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
Nov 25, 2020
29 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 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
The geometry and numerics of first order ODEs
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Teaching ODEs
// Perform Euler's method with n steps over the interval [a,b]
// for y' = f(y,t) with y(t0) = y0.
function euler(f, y0, a, b, n) {
let dt = (b - a) / n;
let ti = a;
let yi = y0;
let pts = [[ti, yi]];
for (let i = 0; i < n; i++) {
yi = yi + f(yi, ti) * dt;
ti = ti + dt;
pts.push([ti, yi]);
}
return pts;
}
pts = euler((y, t) => y, 1, 0, 1, 10)
// Apply the Runge-Kutta method with n steps over an interval
// [a,b] containing t0.
function rk4_1d(f, y0, t0, a, b, n) {
let dt = (b - a) / n;
if (t0 == a) { // Just go forward
return rk4_multi_step_1d(f, y0, a, dt, n);
} else if (b == t0) { // Just go backward
let result = rk4_multi_step_1d(f, y0, b, -dt, n);
result.reverse();
return result;
} else if (a < t0 && t0 < b) { // Go forward and backward and concatenate
let i = Math.floor((t0 - a) / dt);
let backward = rk4_multi_step_1d(f, y0, t0, -dt, i);
backward.reverse();
let forward = rk4_multi_step_1d(f, y0, t0, dt, n - i).slice(1);
return backward.concat(forward);
}
}
function rk4_multi_step_1d(f, y0, t0, dt, n) {
let result = [[t0, y0]];
for (let i = 0; i < n; i++) {
y0 = rk4_single_step_1d(f, y0, t0, dt);
t0 = t0 + dt;
if (Math.abs(y0) < 1000) {
result.push([t0, y0]);
} else {
break;
}
}
return result;
}
function rk4_single_step_1d(f, y0, t0, dt) {
let dt2 = dt / 2;
let k1 = f(y0, t0);
let k2 = f(y0 + k1 * dt2, t0 + dt2);
let k3 = f(y0 + k2 * dt2, t0 + dt2);
let k4 = f(y0 + k3 * dt, t0 + dt);
return y0 + ((k1 + 2 * k2 + 2 * k3 + k4) * dt) / 6;
}
md`Euler's method with 10 steps:`
euler_approx10 = euler(y => y, 1, 0, 1, 10).slice(-1)[0][1]
rk4_approx = rk4_1d(y => y, 1, 0, 0, 1, 10).slice(-1)[0][1]
euler_approx100000 = euler(y => y, 1, 0, 1, 100000).slice(-1)[0][1]
[Math.E - euler_approx100000, Math.E - rk4_approx]
// d3 = require('d3@6')
d3 = require('d3-selection@2', 'd3-scale@3', 'd3-shape@2', 'd3-array@2', 'd3-axis@2')
math = require('mathjs@7.6.0')
import { slider, text, select } from "@jashkenas/inputs"
import { plotter } from '@kjerandp/plotter'
// Store the values of t0 and y0 in this external object so
// they don't get rewritten when we resize R.
inits = ({
t0: null,
y0: null
})
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