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 circleNewtons'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 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
Penrose three coloring
// This is the main function that's called recursively in response to
// a push of the 'Find three coloring' button.
function recolor(r) {
let neighbors = rhombs.filter((r2) => r.neighbors.indexOf(r2.idx) > -1);
let current_color = r.color;
if (neighbors.map((d) => d.color).indexOf(current_color) == -1) {
return current_color;
}
let color_choices, new_color;
if (current_color == 0) {
color_choices = [1, 2];
} else if (current_color == 1) {
color_choices = [0, 2];
} else if (current_color == 2) {
color_choices = [0, 1];
}
let choice = d3.randomUniform()();
let rnorm = norm1(r.centroid);
let smaller_neighbors = neighbors.filter((r2) => norm1(r2.centroid) < rnorm);
if (smaller_neighbors.map((d) => d.color).indexOf(current_color) > -1) {
if (choice < 0.45) {
new_color = color_choices[0];
} else if (choice < 0.9) {
new_color = color_choices[1];
} else {
new_color = current_color;
}
} else {
if (choice < 0.05) {
new_color = color_choices[0];
} else if (choice < 0.1) {
new_color = color_choices[1];
} else {
new_color = current_color;
}
}
return new_color;
}
// Iterate until this returns true
function check_three_color(colors) {
for (let i = 0; i < colors.length; i++) {
if (
rhombs[i].neighbors.map(d => rhombs[d].color).indexOf(rhombs[i].color) >
-1
) {
return false;
}
}
return true;
}
// Assign the colors
function color(c) {
if (c == 0) {
return "#e41a1c";
} else if (c == 1) {
return "#377eb8";
} else {
return "#ffff33";
}
}
triangles = {
let triangles = [S];
for (let i = 0; i < n.value; i++) {
triangles = triangles.map(dissect(i)).reduce(function (a, c) {
return a.concat(c);
}, []);
}
return triangles;
}
phi = (Math.sqrt(5) + 1) / 2
// The initial triangle
S = ({
vertices: [[phi, 0], [phi / 2, Math.sin(Math.PI / 5)], [0, 0]],
type: 'o'
})
function dissect(i) {
let i_dissect = function(T) {
let T1, T2;
let [p, q, r] = T.vertices;
if (T.type == 'a') {
if (i % 2 == 0) {
let new_point = [
(phi * q[0] + p[0]) * (2 - phi),
(phi * q[1] + p[1]) * (2 - phi)
];
T1 = {
vertices: [r, new_point, q],
type: 'a'
};
T2 = {
vertices: [r, new_point, p],
type: 'o'
};
return [T1, T2];
} else {
return [T];
}
} else if (T.type == 'o') {
if (i % 2 == 1) {
let new_point = [
(phi * r[0] + p[0]) * (2 - phi),
(phi * r[1] + p[1]) * (2 - phi)
];
T1 = {
vertices: [r, new_point, q],
type: 'o'
};
T2 = {
vertices: [p, q, new_point],
type: 'a'
};
return [T1, T2];
} else {
return [T];
}
}
};
return i_dissect;
}
// Merge the triangles into rhombs and compute the resulting adjacencies.
// This is the computationally intensive portion.
rhombs = {
// How close two points should be to be considered the same
let tol = 0.01;
// A list to hold all the rhombs
let rhombs = [];
// We'll do a double loop (certainly, this could benefit from a QuadTree).
// i will range over all the triangles; j will range from i+1 to triangles.length
// Of course, if j>i but already matches with a previous triangle, we can cross
// it off our list. That's the purpose of found_indices.
let found_indices = [];
for (let i = 0; i < triangles.length; i++) {
let Ti = triangles[i];
let found = false;
for (let j = i + 1; j < triangles.length; j++) {
if (found_indices.indexOf(i) == -1) {
let Tj = triangles[j];
// Two type a's could join to form a skinny rhomb
if (Ti.type == 'a' && Tj.type == 'a') {
if (
dist1(Ti.vertices[1], Tj.vertices[1]) +
dist1(Ti.vertices[2], Tj.vertices[2]) <
tol
) {
found_indices.push(j);
rhombs.push({
type: 's',
vertices: [
Ti.vertices[2],
Ti.vertices[0],
Ti.vertices[1],
Tj.vertices[0]
]
});
found = true;
break;
}
// Two type o's could join to form a fat rhomb.
} else if (Ti.type == 'o' && Tj.type == 'o') {
if (
dist1(Ti.vertices[0], Tj.vertices[0]) +
dist1(Ti.vertices[2], Tj.vertices[2]) <
tol
) {
found_indices.push(j);
rhombs.push({
type: 'f',
vertices: [
Ti.vertices[2],
Ti.vertices[1],
Ti.vertices[0],
Tj.vertices[1]
]
});
found = true;
break;
}
}
} else {
found = true;
}
}
// We do see some triangles on the border that would match with
// triangles outside the border.
if (!found) {
rhombs.push(Ti);
}
}
// Make sure that all polygons (rhombs and triangles) are positively
// oriented. Should speed up adjacency testing later.
rhombs.forEach(function(r, idx) {
// r.idx = idx;
if (
r.type == 'a' &&
r.vertices.filter(v => Math.abs(v[1]) < tol).length == 2
) {
r.vertices.sort().reverse();
} else if (!leftOf(...r.vertices)) {
r.vertices.reverse();
if (r.type == 'a') {
let vv = r.vertices;
r.vertices = [vv[1], vv[2], vv[0]];
}
}
});
// Use symmetry to extend the tiling. Note that a few of the triangles on the
// bottom edge are 1/2 of a rhomb obtained after reflection.
let bottom_half_rhombs = [];
let upper_rhombs = [];
rhombs.forEach(function(r) {
let vv = r.vertices;
// The half-rhombs are type 'o' or 'a'.
if (r.type == 'o' || r.type == 'a') {
// and their y-coordinates of the first and third vertices are zero
if (Math.abs(vv[0][1]) < tol && Math.abs(vv[2][1]) < tol) {
// if (vv.filter(v => Math.abs(v[1]) < tol).length == 2) {
r.vertices.sort().reverse();
r.vertices.push([vv[1][0], -vv[1][1]]);
r.vertices = vv;
bottom_half_rhombs.push(r);
} else {
// Other 'o's and 'a's are treated more easily
upper_rhombs.push(r);
}
} else {
// All the 'f's and 's's are treated more easily
upper_rhombs.push(r);
}
});
// The upper_rhombs can simply be reflected
let new_rhombs = upper_rhombs.map(function(r) {
let r2 = { type: r.type };
r2.vertices = r.vertices.map(([x, y]) => [x, -y]).reverse();
return r2;
});
rhombs = rhombs.concat(new_rhombs);
// Compute centroids and prepare the neighborhood
rhombs.forEach(function(r, idx) {
r.idx = idx;
let x0 = d3.mean(r.vertices.map(d => d[0]));
let y0 = d3.mean(r.vertices.map(d => d[1]));
r.centroid = [x0, y0];
r.neighbors = [];
r.color = Math.floor(d3.randomUniform(3)());
});
// Compute adjacencies
for (let i = 0; i < rhombs.length; i++) {
let p1 = rhombs[i].vertices;
for (let j = i + 1; j < rhombs.length; j++) {
let p2 = rhombs[j].vertices;
if (adjacent(p1, p2)) {
rhombs[i].neighbors.push(j);
rhombs[j].neighbors.push(i);
}
}
}
return rhombs;
}
function dist1(point1, point2) {
let [a1, b1] = point1;
let [a2, b2] = point2;
return Math.abs(a1 - a2) + Math.abs(b1 - b2);
}
function norm1(p) {
return Math.abs(p[0]) + Math.abs(p[1]);
}
function leftOf([x0, y0], [x1, y1], [x, y]) {
return x1 * (y - y0) + x * (y0 - y1) + x0 * (y1 - y) > 0;
}
function adjacent(p1, p2) {
let tol = 0.01;
for (let i = 0; i < p1.length; i++) {
for (let j = p2.length; j > 0; j--) {
if (
dist1(p1[i], p2[j % p2.length]) +
dist1(p1[(i + 1) % p1.length], p2[j - 1]) <
tol
) {
return true;
}
}
}
return false;
}
svg_width = 0.8 * width
svg_height = (2 * svg_width) / phi ** 2
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