Successfully auto-coloured a single dodecahedron
Mike Lynch
parent
2808b256a2
commit
8a926f0552
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@ -5,13 +5,6 @@ Steps forward -
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2. using this, generate a list of all 120 dodecahedra:
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[ a b c d e f g h i j k l m n o p q r s t ] <- 20 vertices
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Check that each vertex appears in four of these
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Then - either manually start labelling them, or build an interface to help
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with the manual labelling
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1.
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@ -30,7 +23,31 @@ the last five.
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2. have tried manual labelling and it will drive me crazy before I finish
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3. Automated approach based on what I've got so far:
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- should be possible to colour a single dodecahedron from a single face and
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one other vertex (to pick a chirality)
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- write a function to do this - the compound-of-four-tetrahedra map can be
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more or less hard coded: follow the pattern 1-2-3-4-5, 3-4-5-1-2, etc out
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from the inner ring, and map the original face's permutation
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From this:
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- colour the first dodecahedron, picking a chirality
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- the next vertices from each of this dodecahedron's vertices have colours
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which come from the first dodeca
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- these can be used to colour the next layer of dodecahedra
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- and so on
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Alternatively:
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- do it by the discrete Hopf fibration, one fibre at a time
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33
polytopes.js
33
polytopes.js
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@ -341,6 +341,7 @@ function manual_label_120cell(nodes, links) {
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label_nodes(nodes, [217, 413, 457, 361], 4);
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label_nodes(nodes, [313, 157, 461, 505], 5);
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// second dodecahedron needs to have opposite chirality
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label_nodes(nodes, [ 165, 33, 117 ], 1);
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@ -349,10 +350,38 @@ function manual_label_120cell(nodes, links) {
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label_nodes(nodes, [ 341, 37, 513 ], 4);
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label_nodes(nodes, [ 421, 269, 113 ], 5);
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// third
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label_nodes(nodes, [ 45, 101, 181 ], 1);
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label_nodes(nodes, [ 241, 429, 53 ], 2);
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label_nodes(nodes, [ 93, 229 ], 3);
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label_nodes(nodes, [ 173, 437 ], 4);
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label_nodes(nodes, [ 245, 325 ], 5);
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// fourth (id = 3)
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label_nodes(nodes, [ 89, 169, 49 ], 1);
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label_nodes(nodes, [ 321 ], 2);
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label_nodes(nodes, [ 425, 177 ], 3);
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label_nodes(nodes, [ 97, 225 ], 4);
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label_nodes(nodes, [ 41, 433], 5);
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}
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function semiautomatic_label_120cell(nodes, links) {
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const COLOURS = {
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1: [ 313, 157, 461, 505 ],
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2: [ 1, 153, 29, 105 ],
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3: [ 317, 409, 265, 109 ],
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4: [ 221, 337, 25, 509 ],
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5: [ 217, 413, 457, 361 ]
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};
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for( const col in COLOURS ) {
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label_nodes(nodes, COLOURS[col], col);
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}
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}
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@ -362,7 +391,7 @@ export const cell120 = () => {
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const nodes = make_120cell_vertices();
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const links = auto_detect_edges(nodes, 4);
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manual_label_120cell(nodes, links);
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semiautomatic_label_120cell(nodes, links);
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return {
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nodes: nodes,
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132
testbed.js
132
testbed.js
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@ -360,6 +360,107 @@ function make_dodecahedron(faces, f1, f2) {
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return dodecahedron;
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}
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// goal - a version of the above which collates the nodes to
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// a standard 'layout' on the dodecahedron, so that it's then easy
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// to colour them automatically
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function faces_to_dodecahedron(faces, f1, f2) {
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const dodecahedron = [ f1, f2 ];
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// take f1 as the 'center', get the other four around it from f2
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const fs = find_dodeca_mutuals(faces, f1, f2);
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const f3 = fs[0];
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const f6 = fs[1];
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dodecahedron.push(f3);
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const f4 = find_dodeca_next(faces, dodecahedron, f1, f3);
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dodecahedron.push(f4);
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const f5 = find_dodeca_next(faces, dodecahedron, f1, f4);
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dodecahedron.push(f5);
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dodecahedron.push(f6);
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// get the next ring
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const f7 = find_dodeca_next(faces, dodecahedron, f6, f2);
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dodecahedron.push(f7);
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const f8 = find_dodeca_next(faces, dodecahedron, f2, f3);
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dodecahedron.push(f8);
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const f9 = find_dodeca_next(faces, dodecahedron, f3, f4);
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dodecahedron.push(f9);
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const f10 = find_dodeca_next(faces, dodecahedron, f4, f5);
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dodecahedron.push(f10);
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const f11 = find_dodeca_next(faces, dodecahedron, f5, f6);
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dodecahedron.push(f11);
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// get the last
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const f12 = find_dodeca_next(faces, dodecahedron, f7, f8);
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dodecahedron.push(f12);
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return dodecahedron;
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}
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// for three faces, return their common vertex (if they have one)
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function find_dodeca_vertex(f1, f2, f3) {
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const v12 = f1.nodes.filter((n) => f2.nodes.includes(n));
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const v123 = v12.filter((n) => f3.nodes.includes(n));
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if( v123.length === 1 ) {
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return v123[0];
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} else {
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console.log(`warning: faces ${f1.id} ${f2.id} ${f3.id} don't share 1 vertex`);
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return false;
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}
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}
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function dodecahedron_vertices(dodeca) {
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const VERTEX_MAP = [
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[ 0, 1, 5 ],
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[ 0, 1, 2 ],
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[ 0, 2, 3 ],
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[ 0, 3, 4 ],
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[ 0, 4, 5 ],
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[ 1, 5, 6 ],
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[ 1, 2, 7 ],
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[ 2, 3, 8 ],
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[ 3, 4, 9 ],
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[ 4, 5, 10 ],
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[ 1, 6, 7 ],
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[ 2, 7, 8 ],
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[ 3, 8, 9 ],
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[ 4, 9, 10 ],
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[ 5, 6, 10 ],
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[ 6, 7, 11 ],
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[ 7, 8, 11 ],
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[ 8, 9, 11 ],
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[ 9, 10, 11 ],
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[ 6, 10, 11 ],
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];
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return VERTEX_MAP.map((vs) => find_dodeca_vertex(...vs.map((v) => dd[v])));
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}
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function dodecahedron_colours(vertices, left) {
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const PARTITION = [
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1, 2, 3, 4, 5, 3, 4, 5, 1, 2, 5, 1, 2, 3, 4, 2, 3, 4, 5, 1,
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];
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const colours = { 1: [], 2: [], 3: [], 4: [], 5: [] };
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for( let i = 0; i < 20; i++ ) {
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colours[PARTITION[i]].push(vertices[i]);
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}
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return colours;
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}
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// for a face, pick an edge, and then find the other two faces which
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// share this edge. These can be used as the starting points for the
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@ -392,13 +493,20 @@ function dd_fingerprint(dodecahedron) {
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function make_120cell_cells(faces) {
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const dodecas = [];
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const seen = {};
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let i = 1;
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for( const face of faces ) {
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const dds = face_to_dodecahedra(faces, face);
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for( const dd of dds ) {
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const fp = dd_fingerprint(dd);
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if( ! (fp in seen) ) {
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//console.log(`added dodeca ${fp}`);
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dodecas.push(dd);
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const d = {
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id: i,
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faces: dd
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}
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dodeca_nodes(d);
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dodecas.push(d);
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i += 1;
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seen[fp] = 1;
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}
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}
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@ -422,22 +530,20 @@ const cell120 = () => {
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}
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function dodeca_nodes(dd) {
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const ns = new Set();
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for( const face of dd.faces ) {
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for( const node of face.nodes ) {
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ns.add(node);
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}
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}
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dd.nodes = Array.from(ns);
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}
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const nodes = make_120cell_vertices();
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const links = auto_detect_edges(nodes, 4);
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const faces = auto_120cell_faces(links);
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const dodecas = make_120cell_cells(faces);
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const ddfaces = dodecas.map((dd) => dd.map((f) => f.id));
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//console.log(JSON.stringify(ddfaces));
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const dd0 = dodecas[0];
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// for( const dodeca of dodecas ) {
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// console.log(dodeca.map((f) => f.id)
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// }
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//const dodecas = make_120cell_cells(faces);
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