Merge branch 'feature-120-cell-index'

docs/notes_120_cell.md

@@ -5,13 +5,6 @@ Steps forward -
 
 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.
@@ -30,7 +23,31 @@ the last five.
 
 
 
+2. have tried manual labelling and it will drive me crazy before I finish
 
+3. Automated approach based on what I've got so far:
+
+- should be possible to colour a single dodecahedron from a single face and
+  one other vertex (to pick a chirality)
+
+- write a function to do this - the compound-of-four-tetrahedra map can be
+  more or less hard coded: follow the pattern 1-2-3-4-5, 3-4-5-1-2, etc out
+  from the inner ring, and map the original face's permutation
+
+From this:
+
+- colour the first dodecahedron, picking a chirality
+
+- the next vertices from each of this dodecahedron's vertices have colours
+  which come from the first dodeca
+
+- these can be used to colour the next layer of dodecahedra 
+
+- and so on 
+
+Alternatively:
+
+- do it by the discrete Hopf fibration, one fibre at a time
 
 
 

polytopes.js

@@ -303,7 +303,7 @@ function label_faces_120cell(nodes, faces, cfaces, label) {
 }
 
 
-function manual_label_120cell(nodes, links) {
+function basic_auto_label_120cell(nodes, links) {
 
 	const faces = auto_120cell_faces(links);
 	const dodecas = DODECAHEDRA.DODECAHEDRA;
@@ -332,11 +332,118 @@ function manual_label_120cell(nodes, links) {
 
 }
 
+// manual compound-of-tetrahedra colouring 
+
+function manual_label_120cell(nodes, links) {
+	label_nodes(nodes, [1, 153, 29, 105], 1);
+	label_nodes(nodes, [317, 409, 265, 109], 2);
+	label_nodes(nodes, [221, 337, 25, 509], 3);
+	label_nodes(nodes, [217, 413, 457, 361], 4);
+	label_nodes(nodes, [313, 157, 461, 505], 5);
+
+
+	// second dodecahedron needs to have opposite chirality
+
+	label_nodes(nodes, [ 165, 33, 117 ], 1);   
+	label_nodes(nodes, [ 161, 465, 517 ], 2);
+	label_nodes(nodes, [ 417, 469, 365 ], 3);
+	label_nodes(nodes, [ 341, 37, 513 ], 4);
+	label_nodes(nodes, [ 421, 269, 113 ], 5);
+
+	// third
+
+	label_nodes(nodes, [ 45, 101, 181 ], 1);   
+	label_nodes(nodes, [ 241, 429, 53 ], 2);
+	label_nodes(nodes, [ 93, 229 ], 3);
+	label_nodes(nodes, [ 173, 437 ], 4);
+	label_nodes(nodes, [ 245, 325 ], 5);
+
+	// fourth (id = 3)
+
+	label_nodes(nodes, [ 89, 169, 49 ], 1);   
+	label_nodes(nodes, [ 321 ], 2);
+	label_nodes(nodes, [ 425, 177 ], 3);
+	label_nodes(nodes, [ 97,  225 ], 4);
+	label_nodes(nodes, [ 41,  433], 5);
 
 
 
+}
 
+function meridian_label_120cell(nodes, links) {
 
+			const DODECAS = [
+			  [
+			    313,   1, 317, 221, 217, 417,
+			    341, 421, 165, 161, 465, 469,
+			     37, 269,  33, 113, 117, 517,
+			    365, 513
+			  ],
+			  [
+			    513, 365, 517, 117, 113, 577,
+			     15, 581, 565, 561, 397, 399,
+			     85, 389,  81, 301, 303, 213,
+			    293, 209
+			  ],
+			  [
+			    301, 209, 293, 213, 303,
+			    211, 309, 294, 311, 215,
+			    310, 210, 214, 312, 295,
+			    212, 302, 304, 216, 296
+			  ],
+			  [
+			    304, 302, 212, 296, 216, 400,
+			    398,  84, 392,  88,  16, 580,
+			    564, 568, 584, 368, 516, 116,
+			    120, 520
+			  ],
+			  [
+			    368, 516, 116, 120, 520, 272,
+			     36, 468, 472,  40, 164, 420,
+			    344, 424, 168, 220, 316,   4,
+			    320, 224
+			  ],
+			  [
+			    316,   4, 320, 224, 220, 412,
+			    340, 416, 160, 156, 460, 464,
+			     32, 268,  28, 108, 112, 512,
+			    364, 508
+			  ],
+			  [
+			    508, 364, 512, 112, 108, 572,
+			     14, 576, 560, 556, 394, 396,
+			     80, 388,  76, 298, 300, 208,
+			    292, 204
+			  ],
+			  [
+			    300, 298, 204, 292, 208,
+			    206, 202, 306, 291, 308,
+			    290, 305, 203, 207, 307,
+			    289, 201, 297, 299, 205
+			  ],
+			  [
+			    299, 297, 201, 289, 205, 395,
+			    393,  73, 385,  77,  13, 569,
+			    553, 557, 573, 361, 505, 105,
+			    109, 509
+			  ],
+			  [
+			    361, 505, 105, 109, 509, 265,
+			     25, 457, 461,  29, 153, 409,
+			    337, 413, 157, 217, 313,   1,
+			    317, 221
+			  ]
+			]
+
+		let col = 1;
+		for( const dd of DODECAS ) {
+			label_nodes(nodes, dd, 5);
+			col++;
+			if( col > 5 ) {
+				col = 1;
+			}
+	}
+}
 
 
 
@@ -344,7 +451,7 @@ export const cell120 = () => {
 	const nodes  = make_120cell_vertices();
 	const links = auto_detect_edges(nodes, 4);
 
-	manual_label_120cell(nodes, links);
+	meridian_label_120cell(nodes, links);
 
 	return {
 		nodes: nodes,

testbed.js

@@ -361,6 +361,148 @@ function make_dodecahedron(faces, f1, f2) {
 }
 
 
+function faces_to_dodecahedron(faces, f1, f2) {
+	const dodecahedron = [ f1, f2 ];
+
+	// take f1 as the 'center', get the other four around it from f2
+	const fs = find_dodeca_mutuals(faces, f1, f2);
+	const f3 = fs[0];
+	const f6 = fs[1];
+	dodecahedron.push(f3);
+	const f4 = find_dodeca_next(faces, dodecahedron, f1, f3);
+	dodecahedron.push(f4);
+	const f5 = find_dodeca_next(faces, dodecahedron, f1, f4);
+	dodecahedron.push(f5);
+	dodecahedron.push(f6);
+
+	// get the next ring
+
+	const f7 = find_dodeca_next(faces, dodecahedron, f6, f2);
+	dodecahedron.push(f7);
+	const f8 = find_dodeca_next(faces, dodecahedron, f2, f3);
+	dodecahedron.push(f8);
+	const f9 = find_dodeca_next(faces, dodecahedron, f3, f4);
+	dodecahedron.push(f9);
+	const f10 = find_dodeca_next(faces, dodecahedron, f4, f5);
+	dodecahedron.push(f10);
+	const f11 = find_dodeca_next(faces, dodecahedron, f5, f6);
+	dodecahedron.push(f11);
+
+	// get the last
+
+	const f12 = find_dodeca_next(faces, dodecahedron, f7, f8);
+	dodecahedron.push(f12);
+
+	return dodecahedron;
+}
+
+// from a face and one neighbouring node, return a dodecahedron
+
+function face_plus_to_dodecahedron(faces, f1, node) {
+	const neighbours = find_adjacent_faces(faces, f1);
+	const nodens = neighbours.filter((f) => f.nodes.includes(node));
+	return faces_to_dodecahedron(faces, f1, nodens[0]); // does it matter which?
+}
+
+
+
+
+// for three faces, return their common vertex (if they have one)
+
+function find_dodeca_vertex(f1, f2, f3) {
+	const v12 = f1.nodes.filter((n) => f2.nodes.includes(n));
+	const v123 = v12.filter((n) => f3.nodes.includes(n));
+	if( v123.length === 1 ) {
+		return v123[0];
+	} else {
+		console.log(`warning: faces ${f1.id} ${f2.id} ${f3.id} don't share 1 vertex`);
+		return false;
+	}
+}
+
+const VERTEX_MAP = [
+	[ 0, 1, 5 ],
+	[ 0, 1, 2 ],
+	[ 0, 2, 3 ],
+	[ 0, 3, 4 ],
+	[ 0, 4, 5 ],
+	[ 1, 5, 6 ],
+	[ 1, 2, 7 ],
+	[ 2, 3, 8 ],
+	[ 3, 4, 9 ],
+	[ 4, 5, 10 ],
+	[ 1, 6, 7 ],
+	[ 2, 7, 8 ],
+	[ 3, 8, 9 ],
+	[ 4, 9, 10 ],
+	[ 5, 6, 10 ],
+	[ 6, 7, 11 ],
+	[ 7, 8, 11 ],
+	[ 8, 9, 11 ],
+	[ 9, 10, 11 ],
+	[ 6, 10, 11 ],
+];
+
+
+
+function dodecahedron_vertices(faces) {
+	const face_sets = VERTEX_MAP.map((vs) => vs.map((v) => faces[v]));
+	return face_sets.map((fs) => find_dodeca_vertex(...fs));
+}
+
+// p is the permutation of the first face
+
+function dodecahedron_colours(vertices, p) {
+	const LEFT_PART = [
+		1, 2, 3, 4, 5,   3, 4, 5, 1, 2,   5, 1, 2, 3, 4,   2, 3, 4, 5, 1,
+		];
+	const RIGHT_PART = [
+		1, 2, 3, 4, 5,   4, 5, 1, 2, 3,   3, 4, 5, 1, 2,   1, 2, 3, 4, 5,
+		];
+	const part = LEFT_PART;
+	const colours = {};
+	for( let i = 0; i < 20; i++ ) {
+		const v = vertices[i];
+		const colour = p[part[i] - 1];
+		colours[v] = colour;
+	}
+	return colours;
+}
+
+
+// p is the permutation of the first face
+
+function colour_one_dodecahedron(faces, face, node, p) {
+	const dd = face_plus_to_dodecahedron(faces, face, node);
+	const vertices = dodecahedron_vertices(dd);
+	return dodecahedron_colours(vertices, p);
+}
+
+
+// go along a meridian
+
+function meridian(faces, startf, startn) {
+	const dds =  [ face_plus_to_dodecahedron(faces, startf, startn) ];
+
+	while( dds.length < 10 ) {
+		const dd = dds[dds.length - 1];
+		const nextf = dd[11]; // opposite to startf
+		const neighbours = find_adjacent_faces(faces, nextf);
+		const nextnbors = neighbours.filter((f) => !dd.includes(f));
+
+		const nextdd = faces_to_dodecahedron(faces, nextf, nextnbors[0]);
+
+
+		dds.push(nextdd);
+	}
+
+	return dds;
+
+}
+
+
+
+
 // for a face, pick an edge, and then find the other two faces which
 // share this edge. These can be used as the starting points for the
 // first face's two dodecahedra 
@@ -392,13 +534,20 @@ function dd_fingerprint(dodecahedron) {
 function make_120cell_cells(faces) {
 	const dodecas = [];
 	const seen = {};
+	let i = 1;
 	for( const face of faces ) {
 		const dds = face_to_dodecahedra(faces, face);
 		for( const dd of dds ) {
 			const fp = dd_fingerprint(dd);
 			if( ! (fp in seen) ) {
 				//console.log(`added dodeca ${fp}`);
-				dodecas.push(dd);
+				const d = {
+					id: i,
+					faces: dd,
+					nodes: dodecahedron_vertices(dd),
+				}
+				dodecas.push(d);
+				i += 1;
 				seen[fp] = 1;
 			}
 		}
@@ -422,19 +571,20 @@ const cell120 = () => {
 }
 
 
+function dodeca_nodes(dd) {
+	const ns = new Set();
+	for( const face of dd.faces ) {
+		for( const node of face.nodes ) {
+			ns.add(node);
+		}
+	}
+	dd.nodes = Array.from(ns);
+}
 
 
 
 const nodes = make_120cell_vertices();
 const links = auto_detect_edges(nodes, 4);
 const faces = auto_120cell_faces(links);
-const dodecas = make_120cell_cells(faces);
+//const dodecas = make_120cell_cells(faces);
-
-const ddfaces = dodecas.map((dd) => dd.map((f) => f.id));
-
-console.log(JSON.stringify(ddfaces));
-
-// for( const dodeca of dodecas ) {
-// 	console.log(dodeca.map((f) => f.id)
-// }