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Getting towards an auto-dodecahedon detector for the 120-cell

feature-120-cell-index
Mike Lynch 2023-08-19 18:00:19 +10:00
parent aa5501e14a
commit 5a09caef93
3 changed files with 116 additions and 509 deletions
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39
docs/notes_120_cell.md
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@ -1,4 +1,43 @@
Steps forward -
1. algorithm which, given a face, finds the two dodecahedra it belongs to
2. using this, generate a list of all 120 dodecahedra:
[ a b c d e f g h i j k l m n o p q r s t ] <- 20 vertices
Check that each vertex appears in four of these
Then - either manually start labelling them, or build an interface to help
with the manual labelling
1.
For a face: there are five edges, and ten other faces sharing an edge.
These edges are in two sets: one for each dodecahedron. The sets are defined
by them sharing vertices which aren't in the first face.
Go around a set of five, by pairs: for each pair, find the other neighbour -
this gives the next five faces.
There's only one face left, which is defined by the shared other vertices of
the last five.
/// old shit below that didn't work VVVV
Chords: 1.74806 - the 120-cell has 7200 chords of this length Chords: 1.74806 - the 120-cell has 7200 chords of this length

33
indexing_attempts/complicated_vectors.js
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@ -259,3 +259,36 @@ function nice_icosa(nodes, icosa) {
} }
function find_by_chord(nodesid, n, d) {
const EPSILON = 0.02;
return Object.keys(nodesid).filter((n1) => {
const d2 = dist2(nodesid[n1], nodesid[n]);
return Math.abs(d2 - d ** 2) < EPSILON;
});
}
function has_chord(n1, n2, d) {
const d2 = dist2(n1, n2);
const EPSILON = 0.01;
return Math.abs(d2 - d ** 2) < EPSILON;
}
function find_all_chords(nodes) {
const chords = {};
for( let i = 0; i < nodes.length - 1; i++ ) {
for( let j = i + 1; j < nodes.length; j++ ) {
const n1 = nodes[i];
const n2 = nodes[j];
const chord = Math.sqrt(dist2(n1, n2)).toFixed(5);
if( !(chord in chords) ) {
chords[chord] = [];
}
chords[chord].push([n1, n2]);
}
}
return chords;
}

553
testbed.js
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@ -1,8 +1,5 @@
// Utilities for generating sets of coordinates based on //testbed for playing with stuff in node repl
// permutations, even permutations and changes of sign.
// Based on https://www.qfbox.info/epermute
const THREE =require('three'); const THREE =require('three');
@ -169,93 +166,6 @@ function auto_detect_edges(nodes, neighbours, debug=false) {
return links; return links;
} }
// too small and simple to calculate
const cell5 = () => {
const r5 = Math.sqrt(5);
const r2 = Math.sqrt(2) / 2;
return {
nodes: [
{id:1, x: r2, y: r2, z: r2, w: -r2 / r5 },
{id:2, x: r2, y: -r2, z: -r2, w: -r2 / r5 },
{id:3, x: -r2, y: r2, z: -r2, w: -r2 / r5 },
{id:4, x: -r2, y: -r2, z: r2, w: -r2 / r5 },
{id:5, x: 0, y: 0, z: 0, w: 4 * r2 / r5 },
],
links: [
{ id:1, source:1, target: 2},
{ id:2, source:1, target: 3},
{ id:3, source:1, target: 4},
{ id:4, source:1, target: 5},
{ id:5, source:2, target: 3},
{ id:6, source:2, target: 4},
{ id:7, source:2, target: 5},
{ id:8, source:3, target: 4},
{ id:9, source:3, target: 5},
{ id:10, source:4, target: 5},
],
geometry: {
node_size: 0.02,
link_size: 0.02
}
};
};
const cell16 = () => {
let nodes = coordinates([1, 1, 1, 1], 0);
nodes = nodes.filter((n) => n.x * n.y * n.z * n.w > 0);
index_nodes(nodes);
scale_nodes(nodes, 0.75);
const links = auto_detect_edges(nodes, 6);
return {
nodes: nodes,
links: links,
geometry: {
node_size: 0.02,
link_size: 0.02
}
};
};
const tesseract = () => {
const nodes = coordinates([1, 1, 1, 1], 0);
index_nodes(nodes);
scale_nodes(nodes, Math.sqrt(2) / 2);
const links = auto_detect_edges(nodes, 4);
return {
nodes: nodes,
links: links,
geometry: {
node_size: 0.02,
link_size: 0.02
}
};
}
const cell24 = () => {
const nodes = coordinates([0, 0, 1, 1], 0);
index_nodes(nodes);
const links = auto_detect_edges(nodes, 6);
return {
nodes: nodes,
links: links,
geometry: {
node_size: 0.02,
link_size: 0.02
}
};
}
function make_120cell_vertices() { function make_120cell_vertices() {
@ -359,6 +269,49 @@ function auto_120cell_faces(links) {
} }
// trying to go from faces to dodecahedra
function shared_vertices(f1, f2) {
return f1.nodes.filter((f) => f2.nodes.includes(f));
}
function adjacent_faces(f1, f2) {
// adjacent faces which share an edge, not just a vertex
const intersect = shared_vertices(f1, f2);
if( intersect.length < 2 ) {
return false;
}
if( intersect.length > 2 ) {
console.log(`warning: faces ${f1.id} and ${f2.id} have too many common vertices`);
}
return true;
}
function find_adjacent_faces(faces, face) {
const neighbours = faces.filter((f) => f.id !== face.id && adjacent_faces(f, face));
return neighbours;
}
function find_dodeca_mutuals(faces, f1, f2) {
// for any two adjacent faces, find their common neighbours where
// all three share exactly one vertex (this, I think, guarantees that
// all are on the same dodecahedron)
const n1 = find_adjacent_faces(faces, f1);
const n2 = find_adjacent_faces(faces, f2);
const common = n1.filter((f1) => n2.filter((f2) => f1.id === f2.id).length > 0 );
// there's one extra here - the third which has two nodes in common with
// both f1 and f2 - filter it out
const mutuals = common.filter((cf) => {
const shared = cf.nodes.filter((n) => f1.nodes.includes(n) && f2.nodes.includes(n));
return shared.length === 1
});
return mutuals;
}
@ -380,426 +333,8 @@ const cell120 = () => {
function make_600cell_vertices() {
const phi = 0.5 * (1 + Math.sqrt(5));
const nodes = [
coordinates([0, 0, 0, 2], 0),
coordinates([1, 1, 1, 1], 1),
coordinates([phi, 1, 1 / phi, 0], 1, true)
].flat();
index_nodes(nodes);
return nodes;
}
function find_by_chord(nodesid, n, d) {
const EPSILON = 0.02;
return Object.keys(nodesid).filter((n1) => {
const d2 = dist2(nodesid[n1], nodesid[n]);
return Math.abs(d2 - d ** 2) < EPSILON;
});
}
function has_chord(n1, n2, d) {
const d2 = dist2(n1, n2);
const EPSILON = 0.01;
return Math.abs(d2 - d ** 2) < EPSILON;
}
function find_all_chords(nodes) {
const chords = {};
for( let i = 0; i < nodes.length - 1; i++ ) {
for( let j = i + 1; j < nodes.length; j++ ) {
const n1 = nodes[i];
const n2 = nodes[j];
const chord = Math.sqrt(dist2(n1, n2)).toFixed(5);
if( !(chord in chords) ) {
chords[chord] = [];
}
chords[chord].push([n1, n2]);
}
}
return chords;
}
const cell600 = () => {
const nodes = make_600cell_vertices();
const links = auto_detect_edges(nodes, 12);
return {
nodes: nodes,
links: links,
geometry: {
node_size: 0.08,
link_size: 0.02
}
}
}
// bad stuff
function find_chords(chords, n) {
return chords.filter((c) => c[0].id === n.id || c[1].id === n.id);
}
function find_neighbours(chords, n) {
const c = find_chords(chords, n);
return c.map((c) => c[0].id === n.id ? c[1] : c[0])
}
// for a list of pairs [n1, n2] (these are nodes which share a common angle
// from a center), find all the groups of nodes which don't appear in a pair
// together
function partition_nodes(pairs) {
let groups = [];
const seen = new Set();
for( const pair of pairs ) {
// both nodes are in a group already
if( seen.has(pair[0]) && seen.has(pair[1]) ) {
continue;
}
let already = false;
// check if either node is already in a group
for( const group of groups ) {
if( group.has(pair[0]) ) {
group.add(pair[1]);
seen.add(pair[1]);
already = true;
continue;
} else if( group.has(pair[1]) ) {
group.has(pair[0]);
seen.has(pair[0]);
already = true;
continue;
}
}
// if neither of the pair was in a former group, start a new group
if( !already ) {
groups.push(new Set(pair));
}
// collapse any groups which now have common elements
groups = collapse_groups(groups);
}
return groups;
}
// given a list of groups, if any have common elements, collapse them
function collapse_groups(groups) {
const new_groups = [ ];
for( group of groups ) {
let collapsed = false;
for( new_group of new_groups ) {
const i = intersection(group, new_group);
if( i.size > 0 ) {
for( const e of group ) {
new_group.add(e);
}
collapsed = true;
break;
}
}
if( !collapsed ) {
new_groups.push(new Set(group));
}
}
return new_groups;
}
function intersection(s1, s2) {
const i = new Set();
for( const e of s1 ) {
if( s2.has(e) ) {
i.add(e)
}
}
return i;
}
function union(s1, s2) {
const u = new Set(s1);
for( const e of s2 ) {
u.add(e);
}
return u;
}
function vector_angle(n1, n2, n3) {
const v1 = new THREE.Vector4(n1.x, n1.y, n1.z, n1.w);
const v2 = new THREE.Vector4(n2.x, n2.y, n2.z, n2.w);
const v3 = new THREE.Vector4(n3.x, n3.y, n3.z, n3.w);
v2.sub(v1);
v3.sub(v1);
const dp = v2.dot(v3);
return Math.acos(dp / ( v2.length() * v3.length()));
}
function neighbour_angles_orig(chords, n) {
const ns = find_neighbours(chords, n);
const angles = {};
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 in angles) ) {
angles[af] = [];
}
angles[af].push([n2.id, n3.id]);
}
}
return angles;
}
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 nodes = make_120cell_vertices();
const links = auto_detect_edges(nodes, 4); const links = auto_detect_edges(nodes, 4);
const faces = auto_120cell_faces(links); const faces = auto_120cell_faces(links);
console.log('links');
for( const link of links ) {
console.log(link);
}
console.log('faces');
for( const face of faces ) {
console.log(face);
}