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.
// 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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