791 lines
17 KiB
JavaScript
791 lines
17 KiB
JavaScript
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// Utilities for generating sets of coordinates based on
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// permutations, even permutations and changes of sign.
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// Based on https://www.qfbox.info/epermute
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const THREE =require('three');
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function pandita(a) {
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const n = a.length;
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for( let k = n - 2; k >= 0; k-- ) {
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if( a[k] < a[k + 1] ) {
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for( let l = n - 1; l >= 0; l-- ) {
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if( a[k] < a[l] ) {
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const tmp = a[k];
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a[k] = a[l];
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a[l] = tmp;
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const revtail = a.slice(k + 1);
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revtail.reverse();
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for( let i = 0; i < revtail.length; i++ ) {
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a[k + 1 + i] = revtail[i];
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}
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return Math.floor(revtail.length / 2) + 1;
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}
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}
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console.log("Shouldn't get here");
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process.exit();
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}
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}
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return false;
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}
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function permutations_old(a) {
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a.sort();
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const ps = [ [...a] ];
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let running = true;
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while( running ) {
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const s = pandita(a);
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if( s ) {
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ps.push([...a]);
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} else {
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running = false;
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}
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}
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return ps;
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}
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function permutations(a) {
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a.sort();
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const ps = [ [...a] ];
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let running = true;
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while( pandita(a) > 0 ) {
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ps.push([...a]);
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}
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return ps;
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}
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function permutations_even(a) {
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a.sort();
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let parity = 'even';
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const ps = [ [...a] ];
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let running = true;
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while( running ) {
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const s = pandita(a);
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if( s ) {
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if( parity === 'even' ) {
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if( s % 2 === 1 ) {
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parity = 'odd';
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}
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} else {
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if( s % 2 === 1 ) {
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parity = 'even';
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}
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}
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if( parity === 'even' ) {
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ps.push([...a]);
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}
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} else {
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running = false;
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}
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}
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return ps;
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}
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// for a given permutation, say [ 1, 1, 0, 0 ], return all
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// of the valid changes of sign, so:
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// [ [1, 1, 0, 0 ], [ -1, 1, 0, 0 ], [ 1, -1, 0, 0 ], [-1, -1, 0, 0 ]]
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// ie don't do it on the zeros
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function expand_sign(a, label) {
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const expanded = [];
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const exv = a.map((v) => v ? [ -v, v ] : [ 0 ]);
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for( const xv of exv[0] ) {
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for( const yv of exv[1] ) {
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for( const zv of exv[2] ) {
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for( const wv of exv[3] ) {
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expanded.push({label: label, x: xv, y:yv, z:zv, w:wv});
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}
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}
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}
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}
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return expanded;
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}
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function coordinates(a, id0=1, even=false) {
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const ps = even ? permutations_even(a) : permutations(a);
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const coords = [];
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for( const p of ps ) {
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const expanded = expand_sign(p, 0);
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coords.push(...expanded);
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}
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return coords;
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}
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function index_nodes(nodes, scale) {
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let i = 1;
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for( const n of nodes ) {
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n["id"] = i;
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i++;
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}
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}
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function scale_nodes(nodes, scale) {
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for( const n of nodes ) {
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for( const a of [ 'x', 'y', 'z', 'w' ] ) {
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n[a] = scale * n[a];
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}
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}
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}
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function dist2(n1, n2) {
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return (n1.x - n2.x) ** 2 + (n1.y - n2.y) ** 2 + (n1.z - n2.z) ** 2 + (n1.w - n2.w) ** 2;
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}
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function auto_detect_edges(nodes, neighbours, debug=false) {
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const seen = {};
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const nnodes = nodes.length;
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const links = [];
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let id = 1;
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for( const n1 of nodes ) {
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const d2 = [];
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for( const n2 of nodes ) {
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d2.push({ d2: dist2(n1, n2), id: n2.id });
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}
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d2.sort((a, b) => a.d2 - b.d2);
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const closest = d2.slice(1, neighbours + 1);
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if( debug ) {
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console.log(`closest = ${closest.length}`);
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console.log(closest);
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}
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for( const e of closest ) {
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const ids = [ n1.id, e.id ];
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ids.sort();
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const fp = ids.join(',');
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if( !seen[fp] ) {
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seen[fp] = true;
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links.push({ id: id, label: 0, source: n1.id, target: e.id });
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id++;
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}
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}
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}
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if( debug ) {
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console.log(`Found ${links.length} edges`)
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}
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return links;
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}
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// too small and simple to calculate
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const cell5 = () => {
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const r5 = Math.sqrt(5);
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const r2 = Math.sqrt(2) / 2;
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return {
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nodes: [
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{id:1, x: r2, y: r2, z: r2, w: -r2 / r5 },
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{id:2, x: r2, y: -r2, z: -r2, w: -r2 / r5 },
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{id:3, x: -r2, y: r2, z: -r2, w: -r2 / r5 },
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{id:4, x: -r2, y: -r2, z: r2, w: -r2 / r5 },
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{id:5, x: 0, y: 0, z: 0, w: 4 * r2 / r5 },
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],
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links: [
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{ id:1, source:1, target: 2},
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{ id:2, source:1, target: 3},
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{ id:3, source:1, target: 4},
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{ id:4, source:1, target: 5},
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{ id:5, source:2, target: 3},
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{ id:6, source:2, target: 4},
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{ id:7, source:2, target: 5},
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{ id:8, source:3, target: 4},
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{ id:9, source:3, target: 5},
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{ id:10, source:4, target: 5},
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],
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geometry: {
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node_size: 0.02,
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link_size: 0.02
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}
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};
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};
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const cell16 = () => {
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let nodes = coordinates([1, 1, 1, 1], 0);
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nodes = nodes.filter((n) => n.x * n.y * n.z * n.w > 0);
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index_nodes(nodes);
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scale_nodes(nodes, 0.75);
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const links = auto_detect_edges(nodes, 6);
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return {
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nodes: nodes,
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links: links,
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geometry: {
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node_size: 0.02,
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link_size: 0.02
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}
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};
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};
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const tesseract = () => {
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const nodes = coordinates([1, 1, 1, 1], 0);
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index_nodes(nodes);
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scale_nodes(nodes, Math.sqrt(2) / 2);
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const links = auto_detect_edges(nodes, 4);
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return {
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nodes: nodes,
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links: links,
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geometry: {
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node_size: 0.02,
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link_size: 0.02
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}
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};
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}
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const cell24 = () => {
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const nodes = coordinates([0, 0, 1, 1], 0);
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index_nodes(nodes);
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const links = auto_detect_edges(nodes, 6);
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return {
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nodes: nodes,
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links: links,
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geometry: {
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node_size: 0.02,
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link_size: 0.02
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}
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};
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}
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function make_120cell_vertices() {
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const phi = 0.5 * (1 + Math.sqrt(5));
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const r5 = Math.sqrt(5);
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const phi2 = phi * phi;
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const phiinv = 1 / phi;
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const phi2inv = 1 / phi2;
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const nodes = [
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coordinates([0, 0, 2, 2], 0),
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coordinates([1, 1, 1, r5], 0),
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coordinates([phi, phi, phi, phi2inv], 0),
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coordinates([phiinv, phiinv, phiinv, phi2], 0),
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coordinates([phi2, phi2inv, 1, 0], 0, true),
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coordinates([r5, phiinv, phi, 0], 0, true),
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coordinates([2, 1, phi, phiinv], 0, true),
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].flat();
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index_nodes(nodes);
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// scale_nodes(nodes, 0.5);
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return nodes;
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}
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// face detection for the 120-cell
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// NOTE: all of these return node ids, not nodes
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// return all the links which connect to a node
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function nodes_links(links, nodeid) {
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return links.filter((l) => l.source === nodeid || l.target === nodeid);
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}
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// filter to remove a link to a given id from a set of links
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function not_to_this(link, nodeid) {
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return !(link.source === nodeid || link.target === nodeid);
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}
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// given nodes n1, n2, return all neighbours of n2 which are not n1
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function unmutuals(links, n1id, n2id) {
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const nlinks = nodes_links(links, n2id).filter((l) => not_to_this(l, n1id));
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return nlinks.map((l) => {
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if( l.source === n2id ) {
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return l.target;
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} else {
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return l.source;
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}
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})
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}
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function fingerprint(ids) {
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const sids = [...ids];
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sids.sort();
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return sids.join(',');
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}
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function auto_120cell_faces(links) {
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const faces = [];
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const seen = {};
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let id = 1;
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for( const edge of links ) {
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const v1 = edge.source;
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const v2 = edge.target;
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const n1 = unmutuals(links, v2, v1);
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const n2 = unmutuals(links, v1, v2);
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const shared = [];
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for( const a of n1 ) {
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const an = unmutuals(links, v1, a);
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for( const d of n2 ) {
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const dn = unmutuals(links, v2, d);
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for( const x of an ) {
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for( const y of dn ) {
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if( x == y ) {
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shared.push([v1, a, x, d, v2])
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}
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}
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}
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}
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}
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if( shared.length !== 3 ) {
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console.log(`Bad shared faces for ${edge.id} ${v1} ${v2}`);
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}
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for( const face of shared ) {
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const fp = fingerprint(face);
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if( !seen[fp] ) {
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faces.push({ id: id, nodes: face });
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id++;
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seen[fp] = true;
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}
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}
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}
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return faces;
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}
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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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return {
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nodes: nodes,
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links: links,
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geometry: {
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node_size: 0.02,
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link_size: 0.02
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}
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}
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}
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function make_600cell_vertices() {
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const phi = 0.5 * (1 + Math.sqrt(5));
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const nodes = [
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coordinates([0, 0, 0, 2], 0),
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coordinates([1, 1, 1, 1], 1),
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coordinates([phi, 1, 1 / phi, 0], 1, true)
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].flat();
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index_nodes(nodes);
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return nodes;
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}
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function find_by_chord(nodesid, n, d) {
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const EPSILON = 0.02;
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return Object.keys(nodesid).filter((n1) => {
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const d2 = dist2(nodesid[n1], nodesid[n]);
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return Math.abs(d2 - d ** 2) < EPSILON;
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});
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}
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function has_chord(n1, n2, d) {
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const d2 = dist2(n1, n2);
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const EPSILON = 0.01;
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return Math.abs(d2 - d ** 2) < EPSILON;
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}
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function find_all_chords(nodes) {
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const chords = {};
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for( let i = 0; i < nodes.length - 1; i++ ) {
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for( let j = i + 1; j < nodes.length; j++ ) {
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const n1 = nodes[i];
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const n2 = nodes[j];
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const chord = Math.sqrt(dist2(n1, n2)).toFixed(5);
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if( !(chord in chords) ) {
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chords[chord] = [];
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}
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chords[chord].push([n1, n2]);
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}
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}
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return chords;
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}
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const cell600 = () => {
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const nodes = make_600cell_vertices();
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const links = auto_detect_edges(nodes, 12);
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return {
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nodes: nodes,
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links: links,
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geometry: {
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node_size: 0.08,
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link_size: 0.02
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}
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}
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}
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// bad stuff
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function find_chords(chords, n) {
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return chords.filter((c) => c[0].id === n.id || c[1].id === n.id);
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}
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function find_neighbours(chords, n) {
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const c = find_chords(chords, n);
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return c.map((c) => c[0].id === n.id ? c[1] : c[0])
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}
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// for a list of pairs [n1, n2] (these are nodes which share a common angle
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// from a center), find all the groups of nodes which don't appear in a pair
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// together
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function partition_nodes(pairs) {
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let groups = [];
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const seen = new Set();
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for( const pair of pairs ) {
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// both nodes are in a group already
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if( seen.has(pair[0]) && seen.has(pair[1]) ) {
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continue;
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}
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let already = false;
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// check if either node is already in a group
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for( const group of groups ) {
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if( group.has(pair[0]) ) {
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group.add(pair[1]);
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seen.add(pair[1]);
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already = true;
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continue;
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} else if( group.has(pair[1]) ) {
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group.has(pair[0]);
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seen.has(pair[0]);
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already = true;
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continue;
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}
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}
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// if neither of the pair was in a former group, start a new group
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if( !already ) {
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groups.push(new Set(pair));
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}
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// collapse any groups which now have common elements
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groups = collapse_groups(groups);
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}
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return groups;
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}
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// given a list of groups, if any have common elements, collapse them
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function collapse_groups(groups) {
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const new_groups = [ ];
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for( group of groups ) {
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let collapsed = false;
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for( new_group of new_groups ) {
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const i = intersection(group, new_group);
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if( i.size > 0 ) {
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for( const e of group ) {
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new_group.add(e);
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}
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collapsed = true;
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break;
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}
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}
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if( !collapsed ) {
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new_groups.push(new Set(group));
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}
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}
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return new_groups;
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}
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function intersection(s1, s2) {
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const i = new Set();
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for( const e of s1 ) {
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if( s2.has(e) ) {
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i.add(e)
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}
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}
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return i;
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}
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function union(s1, s2) {
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const u = new Set(s1);
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for( const e of s2 ) {
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u.add(e);
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}
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return u;
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}
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function vector_angle(n1, n2, n3) {
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const v1 = new THREE.Vector4(n1.x, n1.y, n1.z, n1.w);
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const v2 = new THREE.Vector4(n2.x, n2.y, n2.z, n2.w);
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const v3 = new THREE.Vector4(n3.x, n3.y, n3.z, n3.w);
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v2.sub(v1);
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v3.sub(v1);
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const dp = v2.dot(v3);
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return Math.acos(dp / ( v2.length() * v3.length()));
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}
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function neighbour_angles_orig(chords, n) {
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const ns = find_neighbours(chords, n);
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const angles = {};
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for( let i = 0; i < ns.length - 1; i++ ) {
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for( let j = i + 1; j < ns.length; j++ ) {
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const n2 = ns[i];
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const n3 = ns[j];
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const a = THREE.MathUtils.radToDeg(vector_angle(n, n2, n3));
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const af = (a).toFixed(3);
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if( ! (af in angles) ) {
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angles[af] = [];
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}
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angles[af].push([n2.id, n3.id]);
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}
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}
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return angles;
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}
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|
|
function neighbour_angles(chords, n, angle) {
|
|
const ns = find_neighbours(chords, n);
|
|
const pairs = [];
|
|
for( let i = 0; i < ns.length - 1; i++ ) {
|
|
for( let j = i + 1; j < ns.length; j++ ) {
|
|
const n2 = ns[i];
|
|
const n3 = ns[j];
|
|
const a = THREE.MathUtils.radToDeg(vector_angle(n, n2, n3));
|
|
const af = (a).toFixed(3);
|
|
if( af === angle ) {
|
|
pairs.push([n2.id, n3.id]);
|
|
}
|
|
}
|
|
}
|
|
return pairs;
|
|
}
|
|
|
|
|
|
function make_120_partition(nodes, n) {
|
|
const chords = find_all_chords(nodes);
|
|
const chord3 = chords["1.74806"]; // these are edges of the 600-cells;
|
|
const pairs60 = neighbour_angles(chord3, n, "60.000");
|
|
const icosas = partition_nodes(pairs60);
|
|
|
|
n.label = 1;
|
|
const angles = icosa_nodes(nodes, icosas[0]);
|
|
label_120_partition_r(nodes, chord3, 1, n, angles);
|
|
}
|
|
|
|
// recursive function to label a single 600-cell vertex partition of the
|
|
// 120-cell by following icosahedral nets
|
|
// this doesn't work! completely - labels only 108-112
|
|
|
|
function label_120_partition_r(nodes, chords, label, origin, neighbours) {
|
|
console.log(`label_120_partition_r ${origin.id}`);
|
|
console.log(neighbours.map((n) => n.id).join(', '));
|
|
|
|
// first try to label everything
|
|
const unlabelled = [];
|
|
for( const n of neighbours ) {
|
|
if( n.label === 0 ) {
|
|
console.log(`Labelled ${n.id} ${label}`);
|
|
n.label = label;
|
|
unlabelled.push(n);
|
|
} else if( n.label !== label ) {
|
|
console.log(`node ${n.id} is already in group ${n.label}`);
|
|
//return false;
|
|
}
|
|
}
|
|
for( const n of unlabelled ) {
|
|
// the angles represent two icosahedral pyramids - partition them and
|
|
// pick the one which is at 60 to the edge we arrived on
|
|
//console.log(`looking for more neighbors for ${n}`);
|
|
const pairs60 = neighbour_angles(chords, n, "60.000");
|
|
const icosas = partition_nodes(pairs60);
|
|
const icosa = choose_icosa(nodes, origin, n, icosas);
|
|
const icosa_n = icosa_nodes(nodes, icosa);
|
|
console.log(`recursing to ${nice_icosa(nodes,icosa)}`);
|
|
return label_120_partition_r(nodes, chords, label, n, icosa_n);
|
|
}
|
|
}
|
|
|
|
// given a pair of icosa-sets, pick the one which is at the right angle to
|
|
// the incoming vector
|
|
|
|
function choose_icosa(nodes, origin, n1, icosas) {
|
|
for( const icosa of icosas ) {
|
|
const inodes = icosa_nodes(nodes, icosa);
|
|
const a60 = inodes.map((ni) => {
|
|
const a = THREE.MathUtils.radToDeg(vector_angle(n1, origin, ni));
|
|
return a.toFixed(3);
|
|
});
|
|
if( a60.filter((a) => a === "60.000").length > 0 ) {
|
|
return icosa;
|
|
}
|
|
}
|
|
console.log("No icosa found!");
|
|
return undefined;
|
|
}
|
|
|
|
function icosa_nodes(nodes, icosa) {
|
|
return Array.from(icosa).map((nid) => node_by_id(nodes, nid)).sort((a, b) => a.id - b.id);
|
|
}
|
|
|
|
function node_by_id(nodes, nid) {
|
|
const ns = nodes.filter((n) => n.id === nid);
|
|
return ns[0];
|
|
}
|
|
|
|
|
|
function enumerate_icosas(nodes) {
|
|
const chords = find_all_chords(nodes);
|
|
const chord3 = chords["1.74806"]; // these are edges of the 600-cells;
|
|
|
|
for( const n of nodes ) {
|
|
const pairs60 = neighbour_angles(chord3, n, "60.000");
|
|
const icosas = partition_nodes(pairs60);
|
|
for( const icosa of icosas ) {
|
|
const inodes = icosa_nodes(nodes, icosa);
|
|
console.log(icosa_to_csv(n.id, inodes).join(','));
|
|
}
|
|
}
|
|
}
|
|
|
|
|
|
function icosa_to_csv(nid, icosa) {
|
|
const cols = [ nid ];
|
|
const ia = icosa.map((n) => n.id);
|
|
for( let i = 1; i < 601; i++ ) {
|
|
if( ia.includes(i) ) {
|
|
cols.push(i);
|
|
} else {
|
|
cols.push('')
|
|
}
|
|
}
|
|
return cols;
|
|
}
|
|
|
|
|
|
function start_icosas(nodes, chords, origin) {
|
|
const pairs60 = neighbour_angles(chords, origin, "60.000");
|
|
return partition_nodes(pairs60).map((i) => nice_icosa(nodes, i));
|
|
}
|
|
|
|
|
|
|
|
function next_icosa(nodes, chords, origin, nid) {
|
|
const n = node_by_id(nodes, nid);
|
|
const pairs60 = neighbour_angles(chords, n, "60.000");
|
|
const icosas = partition_nodes(pairs60);
|
|
const icosa = choose_icosa(nodes, origin, n, icosas);
|
|
|
|
return nice_icosa(nodes, icosa);
|
|
}
|
|
|
|
function nice_icosa(nodes, icosa) {
|
|
return icosa_nodes(nodes, icosa).map((n) => n.id).join(', ');
|
|
}
|
|
|
|
|
|
// New approach with tetrahedral coloring
|
|
|
|
function find_edges(links, nid) {
|
|
return links.filter((l) => l.source === nid || l.target === nid );
|
|
}
|
|
|
|
|
|
function find_adjacent(links, nid) {
|
|
return find_edges(links, nid).map((l) => {
|
|
if( l.source === nid ) {
|
|
return l.target;
|
|
} else {
|
|
return l.source;
|
|
}
|
|
});
|
|
}
|
|
|
|
function iterate_graph(nodes, links, n, fn) {
|
|
const queue = [];
|
|
const seen = {};
|
|
const nodes_id = {};
|
|
nodes.map((n) => nodes_id[n.id] = n);
|
|
|
|
queue.push(n.id);
|
|
seen[n.id] = true;
|
|
fn(n);
|
|
|
|
while( queue.length > 0 ) {
|
|
const v = queue.shift();
|
|
find_adjacent(links, v).map((aid) => {
|
|
if( !(aid in seen) ) {
|
|
seen[aid] = true;
|
|
queue.push(aid);
|
|
fn(nodes_id[aid]);
|
|
}
|
|
})
|
|
}
|
|
}
|
|
|
|
|
|
|
|
|
|
// stupid tetrahedral labelling
|
|
// keeps getting stuck
|
|
|
|
|
|
function naive_label_120cell(nodes, links, n) {
|
|
const nodes_id = {};
|
|
nodes.map((n) => nodes_id[n.id] = n);
|
|
iterate_graph(nodes, links, nodes[0], (n) => {
|
|
const cols = new Set();
|
|
const nbors = find_adjacent(links, n.id);
|
|
for( const nb of nbors ) {
|
|
if( nodes_id[nb].label > 0 ) {
|
|
cols.add(nodes_id[nb].label);
|
|
}
|
|
for( const nb2 of find_adjacent(links, nb) ) {
|
|
if( nb2 !== n.id && nodes_id[nb].label > 0 ) {
|
|
cols.add(nodes_id[nb2].label);
|
|
}
|
|
}
|
|
}
|
|
const pcols = [ 1, 2, 3, 4, 5 ].filter((c) => !cols.has(c));
|
|
if( pcols.length < 1 ) {
|
|
console.log(`Got stuck, no options at ${n.id}`);
|
|
return false;
|
|
} else {
|
|
n.label = pcols[0];
|
|
console.log(`found ${pcols.length} colors for node ${n.id}`);
|
|
console.log(`applied ${pcols[0]} to node ${n.id}`);
|
|
return true;
|
|
}
|
|
});
|
|
}
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
const nodes = make_120cell_vertices();
|
|
const links = auto_detect_edges(nodes, 4);
|
|
|
|
|