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Stuck an abstraction between permutations and real coordinates for 

the 600-cell so that I can label them
experiments-120-cell
Mike Lynch 2023-08-04 11:10:57 +10:00
parent bee04c7e65
commit 0fdf8a40e4
2 changed files with 129 additions and 113 deletions
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43
permute_testbed.js
View File

@ -324,22 +324,10 @@ function find_peers(nodesid, seen, n, d) {
} }
function peer(n1, n2, d) {
const d2 = dist2(n1, n2);
const EPSILON = 0.01;
function partition_from_node(nodesid, start, d) { return Math.abs(d2 - d ** 2) < EPSILON;
const group = [];
const looping = true;
let ns = find_peers(nodesid, group, start, d);
console.log(`0 ${ns}`);
let i = 1;
while( ns.length > 0 && i < ns.length ) {
group.push(...ns);
ns = find_peers(nodesid, group, ns[i], d);
console.log(`${i} ${ns}`);
i++;
}
return group;
} }
@ -351,15 +339,27 @@ function partition_nodes_by_distance(nodes, d) {
for( const node of nodes ) { for( const node of nodes ) {
nodesid[node.id] = node; nodesid[node.id] = node;
} }
let tick = 0;
while( Object.keys(nodesid).length > 0 ) { while( Object.keys(nodesid).length > 0 ) {
const start = Object.keys(nodesid)[0]; const start = Object.keys(nodesid)[0];
console.log(`Start node = ${start}`); console.log(`Start node = ${start}`);
const group = partition_from_node(nodesid, start, d); const group = [ start ];
if( group.length < 1 ) { for( const n2 of Object.keys(nodesid) ) {
console.log("something went wrong, empty neighbour group"); console.log(`group [ ${group} ] / n2 ${n2}`);
break; const n3 = [];
if( !group.includes(n2) ) {
console.log(`group: ${group}`);
for( const g of group ) {
if( peer(nodesid[g], nodesid[n2], d) ) {
console.log(`Match`);
//n3.push(n2);
} }
}
}
console.log(`n3 = ${n3}`);
group.push(...n3);
}
process.exit();
for( const g of group ) { for( const g of group ) {
delete nodesid[g]; delete nodesid[g];
} }
@ -387,5 +387,6 @@ const cell600 = () => {
} }
const nodes = make_600cell_vertices();

197
polytopes.js
View File

@ -180,111 +180,126 @@ export const cell120 = () => {
} }
} }
// Schoute's partition via https://arxiv.org/abs/1010.4353
const partition600 = {
"2,0,0,0": 1,
"0,2,0,0": 1,
"0,0,2,0": 1,
"0,0,0,2": 1,
"1,1,1,1": 1,
"1,1,-1,-1": 1,
"1,-1,1,-1": 1,
"1,-1,-1,1": 1,
"1,-1,-1,-1": 1,
"1,-1,1,1": 1,
"1,1,-1,1": 1,
"1,1,1,-1": 1,
"k,0,-t,-1": 2,
"0,k,1,-t": 2,
"t,-1,k,0": 2,
"1,t,0,k": 2,
"t,k,0,-1": 2,
"1,0,k,t": 2,
"k,-t,-1,0": 2,
"0,1,-t,k": 2,
"1,k,t,0": 2,
"t,0,-1,k": 2,
"0,t,-k,-1": 2,
"k,-1,0,-t": 2,
"t,0,1,k": 3,
"0,t,-k,1": 3,
"1,-k,-t,0": 3,
"k,1,0,-t": 3,
"0,k,1,t": 3,
"t,1,-k,0": 3,
"k,0,t,-1": 3,
"1,-t,0,k": 3,
"t,-k,0,-1": 3,
"0,1,-t,-k": 3,
"1,0,-k,t": 3,
"k,t,1,0": 3,
"t,0,-1,k": 4,
"0,t,k,-1": 4,
"1,-k,t,0": 4,
"k,1,0,t": 4,
"t,1,k,0": 4,
"0,k,-1,-t": 4,
"1,-t,0,-k": 4,
"k,0,-t,1": 4,
"0,1,t,k": 4,
"t,-k,0,1": 4,
"k,t,-1,0": 4,
"1,0,k,-t": 4,
"k,0,t,1": 5,
"0,k,-1,t": 5,
"t,-1,-k,0": 5,
"1,t,0,-k": 5,
"1,0,-k,-t": 5,
"t,k,0,1": 5,
"0,1,t,-k": 5,
"k,-t,1,0": 5,
"t,0,1,-k": 5,
"1,k,-t,0": 5,
"k,-1,0,t": 5,
"0,t,k,1": 5
};
// permute unique indices ->
// use these to label nodes ->
// assign actual values to indices ->
function map_coord(i, coords, values) {
if( i >= 0 ) {
return values[coords[i]];
}
return -values[coords[-i]];
}
function make_600cell_vertices() { function make_600cell_vertices() {
const phi = 0.5 * (1 + Math.sqrt(5)); const coords = {
0: '0',
1: '1',
2: '2',
3: 't',
4: 'k'
};
const t = 0.5 * (1 + Math.sqrt(5));
const values = {
'0': 0,
'1': 1,
'2': 2,
't': t,
'k': 1 / t
};
const nodes = [ const nodes = [
PERMUTE.coordinates([0, 0, 0, 2], 0), PERMUTE.coordinates([0, 0, 0, 2], ),
PERMUTE.coordinates([1, 1, 1, 1], 1), PERMUTE.coordinates([1, 1, 1, 1], 0),
PERMUTE.coordinates([phi, 1, 1 / phi, 0], 1, true) PERMUTE.coordinates([3, 1, 4, 0], 1, true)
].flat(); ].flat();
for( const n of nodes ) {
for( const a of [ 'x', 'y', 'z', 'w'] ) {
n[a] = map_coord(n[a], coords, values);
}
}
index_nodes(nodes); index_nodes(nodes);
const groups = partition_nodes_by_distance(nodes, 2); //const groups = partition_nodes_by_distance(nodes, 2);
scale_nodes(nodes, 0.75); scale_nodes(nodes, 0.75);
return nodes; return nodes;
} }
// Trying to do this with an algorithm, not just a lookup table.
// I think that all of the vertices which belong to the same 24-cell group
// are 2 units apart.
function partition_nodes_by_distance_bad(nodes, d) {
const groups = [];
const EPSILON = 0.002;
for( const n1 of nodes ) {
let matched = false;
for( const group of groups ) {
for( const n2 of group ) {
const d2 = dist2(n1, n2);
// console.log(`comparing ${n1.id} ${n2.id}`);
// console.log(n1);
// console.log(n2);
// console.log(`${d2} ${Math.abs(d2 - d**2)}`);
if( Math.abs(d2 - d ** 2) < EPSILON ) {
group.push(n1);
matched = true;
// console.log(`Matched ${n1.id} ${n2.id} ${d2}`);
break;
}
}
if( matched ) {
break;
}
}
if(! matched ) {
// console.log(`unmatched node ${n1.id}`);
groups.push([ n1 ]);
}
}
console.log(`Build ${groups.length} groups`);
console.log(groups);
return groups;
}
// find all nodes in nodesid which are d away from n (and are not n)
function nodes_by_distance(nodesid, n, d) {
const EPSILON = 0.02;
const neighbours = Object.keys(nodesid).filter((n1id) => {
if( n1id !== n.id ) {
const d2 = dist2(nodesid[n1id], nodesid[n]);
console.log(`${n} ${n1id} ${d2}`);
return Math.abs(d2 - d ** 2) < EPSILON;
} else {
return false;
}
});
console.log(`neighbours at ${d} ${neighbours}`);
return neighbours;
}
function partition_nodes_by_distance(nodes, d) {
const groups = [];
const nodesid = {};
const EPSILON = 0.02;
for( const node of nodes ) {
nodesid[node.id] = node;
}
while( Object.keys(nodesid).length > 0 ) {
const start = Object.keys(nodesid)[0];
const group = [ start ];
const neighbours = nodes_by_distance(nodesid, n, d).filter((n2) => !(n2 in group));
if( neighbours ) {
group.push(...neighbours);
}
const group = partition_r(nodesid, [ start ], start, d);
console.log(group);
for( const g of group ) {
delete nodesid[g];
}
groups.push(group);
}
}