this.re + z.re, this.im + z.im);
} else if (typeof z == "number") {
return new Complex(this.re + z, this.im);
} else {
return NaN;
}
}
minus(z) {
if (z instanceof Complex) {
return new Complex(this.re - z.re, this.im - z.im);
} else if (typeof z == "number") {
return new Complex(this.re - z, this.im);
} else {
return NaN;
}
}
times(z) {
if (z instanceof Complex) {
return new Complex(
this.re * z.re - this.im * z.im,
this.re * z.im + this.im * z.re
);
} else if (typeof z == "number") {
return new Complex(z * this.re, z * this.im);
} else {
return NaN;
}
}
// Approximate comparison
almost_equal(z, tolerance = 10 ** -20) {
return this.minus(z).abs2() < tolerance;
}
// Check for occurence in an array
indexOf(A, tolerance = 10 ** -10) {
let i = A.length - 1;
while (i >= 0) {
let z = A[i];
if (this.almost_equal(z, tolerance)) {
break;
}
i--;
}
if (i == 0) {
i = -1;
}
return i;
}
};
return complex;
}
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