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.
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 dimensionInteger 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 revolutionOrbit detection for the Mandelbrot setTracing 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
Inverse iteration for quadratic Julia sets
new QuadraticJuliaSet(0).draw_simple()
new QuadraticJuliaSet(math.complex(-0.62, 0.41)).draw_simple({ depth })
new QuadraticJuliaSet(math.complex(-0.62, 0.41)).draw()
QuadraticJuliaSet = {
let QuadraticJuliaSet = class QuadraticJuliaSet {
constructor(c) {
this.c = c;
}
};
QuadraticJuliaSet.prototype.draw_simple = function (opts = {}) {
let {
depth = 12,
width = 512,
height = 512,
extent = [
[-2, 2],
[-2, 2]
]
} = opts;
let c = this.c;
let xmin = extent[0][0];
let xmax = extent[0][1];
let ymin = extent[1][0];
let ymax = extent[1][1];
let pts = [math.complex(0.123, -0.0456)];
for (let i = 0; i < depth; i++) {
pts = pts
.map(function (z) {
let z1 = math.sqrt(math.add(z, math.multiply(c, -1)));
let z2 = math.multiply(z1, -1);
return [z1, z2];
})
.flat();
}
let canvas = d3
.create("canvas")
.attr("width", width)
.attr("height", height);
let ctx = canvas.node().getContext("2d");
ctx.fillStyle = "black";
pts = pts.map(function (z) {
let ij = complex_to_canvas(z, xmin, xmax, ymin, ymax, width, height);
let i = ij[0];
let j = ij[1];
ctx.fillRect(i, j, 1, 1);
});
return canvas.node();
};
QuadraticJuliaSet.prototype.generate_inverse_iterate_matrix = function (
opts = {}
) {
let {
bailout = 5,
extent = [
[-2, 2],
[-2, 2]
],
width = 512,
height = 512
} = opts;
let c = this.c;
let xmin = extent[0][0];
let xmax = extent[0][1];
let ymin = extent[1][0];
let ymax = extent[1][1];
// Set up a matrix to keep track of which
// points have been plotted.
let plot_record = new Array(width + 1);
for (let i = 0; i <= width; i++) {
plot_record[i] = new Array(height + 1);
}
// Initialize the values in the matrix
// to be zero.
for (let row = 0; row <= width; row++) {
for (let col = 0; col <= height; col++) {
plot_record[row][col] = 0;
}
}
// Start the iteration from an intial point.
let z0 = math.complex(0.123, -0.0456);
// And inverse iterate a few times to ensure
// the initial point is close to the Julia set.
for (let i = 0; i < 5; i++) {
z0 = math.sqrt(math.add(z0, math.multiply(c, -1)));
z0 = math.multiply(math.sqrt(math.add(z0, math.multiply(c, -1))), -1);
}
// Now, we're going to use a standard tree construction
// to perform the inverse iteration.
let zNode = new JuliaNode(z0);
let queue = [zNode];
let cnt = 0;
while (queue.length > 0) {
zNode = queue.pop();
let z = zNode.z;
let ij = complex_to_canvas(z, xmin, xmax, ymin, ymax, width, height);
let i = ij[0];
let j = ij[1];
if (plot_record[i][j] < bailout) {
let z1 = math.sqrt(math.add(z, math.multiply(c, -1)));
let z2 = math.multiply(z1, -1);
let left = new JuliaNode(z1);
let right = new JuliaNode(z2);
zNode.left = left;
zNode.right = right;
queue.push(left);
queue.push(right);
plot_record[i][j] = plot_record[i][j] + 1;
}
}
// The output is matrix that tells us which pixels
// are on (value>0) or off (value=0).
return plot_record;
};
QuadraticJuliaSet.prototype.draw = function (opts = {}) {
let {
bailout = 5,
extent = [
[-2, 2],
[-2, 2]
],
width = 512,
height = 512
} = opts;
let all_opts = { bailout, extent, width, height };
let plot_record = this.generate_inverse_iterate_matrix(all_opts);
let canvas = d3
.create("canvas")
.attr("width", width)
.attr("height", height);
let ctx = canvas.node().getContext("2d");
ctx.fillStyle = "black";
let cnts = [];
for (let j = 0; j <= height; j++) {
for (let i = 0; i <= width; i++) {
if (plot_record[i][j] > 0) {
ctx.fillRect(i, j, 1, 1);
}
}
}
return canvas.node();
};
QuadraticJuliaSet.prototype.box_count = function () {
return box_count(
this.generate_inverse_iterate_matrix({
bailout: 5,
width: 2 ** 10,
height: 2 ** 10
})
);
};
return QuadraticJuliaSet;
function xy_to_canvas(x, y, xmin, xmax, ymin, ymax, width, height) {
return [
Math.min(Math.floor(((x - xmin) * width) / (xmax - xmin)), width - 1),
Math.min(Math.floor(((ymax - y) * height) / (ymax - ymin)), height - 1)
];
}
function complex_to_canvas(z, xmin, xmax, ymin, ymax, width, height) {
return xy_to_canvas(z.re, z.im, xmin, xmax, ymin, ymax, width, height);
}
function box_count(matrix) {
let w = matrix.length;
let cnts = [];
for (let s = 1; s <= w; s = 2 * s) {
let boxes = [];
let cnt = 0;
for (let i = 0; s * (i + 1) < w; i++) {
for (let j = 0; s * (j + 1) < w; j++) {
if (check_box(i, j, s)) {
cnt = cnt + 1;
boxes.push([i, j]);
}
}
}
cnts.push({ s: s, cnt: cnt, boxes: boxes });
}
function check_box(i, j, s) {
for (let i0 = i * s; i0 < (i + 1) * s; i0++) {
for (let j0 = j * s; j0 < (j + 1) * s; j0++) {
if (matrix[i0][j0] > 0) {
return true;
}
}
}
return false;
}
return cnts;
}
}
JuliaNode = class JuliaNode {
constructor(z) {
this.z = z;
}
}
math = require("mathjs")
// julia_dim2 = julia_dim
// .filter((a) => a.C < 0.2501)
// .concat([{ C: 0.2505, Dim: undefined }])
// .concat(julia_dim.filter((a) => a.C > 0.2505))
// julia_dim = FileAttachment("julia_dim.csv").csv({ typed: true })
ss = require("simple-statistics")
{
// let div = d3.create("div").style("width", "600px");
let canvas = J.draw({
bailout: 25,
width: 1000,
height: 250,
extent: [
[-2, 2],
[-0.5, 0.5]
]
});
d3.select(canvas).style("max-width", "100%");
//div.append(() => canvas);
return canvas;
}
counts
? ss.linearRegression(counts.map((o) => [-Math.log(o.s), Math.log(o.cnt)]))
: 0
counts = J.box_count({
bailout: 25,
width: 1000,
height: 1000
})
J = new QuadraticJuliaSet(c)
// c = math.complex(-0.122561, 0.744861)
// c = math.complex(0.3, 0.025)
c = math.complex(-1.76, 0.005)
Purpose-built for displays of data
Observable is your go-to platform for exploring data and creating expressive data visualizations. Use reactive JavaScript notebooks for prototyping and a collaborative canvas for visual data exploration and dashboard creation.
