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.
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Dec 4, 2023
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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 intersectionsMandelbrot 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 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
Box-counting dimension examples
Also listed in…
Fractals
function box_count(canvas) {
let ctx = canvas.getContext("2d");
let im_data = ctx.getImageData(0, 0, canvas.width, canvas.height);
let cnts = [];
for (let s = 1; s <= d3.min([canvas.width, canvas.height]); s = 2 * s) {
let boxes = [];
let cnt = 0;
for (let i = 0; s * (i + 1) <= canvas.height; i++) {
for (let j = 0; s * (j + 1) <= canvas.width; 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++) {
let ij0 = 4 * (j0 + i0 * canvas.width);
if (
// im_data.data[ij0] == 0 &&
// im_data.data[ij0 + 1] == 0 &&
// im_data.data[ij0 + 2] == 0 &&
// im_data.data[ij0 + 3] == 255
im_data.data[ij0] < 100 &&
im_data.data[ij0 + 1] < 100 &&
im_data.data[ij0 + 2] < 100 &&
im_data.data[ij0 + 3] > 155
) {
return true;
}
}
}
return false;
}
return cnts;
}
regression = ss.linearRegression(
fractals.cnts.map((o) => [-Math.log(o.s), Math.log(o.cnt)])
)
fractals = {
let canvas;
if (example == "Sierpinski triangle") {
let sierpIFS = new IteratedFunctionSystem([
scale(1 / 2),
scale(1 / 2, [1, 0]),
scale(1 / 2, [1 / 2, Math.sqrt(3) / 2])
]);
canvas = sierpIFS.render_deterministic({
max_depth: 7,
init: [
[0, 0],
[1, 0],
[1 / 2, Math.sqrt(3) / 2]
],
extent: [
[0, 1],
[-0.1, 0.9]
],
image_width: 512,
image_height: 512
});
} else if (example == "Sparse spiral") {
let ifs = new IteratedFunctionSystem([
scale(0.95).compose(rotate(15 * degree)),
scale(0.15, [1, 0])
]);
canvas = ifs.render_stochastic({
n: 20000,
image_width: 1024,
image_height: 1024
});
} else if (example == "Phylo-spiral") {
let ifs = new IteratedFunctionSystem([
scale(0.99).compose(rotate(137.5 * degree)),
scale(0.1, [1, 0])
]);
canvas = ifs.render_stochastic({
n: 100000,
image_width: 1024,
image_height: 1024
});
} else if (example == "Koch curve") {
let f0 = scale(1 / 3, [0, 0]);
let f1 = shift([1 / 3, 0])
.compose(scale(1 / 3))
.compose(rotate(Math.PI / 3));
let f2 = shift([1 / 2, Math.sqrt(3) / 6])
.compose(scale(1 / 3))
.compose(rotate(-Math.PI / 3));
let f3 = scale(1 / 3, [1, 0]);
canvas = new IteratedFunctionSystem([f0, f1, f2, f3]).render_deterministic({
max_depth: 6,
init: [
[0, 0],
[1, 0]
],
extent: [
[0, 1],
[0, 1 / 2]
],
image_width: 512,
image_height: 256
});
} else if (example == "Kieswetter's curve") {
canvas = new IteratedFunctionSystem([
new AffineFunction([
[
[1 / 4, 0],
[0, -1 / 2]
],
[0, 0]
]),
new AffineFunction([
[
[1 / 4, 0],
[0, 1 / 2]
],
[1 / 4, -1 / 2]
]),
new AffineFunction([
[
[1 / 4, 0],
[0, 1 / 2]
],
[1 / 2, 0]
]),
new AffineFunction([
[
[1 / 4, 0],
[0, 1 / 2]
],
[3 / 4, 1 / 2]
])
]).render_deterministic({
max_depth: 6,
init: [
[0, 0],
[1, 1]
],
extent: [
[0, 1],
[-1, 1]
],
image_width: 512,
image_height: 1024
});
} else if (example == "Barnsley's fern") {
canvas = new IteratedFunctionSystem([
[
[
[0.85, 0.04],
[-0.04, 0.85]
],
[0, 1.6]
],
[
[
[-0.15, 0.28],
[0.26, 0.24]
],
[0, 0.44]
],
[
[
[0.2, -0.26],
[0.23, 0.22]
],
[0, 1.6]
],
[
[
[0, 0],
[0, 0.16]
],
[0, 0]
]
]).render_stochastic({ n: 80000, image_width: 512, image_height: 1024 });
} else if (example == "Line segment") {
canvas = d3.create("canvas").attr("width", 1024).attr("height", 256).node();
let ctx = canvas.getContext("2d");
ctx.beginPath();
ctx.moveTo(20, 200);
ctx.lineTo(1000, 20);
ctx.lineWidth = 1.5;
ctx.stroke();
ctx.closePath();
} else if (example == "Squiggle") {
let w = 1024;
let h = 512;
let xmin = 0;
let xmax = 13;
let ymin = 0;
let ymax = 7;
let x_scale = d3.scaleLinear().domain([xmin, xmax]).range([0, w]);
let y_scale = d3.scaleLinear().domain([ymin, ymax]).range([h, 0]);
canvas = d3.create("canvas").attr("width", w).attr("height", h).node();
let ctx = canvas.getContext("2d");
ctx.beginPath();
ctx.moveTo(x_scale(0), y_scale(0));
for (let t = 0; t <= 2 * Math.PI; t = t + Math.PI / 100) {
ctx.lineTo(x_scale(Math.sin(3 * t) + 2 * t), y_scale(t));
}
ctx.lineWidth = 1.5;
ctx.stroke();
ctx.closePath();
} else if (example == "Solid circle") {
canvas = d3.create("canvas").attr("width", 512).attr("height", 512).node();
let ctx = canvas.getContext("2d");
ctx.beginPath();
ctx.arc(250, 255, 200, 0, 2 * Math.PI);
ctx.fill();
} else if (example == "British coast") {
let pts = british_coast.features[0].geometry.coordinates;
let xpts = pts.map((pt) => pt[0]);
let xmin = Math.floor(d3.min(xpts));
let xmax = Math.ceil(d3.max(xpts));
let ypts = pts.map((pt) => pt[1]);
let ymin = Math.floor(d3.min(ypts));
let ymax = Math.ceil(d3.max(ypts));
let w = 512;
let h = (w * (ymax - ymin)) / (xmax - xmin);
canvas = d3.create("canvas").attr("width", w).attr("height", h).node();
let x_scale = d3.scaleLinear().domain([xmin, xmax]).range([0, w]);
let y_scale = d3.scaleLinear().domain([ymin, ymax]).range([h, 0]);
let ctx = canvas.getContext("2d");
ctx.beginPath();
ctx.moveTo(x_scale(xpts[0]), y_scale(ypts[0]));
for (let i = 1; i < pts.length; i++) {
ctx.lineTo(x_scale(xpts[i]), y_scale(ypts[i]));
}
ctx.stroke();
ctx.closePath();
} else if (example == "Trees") {
let bits = await FileAttachment("tree_bits.txt").text();
bits = bits.split(";").map((s) => s.split("").map((c) => parseInt(c)));
let w = bits[0].length;
let h = bits.length;
canvas = d3.create("canvas").node();
canvas.width = w;
canvas.height = h;
let ctx = canvas.getContext("2d");
for (let i = 0; i < h; i++) {
for (let j = 0; j < w; j++) {
if (bits[i][j] == 0) {
ctx.rect(j, i, 1, 1);
}
}
}
ctx.fill();
}
return {
canvas: canvas,
cnts: box_count(canvas),
dataURL: canvas.toDataURL()
};
}
british_coast = {
let map_file = await FileAttachment("british_coastline.json").json();
let coast = topojson.feature(map_file, map_file.objects.ne_10m_coastline);
return coast;
}
import {
IteratedFunctionSystem,
AffineFunction,
shift,
scale,
rotate,
degree
} from "@mcmcclur/iteratedfunctionsystem-class"
ss = require("simple-statistics")
topojson = require("topojson-client@3")
MathJax = require("https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-svg.js").catch(
() => window["MathJax"]
)
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