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
Sep 10, 2022
7 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 revolutionOrbit detection for the Mandelbrot setTracing a path on a spherePlot for mathematiciansFunctions of two variablesPartial derivativesDijkstra's algorithm on an RGGUnfolding 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
Gradient ascent
// Implementation of the peaks function
function f([x, y]) {
return (
3 * Math.exp(-(x ** 2 + (y + 1) ** 2)) * (1 - x) ** 2 -
Math.exp(-((x + 1) ** 2 + y ** 2)) / 3 -
10 * Math.exp(-(x ** 2 + y ** 2)) * (x / 5 - x ** 3 - y ** 5)
);
}
// The gradient of the peaks function
function grad([x, y]) {
return [
-6 * Math.exp(-x * x - (1 + y) ** 2) * (1 - x) -
6 * Math.exp(-x * x - (1 + y) ** 2) * (1 - x) ** 2 * x +
(2 * Math.exp(-(1 + x) * (1 + x) - y ** 2) * (1 + x)) / 3 -
10 * Math.exp(-x * x - y ** 2) * (1 / 5 - 3 * x ** 2) +
20 * Math.exp(-x * x - y ** 2) * x * (x / 5 - x ** 3 - y ** 5),
(2 * Math.exp(-(1 + x) * (1 + x) - y ** 2) * y) / 3 +
50 * Math.exp(-x * x - y ** 2) * y ** 4 -
6 * Math.exp(-x * x - (1 + y) ** 2) * (1 - x) ** 2 * (1 + y) +
20 * Math.exp(-x * x - y ** 2) * y * (x / 5 - x ** 3 - y ** 5)
];
}
// To compute the gradient ascent from a starting point,
// we'll literally follow the gradient at each point.
function compute_path([x0, y0]) {
const tol = 0.01;
const dt = 0.01;
let path = [[x0, y0]];
let step = grad([x0, y0]);
let cnt = 0;
while (Math.abs(step[0]) + Math.abs(step[1]) > tol && cnt < 10000) {
let [x, y] = path[path.length - 1];
let x_new = x + step[0] * dt;
let y_new = y + step[1] * dt;
path.push([x_new, y_new]);
step = grad([x_new, y_new]);
cnt = cnt + 1;
}
return path;
}
grid = {
let a = -2.5;
let b = 2.5;
const m = 250;
let dx = (b - a) / m;
let c = -2.5;
let d = 2.5;
const n = 250;
let dy = (d - c) / n;
let grid = d3.range((m + 1) * (n + 1)).map(function(k) {
let i = k % (m + 1);
let x = a + i * dx;
let j = Math.floor(k / (m + 1));
let y = d - j * dy;
return [x, y];
});
grid.m = m;
grid.n = n;
grid.a = a;
grid.b = b;
grid.c = c;
grid.d = d;
return grid;
}
// A lot of magic happens here!
contours = d3
.contours()
.size([grid.m + 1, grid.n + 1])
.thresholds(thresholds)(grid.map(f))
.map(transform)
// Transform the contours by mapping from grid coordinates (indices)
// to svg coordinates (px).
function transform({ type, value, coordinates }) {
let i_scale = d3
.scaleLinear()
.domain([0, grid.m + 1])
.range([0, svg_width]);
let j_scale = d3
.scaleLinear()
.domain([0, grid.n + 1])
.range([0, svg_height]);
return {
type,
value,
coordinates: coordinates.map(rings => {
return rings.map(points => {
return points.map(([i, j]) => [i_scale(i), j_scale(j)]);
});
})
};
}
// Contour placement
thresholds = d3.range(-8, 9, 1 / 2)
// Put it all together!
function make_gradient_ascent_pic() {
const svg = d3
.create("svg")
.attr("class", "contour_plot")
.attr("viewBox", [0, 0, svg_width, svg_height])
.style("max-width", `${svg_width}px`);
svg
.append("g")
.attr("stroke", "#000")
.attr("stroke-opacity", 0.8)
.selectAll("path")
.data(contours)
.join("path")
.attr("fill", (d) => color(-d.value))
.attr("d", d3.geoPath(null));
let x_scale = d3.scaleLinear().domain([grid.a, grid.b]).range([0, svg_width]);
let y_scale = d3
.scaleLinear()
.domain([grid.c, grid.d])
.range([svg_height, 0]);
let pts_to_path = d3
.line()
.x((d) => x_scale(d[0]))
.y((d) => y_scale(d[1]));
let ascent = svg.append("g");
svg
.on("pointerenter", function (evt) {
let [w, h] = d3.pointer(evt);
let x = x_scale.invert(w);
let y = y_scale.invert(h);
let path = compute_path([x, y]);
ascent
.selectAll("path")
.data([path])
.enter()
.append("path")
.attr("d", pts_to_path)
.attr("stroke", "black")
.attr("stroke-width", 4)
.attr("fill", "none");
ascent
.append("circle")
.attr("id", "start")
.attr("cx", x_scale(path[0][0]))
.attr("cy", y_scale(path[0][1]))
.attr("r", 5)
.attr("fill", "#5f5");
ascent
.append("circle")
.attr("id", "fin")
.attr("cx", x_scale(path[path.length - 1][0]))
.attr("cy", y_scale(path[path.length - 1][1]))
.attr("r", 5)
.attr("fill", "#f55");
})
.on("touchstart", (e) => e.preventDefault())
.on("pointermove", function (evt) {
let [w, h] = d3.pointer(evt);
let x = x_scale.invert(w);
let y = y_scale.invert(h);
let path = compute_path([x, y]);
ascent.select("path").data([path]).join("path").attr("d", pts_to_path);
ascent
.selectAll("circle#start")
.data([path])
.join("circle")
.attr("cx", (d) => x_scale(d[0][0]))
.attr("cy", (d) => y_scale(d[0][1]));
ascent
.selectAll("circle#fin")
.data([path])
.join("circle")
.attr("cx", (d) => x_scale(d[d.length - 1][0]))
.attr("cy", (d) => y_scale(d[d.length - 1][1]));
})
.on("pointerleave", function () {
ascent.selectAll("*").remove();
});
return svg.node();
}
color = d3
.scaleLinear()
.domain(d3.extent(thresholds))
.interpolate(d => d3.interpolateBlues)
svg_width = width < 800 ? width : 800
// Specifies the height of the SVG in px
svg_height = svg_width
// d3 = require('d3@5')
x3dom = require("x3dom@1.7").catch(() => window["x3dom"])
html`<style>
canvas {
outline: none;
}
</style>`
static_pic = {
let pic = d3.select(make_gradient_ascent_pic());
pic.on("pointerenter", null);
pic.on("pointerleave", null);
pic.on("pointeremove", null);
return pic.node();
}
FileAttachment("peaks.x3d").text()
One platform to build and deploy the best data apps
Experiment and prototype by building visualizations in live JavaScript notebooks. Collaborate with your team and decide which concepts to build out.
Use Observable Framework to build data apps locally. Use data loaders to build in any language or library, including Python, SQL, and R.
Seamlessly deploy to Observable. Test before you ship, use automatic deploy-on-commit, and ensure your projects are always up-to-date.