29:null 2a:["$","title",null,{"children":"Indra's Pearls | Nicolas Garcia Belmonte | Observable"}] 2f:T423,computeBFS = (gens, inv, circ, plot, opt) => { let {levmax=5, flag=false} = opt || {}; const group = []; const tag = []; const num = []; let lev = 0; for (let i = 0; i < gens.length; i++) { group[i] = gens[i]; tag[i] = i; plot(circ[i], lev, group[i], tag[i], i); } num[0] = 0; num[1] = gens.length; for (lev = 1; lev < levmax; lev++) { let inew = num[lev]; for (let iold = num[lev -1]; iold < num[lev]; iold++) { for (let j = 0; j < gens.length; j++) { if (j == inv[tag[iold]]) { continue; } group[inew] = math.multiply(group[iold], gens[j]); group[inew].lev = lev; tag[inew] = j; inew++; } } num[lev+1] = inew; } //plot for (let i = 0; i < num[levmax]; i++) { for (let j = 0; j < gens.length; j++) { // if (j == inv[tag[i]]) { // continue; // } if (group[i]) { let newcirc = mobiusOnCircle(group[i], circ[j]); plot(newcirc, group[i].lev, group[i], tag[i], j); } } } }30:T4aa,{ let s = sSlider; let t = tSlider; let a = math.matrix([[math.complex(s + t, 0), math.complex(-2 * s * t, 0)], [math.complex(-2, 0), math.complex(s + t, 0)]]); let A = math.inv(a); let b = math.matrix([[math.complex(s + t, 0), math.complex(2, 0)], [math.complex(2 * s * t, 0), math.complex(s + t, 0)]]); let B = math.inv(b); let gens = [a, b, A, B]; let inv = [2, 3, 0, 1]; let circ = [ [math.complex((t + s) / 2, 0), (t - s) / 2], [math.complex((-1/t - 1/s) / 2, 0), (1/s - 1/t) / 2], [math.complex((-t - s) / 2, 0), (t - s) / 2], [math.complex((1/t + 1/s) / 2, 0), (1/s - 1/t) / 2] ]; let width = 800; let margin = 100; let innerWidth = width - margin*2; let ctx = DOM.context2d(width, width/2); let scale = (val) => ((val - (-1/s)) * s / 2) * innerWidth + margin ; let plot = ([{re, im}, radius]) => { let center = scale(re); let newRadius = radius * innerWidth / (2 * s); ctx.strokeWidth = '1px'; ctx.fillStyle = 'none'; ctx.strokeStyle = 'black'; ctx.beginPath(); ctx.arc(center, width/4, newRadius, 0, Math.PI * 2); ctx.stroke(); } computeBFS(gens, inv, circ, plot); return ctx.canvas; }31:T4e4, plotBFS = (ctx, scale=140, opt) => { let {lineWidth=0.1, strokeStyle='#555', fill=true, shadowColor='rgba(0, 0, 0, 0.8)', shadowBlur=3 } = opt || {}; let colorScales = [ ["#f7fbff","#e1edf8","#cadef0","#abcfe6","#82badb","#59a1cf","#3787c0","#1c6aaf","#0b4d94","#08306b"].reverse(), ["#f7fcf5","#e6f5e1","#cdebc7","#addea7","#88cd87","#5db96b","#38a055","#1b843f","#04672b","#00441b"].reverse(), ["#fff5eb","#fee7cf","#fdd4ab","#fdb97e","#fd9c51","#f77d2a","#e85e0e","#cc4503","#a33503","#7f2704"].reverse(), ["#fcfbfd","#f0eef6","#dedded","#c6c6e1","#abaad1","#918dc2","#796eb2","#65489f","#52238d","#3f007d"].reverse() ]; ctx.shadowColor = shadowColor; ctx.shadowBlur = shadowBlur; let plot = ([{re, im}, radius], level, group, tag, elem) => { let centerX = re * scale; let centerY = im * scale; let newRadius = radius * scale; ctx.strokeWidth = lineWidth + 'px'; ctx.strokeStyle = strokeStyle; ctx.fillStyle = colorScales[0][level % colorScales[elem].length]; ctx.beginPath(); ctx.arc(centerX, centerY, newRadius, 0, Math.PI * 2); if (fill) { ctx.fill(); } else { ctx.stroke(); } }; return plot; }32:T71f,{ let sin = Math.sin; let cos = Math.cos; let tan = Math.tan; let b = math.matrix([[math.complex(xSlider, 0), math.complex(y, 0)], [math.complex(y, 0), math.complex(xSlider, 0)]]); let B = math.inv(b); let a = math.matrix([[math.complex(uSlider, 0), math.complex(0, k * v)], [math.complex(0, -v/k), math.complex(uSlider, 0)]]); let A = math.inv(a); let gens = [a, b, A, B]; let u = uSlider; let x = xSlider; let circ = [ [math.complex(0, -k*u/v), k/v], [math.complex(-x/y, 0), 1/y], [math.complex(0, k*u/v), k/v], [math.complex(x/y, 0), 1/y], ]; let fixBegin = [ fixedPoint(math.multiply(gens[1], math.multiply(gens[2], math.multiply(gens[3], gens[0])))), fixedPoint(math.multiply(gens[2], math.multiply(gens[3], math.multiply(gens[0], gens[1])))), fixedPoint(math.multiply(gens[3], math.multiply(gens[0], math.multiply(gens[1], gens[2])))), fixedPoint(math.multiply(gens[0], math.multiply(gens[1], math.multiply(gens[2], gens[3])))) ]; let fixEnd = [ fixedPoint(math.multiply(gens[3], math.multiply(gens[2], math.multiply(gens[1], gens[0])))), fixedPoint(math.multiply(gens[0], math.multiply(gens[3], math.multiply(gens[2], gens[1])))), fixedPoint(math.multiply(gens[1], math.multiply(gens[0], math.multiply(gens[3], gens[2])))), fixedPoint(math.multiply(gens[2], math.multiply(gens[1], math.multiply(gens[0], gens[3])))) ]; let width = 700; let ctx = DOM.context2d(width, width, 2); ctx.translate(width/2, width/2); ctx.rotate(Math.PI/2); computeLimitSet(gens, fixEnd, plotLimitSet(ctx), fixBegin[0][0], { epsilon: 0.0001, levmax: 10 }); // ctx.globalCompositeOperation = 'xor'; ctx.canvas.style.backgroundColor = '#2A1A21'; ctx.canvas.style.filter = 'drop-shadow(0 0 2px #555)'; return ctx.canvas; }33:T501,computeDFS = (gens, circ, plot, {epsilon, levmax}) => { const fixed = gens.map(fixedPoint); const genLen = gens.length; // const epsilon = 0.005; // const levmax = 4; for (let k = 0; k < 4; k++) { exploreTree(gens[k], k, 0); } function exploreTree(gen, n, lev) { for (let k = n -1; k <= n+1; k++) { let index = k % genLen; index = index < 0 ? (index + genLen) : index; let mat = math.multiply(gen, gens[index]); let circleTransform = mobiusOnCircle(mat, circ[index]); let [c, r] = circleTransform; if (r < epsilon || lev >= levmax) { let p1 = mobiusOnPoint(mat, fixed[index][0]); plot(p1, circleTransform, { colorScale: 0, level: lev, inner: false, levmax }); // let p2 = mobiusOnPoint(mat, fixed[index][1]); // plot(p2, circleTransform, lev, false); } else { // let p1 = mobiusOnPoint(mat, fixed[index][0]); // plot(p1, circleTransform, { // colorScale: 2, // level: lev, // inner: true, // levmax // }); // let p2 = mobiusOnPoint(mat, fixed[index][1]); // plot(p2, circleTransform, lev, true); exploreTree(mat, k, lev + 1); } } } }34:T9fa,computeLimitSet = (gens, circ, plot, beg, {epsilon, levmax, fixEndGasket}) => { const fixed = gens.map(fixedPoint); const genLen = gens.length; // const epsilon = 0.0001; // const levmax = 10; let oldPoint = beg; let newPoint = false; let firstPoint = beg; let pointList = []; let points = [beg]; for (let k = 0; k < 4; k++) { exploreTree(gens[k], k, 0); } if (points.length > 1) { pointList.push(points); } plot(pointList, { colorScale: 3, level: levmax, inner: false, levmax }); function exploreTree(gen, n, lev) { for (let k = n -1; k <= n+1; k++) { let index = k % genLen; index = index < 0 ? (index + genLen) : index; let mat = math.multiply(gen, gens[index]); let p1; let p2; if (fixEndGasket) { p1 = mobiusOnPoint(mat, circ[index][1][0]); p2 = mobiusOnPoint(mat, circ[index][1][1]); } else { p1 = mobiusOnPoint(mat, circ[index][0]); p2 = mobiusOnPoint(mat, circ[index][1]); } let dist = Infinity; if (oldPoint) { let d1 = math.abs(math.subtract(oldPoint, p1)); let d2 = math.abs(math.subtract(oldPoint, p2)); if (d1 < d2) { newPoint = p1; } else { newPoint = p2; } dist = Math.min(d1, d2); } else { newPoint = p1; oldPoint = newPoint; firstPoint = newPoint; points.push(firstPoint); } if (fixEndGasket) { let z = circ[index].map(c => mobiusOnPoint(mat, c[0])); let d1 = math.abs(math.subtract(z[0], z[1])); let d2 = math.abs(math.subtract(z[1], z[2])); if (d1 < epsilon && d2 < epsilon || lev >= levmax) { if (lev >= levmax && (d1 >= epsilon || d2 >= epsilon)) { if (points.length > 1) { pointList.push(points); } points = [newPoint]; oldPoint = newPoint; return; } points.push(z[0], z[1], z[2]); oldPoint = z[2]; } else { exploreTree(mat, k, lev + 1); } } else { if (dist < epsilon || lev >= levmax) { if (lev >= levmax && dist >= epsilon) { if (points.length > 1) { pointList.push(points); } points = [newPoint]; oldPoint = newPoint; return; } points.push(newPoint); oldPoint = newPoint; } else { exploreTree(mat, k, lev + 1); } } } } }35:T528,plotDFS = (ctx) => { let colorScales = [ ["#f7fbff","#e1edf8","#cadef0","#abcfe6","#82badb","#59a1cf","#3787c0","#1c6aaf","#0b4d94","#08306b"].reverse(), ["#f7fcf5","#e6f5e1","#cdebc7","#addea7","#88cd87","#5db96b","#38a055","#1b843f","#04672b","#00441b"].reverse(), ["#fff5eb","#fee7cf","#fdd4ab","#fdb97e","#fd9c51","#f77d2a","#e85e0e","#cc4503","#a33503","#7f2704"].reverse(), ["#fcfbfd","#f0eef6","#dedded","#c6c6e1","#abaad1","#918dc2","#796eb2","#65489f","#52238d","#3f007d"].reverse() ]; let scale = 80; // ctx.shadowBlur = 2; let plot = ({re, im}, [p, r], {level, levmax, inner, colorScale=0}) => { let centerX = re * scale; let centerY = im * scale; let newRadius = r * scale; // let level = Math.floor(80 / newRadius * 10); let colorIndex = Math.floor(10 * (levmax - level) / levmax); ctx.fillStyle = colorScales[colorScale][colorIndex % colorScales[3].length]; ctx.strokeStyle = ctx.fillStyle; // ctx.shadowColor = ctx.fillStyle; ctx.lineWidth = 5 * (levmax - level) / levmax; if (inner) { ctx.beginPath(); ctx.arc(centerX, centerY, newRadius, 0, Math.PI * 2); ctx.stroke(); } else { ctx.beginPath(); ctx.arc(centerX, centerY, newRadius, 0, Math.PI * 2); ctx.stroke(); } }; return plot; }36:T591,plotLimitSet = (ctx, scale=80) => { let colorScales = [ ["#f7fbff","#e1edf8","#cadef0","#abcfe6","#82badb","#59a1cf","#3787c0","#1c6aaf","#0b4d94","#08306b"].reverse(), ["#f7fcf5","#e6f5e1","#cdebc7","#addea7","#88cd87","#5db96b","#38a055","#1b843f","#04672b","#00441b"].reverse(), ["#fff5eb","#fee7cf","#fdd4ab","#fdb97e","#fd9c51","#f77d2a","#e85e0e","#cc4503","#a33503","#7f2704"].reverse(), ["#fcfbfd","#f0eef6","#dedded","#c6c6e1","#abaad1","#918dc2","#796eb2","#65489f","#52238d","#3f007d"].reverse() ]; let moved = false; let plot = (pointList, {level, levmax, inner, colorScale=0}) => { let colorIndex = 5; ctx.globalAlpha = 0.95; ctx.fillStyle = '#E17DAA';//colorScales[colorScale][colorIndex % colorScales[colorScale].length]; ctx.strokeStyle = ctx.fillStyle; // ctx.shadowColor = ctx.fillStyle; // ctx.shadowBlur = 5; ctx.lineWidth = 1;//5 * (levmax - level) / levmax; ctx.beginPath(); pointList.forEach((points, index) => { points.forEach(({re, im}, i) => { let centerX = re * scale; let centerY = im * scale; if (index === 0) { ctx.moveTo(centerX, centerY); return; } ctx.lineTo(centerX, centerY); // ctx.beginPath(); // ctx.arc(centerX, centerY, 2, 0, Math.PI * 2); // ctx.stroke(); }); }); ctx.closePath(); ctx.fill(); }; return plot; }37:T429,findCircle = (x1, y1, x2, y2, x3, y3) => { var x12 = (x1 - x2); var x13 = (x1 - x3); var y12 =( y1 - y2); var y13 = (y1 - y3); var y31 = (y3 - y1); var y21 = (y2 - y1); var x31 = (x3 - x1); var x21 = (x2 - x1); //x1^2 - x3^2 var sx13 = Math.pow(x1, 2) - Math.pow(x3, 2); // y1^2 - y3^2 var sy13 = Math.pow(y1, 2) - Math.pow(y3, 2); var sx21 = Math.pow(x2, 2) - Math.pow(x1, 2); var sy21 = Math.pow(y2, 2) - Math.pow(y1, 2); var f = ((sx13) * (x12) + (sy13) * (x12) + (sx21) * (x13) + (sy21) * (x13)) / (2 * ((y31) * (x12) - (y21) * (x13))); var g = ((sx13) * (y12) + (sy13) * (y12) + (sx21) * (y13) + (sy21) * (y13)) / (2 * ((x31) * (y12) - (x21) * (y13))); var c = -(Math.pow(x1, 2)) - Math.pow(y1, 2) - 2 * g * x1 - 2 * f * y1; // eqn of circle be // x^2 + y^2 + 2*g*x + 2*f*y + c = 0 // where centre is (h = -g, k = -f) and radius r // as r^2 = h^2 + k^2 - c var h = -g; var k = -f; var sqr_of_r = h * h + k * k - c; // r is the radius var r = Math.sqrt(sqr_of_r); return [math.complex(h, k), r]; };38:Ta4d,{ let b = math.matrix([[math.complex(1, -1), math.complex(1, 0)], [math.complex(1, 0), math.complex(1, 1)]]); let B = math.inv(b); let a = math.matrix([[math.complex(1, 0), math.complex(0, 0)], [math.complex(0, -2), math.complex(1, 0)]]); let A = math.inv(a); let gens = [a, b, A, B]; let inv = [2, 3, 0, 1]; let cA = findCircle(0, 0, -0.2, -0.4, 0.2, -0.4); //console.log('nico', cA); let circ = [ [math.complex(0, 0), 0], [math.complex(1, -1), 1], cA, [math.complex(-1, -1), 1] ]; let fixBegin = [ fixedPoint(math.multiply(gens[1], math.multiply(gens[2], math.multiply(gens[3], gens[0])))), fixedPoint(math.multiply(gens[2], math.multiply(gens[3], math.multiply(gens[0], gens[1])))), fixedPoint(math.multiply(gens[3], math.multiply(gens[0], math.multiply(gens[1], gens[2])))), fixedPoint(math.multiply(gens[0], math.multiply(gens[1], math.multiply(gens[2], gens[3])))) ]; let fixEnd = [ fixedPoint(math.multiply(gens[3], math.multiply(gens[2], math.multiply(gens[1], gens[0])))), fixedPoint(math.multiply(gens[0], math.multiply(gens[3], math.multiply(gens[2], gens[1])))), fixedPoint(math.multiply(gens[1], math.multiply(gens[0], math.multiply(gens[3], gens[2])))), fixedPoint(math.multiply(gens[2], math.multiply(gens[1], math.multiply(gens[0], gens[3])))) ]; let fp = (gen) => { gen.reverse(); let mat = gen.shift(); while (gen.length) { mat = math.multiply(gen.shift(), mat); } return fixedPoint(mat); }; let fixEndGasket = [ [ fp([b, A, B, a]), fp([a]), fp([B, A, b, a]) ], [ fp([A, B, a, b]), fp([b]), fp([a, B, A, b]) ], [ fp([B, a, b, A]), fp([A]), fp([b, a, B, A]) ], [ fp([a, b, A, B]), fp([B]), fp([A, b, a, B]) ] ]; let width = 500; let scale = 290/700*width; let ctx = DOM.context2d(width, width, 2); ctx.translate(width/2, width/2); ctx.fillStyle = 'rgba(0, 0, 0, 0)';//'#2A1A21'; ctx.fillRect(-width/2, -width/2, width, width); computeBFS(gens, inv, circ, plotBFS(ctx, scale, { fill: false, shadowColor: 'none', shadowBlur: 0, strokeWidth: 0.01, strokeStyle: '#555' }), {levmax: 5}); computeLimitSet(gens, fixEnd/*fixEndGasket*/, plotLimitSet(ctx, scale), fixBegin[0][0], { epsilon: 0.001, levmax: 30, fixEndGasket: false//true }); // ctx.globalCompositeOperation = 'xor'; // ctx.canvas.style.backgroundColor = '#2A1A21'; // ctx.canvas.style.filter = 'drop-shadow(0 0 2px rgba(0, 0, 0, 0.3))'; return ctx.canvas; }39:Tba3,{ // let ta = math.complex(1.91, 0.05); // let tb = math.complex(1.91, 0.05); let conj = conjVals[chooseConj]; let [ta, tb] = plotTaTb[+chooseParams]; let a = 1; let b = math.multiply(math.multiply(ta, tb), math.complex(-1, 0)); let c = math.add(math.multiply(ta, ta), math.multiply(tb, tb)); let tab = math.divide(math.add(math.multiply(b, math.complex(-1, 0)), math.sqrt(math.subtract(math.multiply(b, b), math.multiply(math.complex(4, 0), c)))), math.complex(2, 0)); let z0num = math.multiply(math.subtract(tab, math.complex(2, 0)), tb); let z0den = math.add(math.subtract(math.multiply(tb, tab), math.multiply(ta, math.complex(2, 0))), math.multiply(tab, math.complex(0, 2))); let z0 = math.divide(z0num, z0den); b = math.matrix([ [ math.divide(math.subtract(tb, math.complex(0, 2)), math.complex(2, 0)), math.divide(tb, math.complex(2, 0)) ], [ math.divide(tb, math.complex(2, 0)), math.divide(math.add(tb, math.complex(0, 2)), math.complex(2, 0)) ] ]); let ab = math.matrix([ [ math.divide(tab, math.complex(2, 0)), math.divide(math.subtract(tab, math.complex(2, 0)), math.multiply(z0, math.complex(2, 0))) ], [ math.divide(math.multiply(math.add(tab, math.complex(2, 0)), z0), math.complex(2, 0)), math.divide(tab, math.complex(2, 0)) ] ]); a = math.multiply(ab, math.inv(b)); let A = math.inv(a); let B = math.inv(b); let gens = [a, b, A, B].map(mat => math.multiply(conj, math.multiply(mat, math.inv(conj)))); let inv = [2, 3, 0, 1]; let fixBegin = [ fixedPoint(math.multiply(gens[1], math.multiply(gens[2], math.multiply(gens[3], gens[0])))), fixedPoint(math.multiply(gens[2], math.multiply(gens[3], math.multiply(gens[0], gens[1])))), fixedPoint(math.multiply(gens[3], math.multiply(gens[0], math.multiply(gens[1], gens[2])))), fixedPoint(math.multiply(gens[0], math.multiply(gens[1], math.multiply(gens[2], gens[3])))) ].map(p => [mobiusOnPoint(conj, p[0]), mobiusOnPoint(conj, p[1])]); let fixEnd = [ fixedPoint(math.multiply(gens[3], math.multiply(gens[2], math.multiply(gens[1], gens[0])))), fixedPoint(math.multiply(gens[0], math.multiply(gens[3], math.multiply(gens[2], gens[1])))), fixedPoint(math.multiply(gens[1], math.multiply(gens[0], math.multiply(gens[3], gens[2])))), fixedPoint(math.multiply(gens[2], math.multiply(gens[1], math.multiply(gens[0], gens[3])))) ].map(p => [mobiusOnPoint(conj, p[0]), mobiusOnPoint(conj, p[1])]); let width = 300; let height = width; let scale = 180/700*width; let ctx = DOM.context2d(width, height, 2); ctx.translate(width/2, height / 2); ctx.fillStyle = '#2A1A21'; ctx.fillRect(-width/2, -height/2, width, height); computeLimitSet(gens, fixEnd/*fixEndGasket*/, plotLimitSet(ctx, scale), fixBegin[0][0], { epsilon: 0.001, levmax: 15, fixEndGasket: false//true }); // return path; return ctx.canvas; } 3a:T8a9,maskitParams = { // debugger; // example page 260 // const μ = math.complex(0.05, 1.91); // const t_a_1 = math.multiply(-1, math.multiply(μ, math.complex(0, 1))); // const t_a = math.complex(2, 0.05); const t_a = math.complex(1.95, 0.07); // const t_a = math.complex(2, 0); // matrices const a = math.matrix([ [t_a, math.complex(0, -1)], [math.complex(0, -1), 0] ]); const b = math.matrix([ [1, 2], [0, 1] ]); let A = math.inv(a); let B = math.inv(b); let gens = [a, b, A, B]; let inv = [2, 3, 0, 1]; let fixBegin = [ fixedPoint(math.multiply(gens[1], math.multiply(gens[2], math.multiply(gens[3], gens[0])))), fixedPoint(math.multiply(gens[2], math.multiply(gens[3], math.multiply(gens[0], gens[1])))), fixedPoint(math.multiply(gens[3], math.multiply(gens[0], math.multiply(gens[1], gens[2])))), fixedPoint(math.multiply(gens[0], math.multiply(gens[1], math.multiply(gens[2], gens[3])))) ]; let fixEnd = [ fixedPoint(math.multiply(gens[3], math.multiply(gens[2], math.multiply(gens[1], gens[0])))), fixedPoint(math.multiply(gens[0], math.multiply(gens[3], math.multiply(gens[2], gens[1])))), fixedPoint(math.multiply(gens[1], math.multiply(gens[0], math.multiply(gens[3], gens[2])))), fixedPoint(math.multiply(gens[2], math.multiply(gens[1], math.multiply(gens[0], gens[3])))) ]; let fixMat = [ fixedPoint(a), fixedPoint(A), // fixedPoint(b), // fixedPoint(B), ]; let bbox = { minX: Infinity, minY: Infinity, maxX: -Infinity, maxY: -Infinity }; const updateBBox = (a) => { if (a.re < bbox.minX) { bbox.minX = a.re; } if (a.im < bbox.minY) { bbox.minY = a.im; } if (a.re > bbox.maxX) { bbox.maxX = a.re; } if (a.im > bbox.maxY) { bbox.maxY = a.im; } }; fixBegin.forEach(([a, b]) => { updateBBox(a); // updateBBox(b); }); // fixEnd.forEach(([a, b]) => { // updateBBox(a); // updateBBox(b); // }); fixMat.forEach(([a, b]) => { updateBBox(a); // updateBBox(b); }); return { fixBegin, fixEnd, fixMat, bbox, a, b, A, B, gens, inv, t_a }; }3b:Teaa,{ let { fixBegin, fixEnd, fixMat, bbox, a, b, A, B, gens, inv, t_a } = maskitParams; let dx = bbox.maxX - bbox.minX; let dy = bbox.maxY - bbox.minY; let width = 200; let phases = 1; let scale = width / (Math.max(dx, dy)); let height = width / dx * dy; let ctx = DOM.context2d(width*phases, height, 2); ctx.fillStyle = '#2A1A21'; ctx.fillRect(0, 0, width*phases, height); ctx.translate(width/2 +4, 0); computeLimitSet(gens, fixEnd/*fixEndGasket*/, plotLimitSet(ctx, scale), fixBegin[0][0], { epsilon: 0.005, levmax: 15, fixEndGasket: false//true }); fixBegin.forEach(([a, b]) => { const r1 = 15; ctx.fillStyle = 'pink'; ctx.beginPath(); ctx.arc(a.re * scale, a.im * scale, r1, 0, Math.PI*2); ctx.fill(); ctx.closePath(); }); fixMat.forEach(([a, b]) => { const r1 = 5; ctx.fillStyle = 'orange'; ctx.beginPath(); ctx.arc(a.re * scale, a.im * scale, r1, 0, Math.PI*2); ctx.fill(); ctx.closePath(); }); // render exponential function const Y = (A, K, B, v, Q, C) => { return (t) => A + (K - A) / Math.pow((C + Q * Math.exp(-B*t)), 1/v); }; const control1 = fixMat[0][0].re < fixMat[1][0].re ? fixMat[0][0] : fixMat[1][0]; const control2 = fixMat[0][0].re > fixMat[1][0].re ? fixMat[0][0] : fixMat[1][0]; const mid = math.divide(math.add(control1, control2), 2); const delta = 0.001; // just one limit point for a and A means a straight line if (math.distance(control1.toVector(), control2.toVector()) < delta) { ctx.fillStyle = 'rgba(0, 0, 0, 0.5)'; ctx.beginPath(); ctx.moveTo(bbox.minX * scale, bbox.minY * scale); ctx.lineTo(bbox.minX * scale, control1.im * scale); ctx.lineTo(bbox.maxX * scale, control1.im * scale); ctx.lineTo(bbox.maxX * scale, bbox.minY * scale); ctx.fill(); ctx.closePath(); return ctx.canvas; } // find B // from 1-30 get the closest B let versor = math.subtract(control1, control2); versor = math.divide(versor, math.norm(versor)).toVector(); let fix = fixBegin .map(f => f[0]) .sort((a, b) => { let aVec = math.subtract(a, mid); let bVec = math.subtract(b, mid); let aVersor = math.divide(aVec, math.norm(aVec)).toVector(); let bVersor = math.divide(bVec, math.norm(bVec)).toVector(); return math.abs(math.dot(aVersor, versor)) - math.abs(math.dot(bVersor, versor)); }).slice(0, 2); // let fix1, fix2; let fix1 = Math.sign(control1.re) == Math.sign(fix[0].re) ? fix[0] : fix[1]; let fix2 = Math.sign(control2.re) == Math.sign(fix[0].re) ? fix[0] : fix[1]; // probably need some code to verify this const MBegin = [ bbox.minX, (control1.im*0.5 + fix1.im*0.5) ]; const MEnd = [ bbox.maxX, (control2.im*0.5 + fix2.im*0.5) ]; let minDot = Infinity, minB = 1, Bparam = 1; for (; Bparam <= 100; Bparam++) { let yt = Y(MBegin[1], MEnd[1], Bparam, 1, 1, 1); let epsi = 0.001; let ytb = yt((control1.re + control2.re) / 2 - epsi); let yte = yt((control1.re + control2.re) / 2 + epsi); let yversor = math.matrix([epsi * 2, yte - ytb]); yversor = math.divide(yversor, math.norm(yversor)); let dot = math.abs(math.dot(versor, yversor)); if (dot < minDot) { minDot = dot; minB = Bparam; } } const ytf = Y(MBegin[1], MEnd[1], minB/2, 1, 1, 1); ctx.fillStyle = 'rgba(0, 0, 0, 0.5)'; ctx.beginPath(); ctx.moveTo(bbox.minX * scale, fix1.im * scale); for (let x = bbox.minX; x <= bbox.maxX; x += (bbox.maxX - bbox.minX) / 100) { let y = ytf(x); ctx.lineTo(x * scale, y * scale); } ctx.lineTo(bbox.maxX * scale, fix1.im * scale); ctx.fill(); ctx.closePath(); return ctx.canvas; }3c:T566,boundary = function (denom) { let fareySeq = nextFareyPQ(denom); // let fareySeq = []; // let n = denom; // while (denom > 1) { // fareySeq.push.apply(fareySeq, nextFareyPQ(denom--)); // } let p = 0; let q = 1; let mu = math.complex(0, 2); let mu2 = math.complex(0, 2); let ans = []; let ans2 = []; ans.push([p, q, mu, 2]); ans2.push([p, q, mu2, -2]); fareySeq.sort((a, b) => a[0]/a[1] - b[0]/b[1]); for (let [p, q] of fareySeq) { let oldMu1 = mu; let traceEqn = getTraceEqn(p, q, 2); mu = newtonSolver(oldMu1, traceEqn, rate); ans.push([p, q, mu, 2]); // let oldMu2 = mu2; // traceEqn = getTraceEqn(p, q, -2); // mu2 = newtonSolver(oldMu2, traceEqn, rate); // ans2.push([p, q, mu2, -2]); } // ans.shift(); // ans.pop(); let steps = ans; let steps2 = [ans2]; // for (let i = 0.1; i < 1; i+=0.1) { // steps.push(steps[steps.length -1].map(([p, q, oldMu], k) => { // let traceEqn = getTraceEqn(p, q, 2+i); // let mu = newtonSolver(oldMu, traceEqn, rate); // return [p, q, mu, 2+i]; // })); // steps2.push(steps2[steps2.length -1].map(([p, q, oldMu], k) => { // let traceEqn = getTraceEqn(p, q, -2+i); // let mu = newtonSolver(oldMu, traceEqn, rate); // return [p, q, mu, -2+i]; // })); // } // steps.push(ans2); return steps;//.slice(1); }3d:T9d9,Not_needed = { let word = getWord(2, 15); // return word; // Function: getWordSequence. // Desc: Returns an array of words in increasing order from the given word. // Input: An initial/seed word. // Output: An array of words in increasing order. // get least significant index. // while index does not overlap more significant index // move least significant index up. // record new number. // else // while next significant index overlaps // take next significant index // if (!next significant index found) // algorithm ends // else // move new significant index up. // move all least significant indices to the bottom of array. // record new number. // repeat. let getWordSequence = (word) => { const wordSize = word.length; // use a more compact data structure to track significant numbers. let indexStart = word.indexOf("B"); let indices = []; while (indexStart > -1) { indices.push(indexStart); indexStart = word.indexOf("B", indexStart+1); } let seedWord = indices.slice(); // record initial word let answers = [seedWord.slice()]; // get least significant index let lastIndex = seedWord.length -1; let counter = 1000000; while (true && counter--) { // while index does not overlap more significant index while ((seedWord.length > 1 && seedWord[lastIndex] > (seedWord[lastIndex -1] + 1)) || (seedWord.length == 1 && seedWord[lastIndex] > 0)) { // move least significant index up. --seedWord[lastIndex]; // record new number answers.push(seedWord.slice()); } lastIndex--; // while next significant index overlaps while (lastIndex > 0 && (seedWord[lastIndex] == (seedWord[lastIndex -1] - 1))) { // take next significant index lastIndex--; } if (lastIndex == 0) { // if (!next significant index found) if (seedWord[lastIndex] == 0) { // algorithm ends return answers; } } // move new significant index up. --seedWord[lastIndex]; // move all least significant indices to the bottom of array. let wordCounter = wordSize; for (let i = seedWord.length -1; i > lastIndex; i--) { seedWord[i] = --wordCounter; } // record new number answers.push(seedWord.slice()); lastIndex = seedWord.length -1; } }; return getWordSequence(word); }3e:[] 2d:{"id":"cde177e22e1a9a6e","slug":null,"title":"Indra's Pearls","thumbnail":null,"access_level":"unlisted","create_time":"$D2021-11-12T19:15:13.627Z","update_time":"$D2024-04-20T13:38:25.927Z","view_time":null,"first_public_time":null,"fork":true,"stars":0,"starred":false,"owner":{"type":"team","id":"fbb2ab89a5f3cda4","login":"philogb","name":"Nicolas Garcia Belmonte","avatar_url":"https://avatars-next.observableusercontent.com/bef72157432760695121834277d169fcc0cfca16f14f1fa4e6ccc99be8a4dce8","tier":"starter_2024"},"version":2775,"publish_version":null,"role":null,"authored":false,"authors":[{"type":"individual","id":"3857442cbbba93b5","login":"philogb","name":"Nicolas Garcia Belmonte","avatar_url":"https://avatars-next.observableusercontent.com/bef72157432760695121834277d169fcc0cfca16f14f1fa4e6ccc99be8a4dce8","tier":null}],"members":[{"type":"individual","id":"3857442cbbba93b5","login":"philogb","name":"Nicolas Garcia Belmonte","avatar_url":"https://avatars-next.observableusercontent.com/bef72157432760695121834277d169fcc0cfca16f14f1fa4e6ccc99be8a4dce8","tier":null,"role":"editor"},{"type":"team","id":"fbb2ab89a5f3cda4","login":"philogb","name":"Nicolas Garcia Belmonte","avatar_url":"https://avatars-next.observableusercontent.com/bef72157432760695121834277d169fcc0cfca16f14f1fa4e6ccc99be8a4dce8","tier":"starter_2024","role":"viewer"}],"comments":[],"custom_thumbnail":false,"trashed":false,"subscribed":null,"forks":1,"collection":null,"type":"notebook","body":{"title":"Indra's Pearls","cells":[{"id":0,"value":"md`# Indra's Pearls\nWriting code for a few of the concepts in Indra's Pearls`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":254,"value":"md`## Circle Inversion\nFormula for circle inversion is given by`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":273,"value":"z \\rightarrow a + \\dfrac{R^2}{z - a}","pinned":false,"hidden":false,"mode":"tex","author":"$undefined","summary":"$undefined"},{"id":324,"value":"invert = (pts) => {\n let a = math.complex(circle.cx, circle.cy);\n let ans = pts.map(p => {\n let inv = math.inv(math.subtract(p, a));\n let mult = math.multiply(inv, math.complex(R * R, 0));\n return math.add(a, mult);\n });\n return ans;\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":296,"value":"viewof R = slider({\n min: 1, \n max: 20, \n step: 0.01, \n value: 1, \n title: \"R\", \n description: \"Radius of circle\"\n})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":308,"value":"viewof tSquare = slider({\n min: 0, \n max: 3, \n step: 0.01, \n value: 0, \n title: \"tSquare\", \n description: \"Square translation\"\n})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":301,"value":"html`\n\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":285,"value":"square = {\n let div = 10;\n let top = [];\n let right = [];\n let bottom = [];\n let left = [];\n \n for (let i = -1; i <= 1; i+=1/div) {\n top.push(math.complex(i, -1));\n right.push(math.complex(1, i));\n bottom.push(math.complex(-i, 1));\n left.push(math.complex(-1, -i));\n }\n let pts = [...top, ...right, ...bottom, ...left];\n return pts;\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":294,"value":"circle = {\n let cy = 0;\n let cr = R;\n let cx = 1 - R;\n \n return {cx, cy, r: cr};\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":311,"value":"transformedSquare = {\n // make the square considerably smaller\n let scale = math.complex(0.2, 0);\n let scaledSquare = square.map(c => math.multiply(scale, c));\n let translation = math.complex(tSquare, 0);\n let translatedSquare = scaledSquare.map(c => math.add(translation, c));\n return {\n pts: translatedSquare,\n svg: getPath(translatedSquare)\n };\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":323,"value":"invertedSquare = {\n let pts = invert(transformedSquare.pts);\n return {\n pts,\n svg: getPath(pts)\n };\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":312,"value":"getPath = (pts) => {\n return `M${pts[0].re},${pts[0].im}` + pts.slice(1).map(p => `L${p.re},${p.im}`).join('');\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":360,"value":"md`## Stereographic Projection\n\nStereographic projection from the Riemann sphere to the complex plane:`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":373,"value":"x = \\dfrac{u}{1 - w} ; y = \\dfrac{v}{1 - w}","pinned":false,"hidden":false,"mode":"tex","author":"$undefined","summary":"$undefined"},{"id":381,"value":"stereoSphere2Plane = (u, v, w) => math.complex(u / (1 - w), v / (1 - w));","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":384,"value":"Stereographic projection from the plane to the sphere:","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":388,"value":"(u, v, w) = \\dfrac{(x, y, x^2 + y^2)}{x^2 + y^2 + 1}","pinned":false,"hidden":false,"mode":"tex","author":"$undefined","summary":"$undefined"},{"id":393,"value":"stereoPlane2Sphere = ({re, im}) => {\n let x = re;\n let y = im;\n let norm = x * x + y * y;\n let den = norm + 1;\n return [x / den, y / den, norm / den];\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":397,"value":"stereoConfig = ({\n distMax: 10,\n divisions: 100,\n count: 10\n})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":471,"value":"viewof loxodromicSlider = slider({\n min: 0, \n max: 1, \n step: 0.001, \n value: 0, \n title: \"Loxodromic\", \n description: \"Apply loxodromic transform\"\n})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":459,"value":"loxodromic = (z) => {\n let ans = math.add(math.multiply(z, 1 - loxodromicSlider), math.multiply(math.divide(math.subtract(z, 1), math.add(z, 1)), loxodromicSlider));\n ans.re = Math.min(1000, Math.abs(ans.re)) * Math.sign(ans.re);\n ans.im = Math.min(1000, Math.abs(ans.im)) * Math.sign(ans.im);\n return ans;\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":428,"value":"html`\n\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":396,"value":"meridians = {\n let angleLen = Math.PI * 2 / stereoConfig.count;\n let distLen = stereoConfig.distMax / stereoConfig.divisions;\n let ans = [];\n for (let angle = 0; angle < Math.PI * 2; angle += angleLen) {\n let pts = [];\n for (let i = 0; i < stereoConfig.divisions; i++) {\n pts.push(math.complex(Math.cos(angle) * i * distLen, Math.sin(angle) * i * distLen));\n }\n ans.push(pts);\n }\n return ans;\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":462,"value":"transformedMeridians = {\n return meridians.map(m => {\n return m.map(loxodromic);\n });\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":420,"value":"meridiansSVG = transformedMeridians.map((m) => {\n return getPath(m);\n}).join('');","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":410,"value":"parallels = {\n let angleLen = Math.PI * 2 / stereoConfig.divisions;\n let distLen = stereoConfig.distMax / stereoConfig.count;\n let ans = [];\n for (let i = 0; i < stereoConfig.count; i++) {\n let pts = [];\n for (let angle = 0; angle < Math.PI * 2; angle += angleLen) {\n pts.push(math.complex(Math.cos(angle) * i * distLen, Math.sin(angle) * i * distLen));\n }\n ans.push(pts);\n }\n return ans;\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":464,"value":"transformedParallels = {\n return parallels.map(m => {\n return m.map(loxodromic);\n });\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":422,"value":"parallelsSVG = transformedParallels.map((m) => {\n return getPath(m);\n}).join('');","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":413,"value":"spiral = {\n let factor = math.complex(1, 0.4);\n let ans = [math.complex(1, 0)];\n let counter = stereoConfig.divisions;\n while (counter--) {\n let first = ans[0];\n first = math.divide(first, factor);\n ans.unshift(first);\n let last = ans[ans.length -1];\n last = math.multiply(last, factor);\n ans.push(last);\n }\n return ans;\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":466,"value":"transformedSpiral = {\n return spiral.map(loxodromic);\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":426,"value":"spiralSVG = getPath(transformedSpiral)","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":524,"value":"md`## Mobius on point\nFormula for Mobius transform is given by`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":529,"value":"T(z) = \\dfrac{az + b}{cz + d}","pinned":false,"hidden":false,"mode":"tex","author":"$undefined","summary":"$undefined"},{"id":533,"value":"mobiusOnPoint = (T, z) => {\n let a = math.subset(T, math.index(0, 0));\n let b = math.subset(T, math.index(0, 1));\n let c = math.subset(T, math.index(1, 0));\n let d = math.subset(T, math.index(1, 1));\n return math.divide(\n math.add(math.multiply(a, z), b),\n math.add(math.multiply(c, z), d)\n );\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":536,"value":"Tr = (T) => math.add(math.subset(T, math.index(0, 0)), math.subset(T, math.index(1, 1)));","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":539,"value":"fixedPoint = (T) => {\n let a = math.subset(T, math.index(0, 0));\n let b = math.subset(T, math.index(0, 1));\n let c = math.subset(T, math.index(1, 0));\n let d = math.subset(T, math.index(1, 1));\n let aMinusd = math.subtract(a, d);\n let trSquared = math.multiply(Tr(T), Tr(T));\n let sqrt = math.sqrt(math.subtract(trSquared, 4));\n let twoC = math.multiply(2, c);\n let zPlus = math.divide(math.add(aMinusd, sqrt), twoC);\n let zMinus = math.divide(math.subtract(aMinusd, sqrt), twoC);\n return [zPlus, zMinus];\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":545,"value":"mobiusOnCircle = (T, circ) => {\n if (circ.length !== 2) {\n return circ;\n }\n let [center, radius] = circ;\n let P = center;\n let r = radius;\n let a = math.subset(T, math.index(0, 0));\n let b = math.subset(T, math.index(0, 1));\n let c = math.subset(T, math.index(1, 0));\n let d = math.subset(T, math.index(1, 1));\n let z = math.subtract(P, math.divide(math.complex(r * r, 0), math.conj(math.add(math.divide(d, c), P))));\n let Q = math.divide(math.add(math.multiply(a, z), b), math.add(math.multiply(c, z), d));\n let pAndR = math.add(P, math.complex(r, 0));\n let s = math.abs(math.subtract(Q, math.divide(math.add(math.multiply(a, pAndR), b), math.add(math.multiply(c, pAndR), d))));\n return [Q, s];\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":556,"value":"md`## Schottky Dance`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":574,"value":"$2f","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":588,"value":"md`### Schottky groups with limit set on the real line`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":592,"value":"a(z) = \\dfrac{(s + t)z - 2st}{-2z + (s + t)} ; b(z) = \\dfrac{(s + t)z + 2}{2stz + (s+t)}","pinned":false,"hidden":false,"mode":"tex","author":"$undefined","summary":"$undefined"},{"id":610,"value":"0 < s < t < 1","pinned":false,"hidden":false,"mode":"tex","author":"$undefined","summary":"$undefined"},{"id":600,"value":"viewof sSlider = slider({\n min: 0.01, \n max: 0.99, \n step: 0.001, \n value: 0.64, \n title: \"s\", \n description: \"s\"\n})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":606,"value":"viewof tSlider = slider({\n min: sSlider, \n max: 0.99, \n step: 0.001,\n value: (sSlider + 0.99) / 2, \n title: \"t\", \n description: \"t\"\n})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":615,"value":"$30","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":697,"value":"md`## θ-Schottky groups`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":703,"value":"viewof thetaSlider = slider({\n min: 0, \n max: Math.PI/3, \n step: 0.001, \n value: Math.PI/4, \n title: \"θ-slider\", \n description: \"θ-slider\"\n})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":816,"value":"{\n let theta = thetaSlider;\n\n let sin = Math.sin;\n let cos = Math.cos;\n let tan = Math.tan;\n \n let b = math.matrix([[math.complex(1/sin(theta), 0), math.complex(cos(theta)/sin(theta), 0)], [math.complex(cos(theta)/sin(theta), 0), math.complex(1/sin(theta), 0)]]);\n let B = math.inv(b);\n let a = math.matrix([[math.complex(1/sin(theta), 0), math.complex(0, cos(theta)/sin(theta))], [math.complex(0, -cos(theta)/sin(theta)), math.complex(1/sin(theta), 0)]]);\n let A = math.inv(a);\n\n let gens = [a, b, A, B];\n let radius = tan(theta);\n let r = 1/cos(theta);\n let circ = [\n [math.complex(0, r), radius],\n [math.complex(r, 0), radius],\n [math.complex(0, -r), radius],\n [math.complex(-r, 0), radius]\n ];\n\n let width = 700;\n let ctx = DOM.context2d(width, width, 4);\n ctx.translate(width/2, width/2);\n let scale = 300;\n\n computeDFS(gens, circ, plotDFS(ctx), {\n epsilon: 0.005,\n levmax: 4\n });\n ctx.canvas.style.backgroundColor = '#2A1A21';\n return ctx.canvas;\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":709,"value":"{\n let theta = thetaSlider;\n\n let sin = Math.sin;\n let cos = Math.cos;\n let tan = Math.tan;\n \n let b = math.matrix([[math.complex(1/sin(theta), 0), math.complex(cos(theta)/sin(theta), 0)], [math.complex(cos(theta)/sin(theta), 0), math.complex(1/sin(theta), 0)]]);\n let B = math.inv(b);\n let a = math.matrix([[math.complex(1/sin(theta), 0), math.complex(0, cos(theta)/sin(theta))], [math.complex(0, -cos(theta)/sin(theta)), math.complex(1/sin(theta), 0)]]);\n let A = math.inv(a);\n \n let gens = [a, b, A, B];\n let inv = [2, 3, 0, 1];\n let radius = tan(theta);\n let r = 1/cos(theta);\n let circ = [\n [math.complex(0, -r), radius],\n [math.complex(-r, 0), radius],\n [math.complex(0, r), radius],\n [math.complex(r, 0), radius]\n ];\n\n let width = 700;\n let ctx = DOM.context2d(width, width, 4);\n ctx.translate(width/2, width/2);\n\n computeBFS(gens, inv, circ, plotBFS(ctx, 250));\n ctx.canvas.style.backgroundColor = '#2A1A21';\n return ctx.canvas;\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":1328,"value":"$31","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":926,"value":"md`## Parametrized kissing Schottky groups`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":931,"value":"b = \\begin{pmatrix}\nx & y \\\\\ny & x\n\\end{pmatrix}\n,\na = \\begin{pmatrix}\nu & ikv \\\\\n\\frac{-iv}{k} & u\n\\end{pmatrix}\n\\newline\nand\n\\newline\nx = \\sqrt{1 + y^2}, u = \\sqrt{1 + v^2}\n\\newline\nwith\n\\newline\n\\dfrac{1}{2}(k + \\dfrac{1}{k}) = \\dfrac{1}{yv}\n","pinned":false,"hidden":false,"mode":"tex","author":"$undefined","summary":"$undefined"},{"id":952,"value":"viewof xSlider = slider({\n min: 1, \n max: Math.PI,\n step: 0.00001, \n value: 1.29, \n title: \"x\",\n description: \"Value for x\"\n})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":955,"value":"viewof uSlider = slider({\n min: 1, \n max: Math.PI,\n step: 0.00001, \n value: Math.SQRT2, \n title: \"u\",\n description: \"Value for u\"\n})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":962,"value":"y = {\n let x = xSlider;\n return x * x - 1; //taking positive root\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":966,"value":"v = {\n let u = uSlider;\n return u * u - 1; //taking positive root\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":972,"value":"k = {\n //k^2 - 2k/(yv) + 1 = 0 \n let a = 1;\n let b = -2 / (y * v);\n let c = 1;\n let b24ac = b * b - 4 * a * c;\n let sqb24ac = Math.sqrt(b24ac);\n let den = 2 * a;\n let rootP =( -b + sqb24ac) / den;\n if (rootP > 0) {\n return rootP;\n }\n return ( -b - sqb24ac) / den;\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":1121,"value":"md`Computing DFS limit set for kissing Schottky circles`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":975,"value":"{\n let sin = Math.sin;\n let cos = Math.cos;\n let tan = Math.tan;\n \n let b = math.matrix([[math.complex(xSlider, 0), math.complex(y, 0)], [math.complex(y, 0), math.complex(xSlider, 0)]]);\n let B = math.inv(b);\n let a = math.matrix([[math.complex(uSlider, 0), math.complex(0, k * v)], [math.complex(0, -v/k), math.complex(uSlider, 0)]]);\n let A = math.inv(a);\n\n let gens = [a, b, A, B];\n let u = uSlider;\n let x = xSlider;\n let circ = [\n [math.complex(0, -k*u/v), k/v],\n [math.complex(-x/y, 0), 1/y],\n [math.complex(0, k*u/v), k/v],\n [math.complex(x/y, 0), 1/y],\n ];\n\n let width = 700;\n let ctx = DOM.context2d(width, width, 2);\n ctx.translate(width/2, width/2);\n ctx.rotate(Math.PI/2);\n computeDFS(gens, circ, plotDFS(ctx), {\n epsilon: 0.005,\n levmax: 4\n });\n // ctx.globalCompositeOperation = 'xor';\n ctx.canvas.style.backgroundColor = '#2A1A21';\n ctx.canvas.style.filter = 'drop-shadow(0 0 2px #555)';\n return ctx.canvas;\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":1126,"value":"md`Computing limit set with lines`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":1127,"value":"$32","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":806,"value":"md`DFS implementation`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":809,"value":"$33","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":1120,"value":"$34","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":1016,"value":"$35","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":1136,"value":"$36","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":1256,"value":"md`## Glowing Gasket`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":1257,"value":"a = \\begin{pmatrix}\n1 & 0 \\\\\n-2i & 1\n\\end{pmatrix}\n\\newline\n and \n\\newline\nb = \\begin{pmatrix}\n1 - i & 1 \\\\\n1 & 1 + i\n\\end{pmatrix}\n","pinned":false,"hidden":false,"mode":"tex","author":"$undefined","summary":"$undefined"},{"id":1750,"value":"md`Find circle from 3 points`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":1737,"value":"$37","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":1753,"value":"md`Find matrices for subgroup of apollonian gasket that make up the modular group`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":1745,"value":"{\n //modular computations\n const b = math.matrix([\n [math.complex(1, -1), math.complex(1, 0)],\n [math.complex(1, 0), math.complex(1, 1)]\n ]);\n const B = math.inv(b);\n const threeP = [\n mobiusOnPoint(B, math.complex(-1, 0)),\n mobiusOnPoint(B, math.complex(0, 0)),\n mobiusOnPoint(B, math.complex(1, 0))\n ];\n const Ba = findCircle(\n threeP[0].re, threeP[0].im,\n threeP[1].re, threeP[1].im,\n threeP[2].re, threeP[2].im\n );\n const BA = mobiusOnCircle(B, [ math.complex(0, -0.25), 0.25]);\n return BA;\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":1270,"value":"$38","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":1366,"value":"md`## Grandma's special parabolic commutator groups`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":1373,"value":"(1) Choose \\space t_a \\space and \\space t_b\n\\newline \\space \\newline\n(2) Solve: x^2 - t_a t_b x + t_a^2 + t_b^2 = 0; t_{ab} = x\n\\newline \\space \\newline\n(3) z_0 = \\dfrac{(t_{ab} - 2) t_b}{t_b t_{ab} - 2 t_a + 2 i t_{ab}}\n\\newline \\space \\newline\na = \\begin{pmatrix}\n\\dfrac{t_a}{2} & \\dfrac{t_a t_{ab} - 2 t_b + 4 i}{(2 t_{ab} + 4) z_0} \\\\\n\\dfrac{(t_a t_{ab} - 2 t_b - 4 i) z_0}{2 t_{ab} - 4} & \\dfrac{t_a}{2}\n\\end{pmatrix}\n\\newline \\space \\newline\nb = \\begin{pmatrix}\n\\dfrac{t_b - 2i}{2} & \\dfrac{t_b}{2} \\\\\n\\dfrac{t_b}{2} & \\dfrac{t_b + 2i}{2}\n\\end{pmatrix}\n\\newline \\space \\newline\nab = \\begin{pmatrix}\n\\dfrac{t_{ab}}{2} & \\dfrac{t_{ab} - 2}{2 z_0} \\\\\n\\dfrac{(t_{ab} + 2) z_0}{2} & \\dfrac{t_{ab}}{2}\n\\end{pmatrix}","pinned":false,"hidden":false,"mode":"tex","author":"$undefined","summary":"$undefined"},{"id":1441,"value":"viewof chooseParams = {\n const dd3 = select({\n title: \"Choose ta & tb\",\n options: plotTaTb.map(([ta, tb], i) => {\n return {label: `ta = (${ta.re}, ${ta.im}); tb = (${tb.re}, ${tb.im})`, value: i};\n }),\n value: \"hmmm\"\n });\n dd3.input.style.fontSize = \"15px\";\n dd3.input.style.marginTop = \"8px\";\n return dd3;\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":1458,"value":"viewof chooseConj = {\n const dd3 = select({\n title: \"Choose a conjugation\",\n options: [\n {label: `Id`, value: 0},\n {label: `1/z`, value: 1},\n {label: `Cayley`, value: 2},\n ],\n });\n dd3.input.style.fontSize = \"15px\";\n dd3.input.style.marginTop = \"8px\";\n return dd3;\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":1461,"value":"conjVals = [\n math.matrix([[1, 0], [0, 1]]),\n math.matrix([[0, 1], [1, 0]]),\n math.matrix([[math.complex(1, 0), math.complex(0, -1)], [math.complex(1, 0), math.complex(0, 1)]])\n];","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":1411,"value":"$39","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":1435,"value":" plotTaTb = [\n [3, 0, 3, 0],\n [2.9, 0, 2.9, 0],\n [2 * Math.SQRT2, 0, 2 * Math.SQRT2, 0],\n [2.7, 0, 2.7, 0],\n [2.2, 0, 2.2, 0],\n [2, 0, 2, 0],\n [3, -0.5, 3, 0.5],\n [2, 0.5, 3, 0],\n [2, 0.5, 2, 0.5],\n [2, 0.2, 2, -0.2],\n [2, 0.05, 3, 0],\n [1.97, 0.05, 3, 0],\n [1.94, 0.05, 3, 0],\n [1.91, 0.05, 3, 0],\n [1.91, 0.05, 1.91, 0.05],\n [2, 0.05, 2, 0],\n [1.889, 0.05, 2, 0]\n ].map(([a, b, c, d]) => [math.complex(a, b), math.complex(c, d)]);","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":1756,"value":"md`## Maskit Parametrization`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":1767,"value":"md`Transforms`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":1759,"value":"a(z) = \\mu + \\frac{1}{z}\n\\newline\nb(z) = z + 2\n\\newline\n\\mu = - \\frac{t_a}{i}","pinned":false,"hidden":false,"mode":"tex","author":"$undefined","summary":"$undefined"},{"id":1770,"value":"md`Matrices`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":1773,"value":"a = \\begin{pmatrix}\nt_a & -i \\\\\n-i & 0\n\\end{pmatrix}\n\\newline \\space \\newline\nb = \\begin{pmatrix}\n1 & 2 \\\\\n0 & 1\n\\end{pmatrix}","pinned":false,"hidden":false,"mode":"tex","author":"$undefined","summary":"$undefined"},{"id":1783,"value":"md`Given μ, compute matrices and inverses, trace and fixed points of commutators`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":1917,"value":"$3a","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":1787,"value":"$3b","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":2128,"value":"# Cusps","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":2032,"value":"md`Compute the right word for p/q`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":2034,"value":"getWord = (p, q) => {\n let start = 1;\n const ans = [];\n let add = true;\n do {\n if (add) {\n if (start + q <= p + q) {\n start += q;\n ans.push('B');\n } else {\n start -= p;\n ans.push('a');\n add = !add;\n }\n } else {\n if (start - p >= 1) {\n start -= p;\n ans.push('a');\n } else {\n start += q;\n ans.push('B');\n add = !add;\n }\n }\n } while (start !== 1);\n return ans.reverse();\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":2108,"value":"getCyclicPermutation = (word) => {\n let words = [word.slice()];\n for (let i = 0; i < word.length -1; i++) {\n let val = word.shift();\n word.push(val);\n words.push(word.slice());\n }\n return words;\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":2118,"value":"sortCyclicPermutation = (perm) => {\n let cycleHash = {};\n perm.forEach(p => {\n let key = p.join('');\n let val = parseInt(p.reduce((acum, val) => acum + (val == 'a'? '0' : '1'), ''), 2);\n cycleHash[key] = val;\n });\n return Object.entries(cycleHash).sort((a, b) => a[1] - b[1]).map(f => f[0].split(''));\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":2133,"value":"## Example usage of functions above","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":2111,"value":"val = {\n let w = getWord(2, 15);\n let perm = getCyclicPermutation(w);\n let ans = sortCyclicPermutation(perm);\n return ans;\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":2135,"value":"# Tracing the Maskit Boundary","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":2159,"value":"`p` is the fraction numerator, `q` is the denominator, `denom` is the fixed denominator.","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":2137,"value":"nextFareyPQ = function(n, denom) {\n let ans = [];\n let [a, b, c, d] = [0, 1, 1, n];\n while (0 <= c && c <= n) {\n let k = Math.floor((n + b) / d);\n [a, b, c, d] = [c, d, k * c - a, k * d - b];\n ans.push([a, b]);\n }\n return ans;\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":2146,"value":"prmEps = 1e-4","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":2149,"value":"valEps = 1e-11","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":2157,"value":"`prm` is a complex number","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":2144,"value":"rate = function(prm, funct) {\n let smidge = 0.1 * prmEps;\n let smidge_re = math.complex(smidge, 0);\n let smidge_im = math.complex(0, smidge);\n let hrate = math.divide(\n math.subtract(funct(math.add(prm, smidge_re)), funct(math.subtract(prm, smidge_re))),\n (math.multiply(math.complex(2, 0), smidge_re)));\n \n let vrate = math.divide(\n math.subtract(funct(math.add(prm, smidge_im)), funct(math.subtract(prm, smidge_im))),\n (math.multiply(math.complex(2, 0), smidge_im)));\n \n return math.multiply(math.complex(0.5, 0), math.add(hrate, vrate));\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":2141,"value":"newtonSolver = function (prm0, funct, rate) {\n const maxItr = 1000;\n for (let i = 0; i < maxItr; i++) {\n let r = rate(prm0, funct);\n let prm = math.subtract(prm0, math.divide(funct(prm0), r));\n let eps = funct(prm0).abs();\n if (eps <= valEps &&\n math.subtract(prm, prm0).abs() <= prmEps) {\n return prm;\n } else {\n prm0 = prm;\n }\n }\n return \"Newton Method's Failed\";\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":2055,"value":"tracePoly = (p, q, t_a, t_B, t_aB) => {\n if (p == 0 && q == 1) return t_a;\n if (p == 1 && q == 0) return t_B;\n\n let [p1, q1] = [0, 1];\n let [p2, q2] = [1, 0];\n let [p3, q3] = [1, 1];\n\n let t_u = t_a;\n let t_v = t_B;\n let t_uv = t_aB;\n let maxIter = 100;\n\n while ((p3/q3 != p/q) && maxIter--) {\n if (p/q < p3/q3) {\n [p2, q2] = [p3, q3];\n [p3, q3] = [p1 + p3, q1 + q3];\n\n let temp = t_uv;\n t_uv = math.subtract(math.multiply(t_u, t_uv), t_v);\n t_v = temp;\n } else {\n [p1, q1] = [p3, q3];\n [p3, q3] = [p3 + p2, q3 + q2];\n\n let temp = t_uv;\n t_uv = math.subtract(math.multiply(t_v, t_uv), t_u);\n t_u = temp;\n }\n }\n\n if (maxIter == 0) {\n console.log(\"Error during Trace function\");\n return \"Error during Trace function\";\n }\n\n return t_uv;\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":2164,"value":"`mu` is a complex number","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":2161,"value":"getTraceEqn = function (p, q, val, mu) {\n return function (mu) {\n return math.add(math.complex(-val, 0), tracePoly(p, q, \n math.multiply(mu, math.complex(0, -1)),\n math.complex(2, 0),\n math.add(math.multiply(mu, math.complex(0, -1)), math.complex(0, 2))));\n };\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":2171,"value":"Will compute the boundary for elements in the farey sequence with denom `denom` and return an array of `[p, q, mu]` numbers.","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":2166,"value":"$3c","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":2175,"value":"Try the boundary function for `denom=11`","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":2177,"value":"boundary_example = {\n let b = boundary(50);\n return b;\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":2309,"value":"Plot Maskit Slice","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":2263,"value":"plot = {\n let b = boundary_example\n .map(([p, q, complex, v]) => ({x: complex.re, y: complex.im, v: p/q, p, q }));\n //let b = boundary_example[5].map(([p, q, complex]) => ({x: complex.re, y: complex.im }));\n return Plot.plot({\n height: 100,\n width: 300,\n // y: {domain: [1.5, 2], range: [100, 0]},\n marks: [\n Plot.areaY(b, {x: \"x\", y: \"y\", stroke: \"none\", fill: \"gray\", z: null, y1: 1.5}),\n ]\n });\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":2316,"value":"## Compute Texture for Cusp","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":2318,"value":"fixedPointsForCusp = {\n // get a random element in the maskit slice\n // let elem = boundary_example.filter(([p, q]) => p == 2 && q == 15)[0];\n let elem = boundary_example.filter(([p, q]) => p/q > 0.13)[8];\n let [p, q, µ] = elem;\n console.log('p, q', [p, q]);\n const t_a = math.multiply(-1, math.multiply(µ, math.complex(0, 1)));\n\n // matrices\n const a = math.matrix([\n [t_a, math.complex(0, -1)],\n [math.complex(0, -1), 0]\n ]);\n \n const b = math.matrix([\n [1, 2],\n [0, 1]\n ]);\n\n const A = math.inv(a);\n const B = math.inv(b);\n \n const lookup = {a, b, A, B};\n \n let word = getWord(p, q);\n console.log('word', word);\n let perm = getCyclicPermutation(word);\n console.log('perm', perm);\n let matrices = sortCyclicPermutation(perm);\n matrices = matrices.map(mlist => mlist.reduce((acum, val) => math.multiply(lookup[val], acum), math.identity(2)));\n let fixedPoints = matrices.map(m => fixedPoint(m)[0]);\n //fixedPoints.sort((a, b) => a.re - b.re);\n return fixedPoints;\n \n \n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":2334,"value":"p = {\nlet fp1 = fixedPointsForCusp.map((fp, i) => ({i, x: fp.re, y: fp.im}));\nlet fp2 = fixedPointsForCusp.map((fp, i) => ({i, x: fp.re, y: fp.im}));\nreturn Plot.plot({\n marks: [\n Plot.line(fp1, {x: \"x\", y: \"y\", stroke: \"red\"}),\n Plot.dot(fp1, {x: \"x\", y: \"y\", stroke: \"green\"})\n ]\n})\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":2056,"value":"*Theory confirms that among all possible solutions to the trace equation (for a fixed p/q), exactly one will be discrete.t These were the transition points we sought. Theory also told us that no value of mu with imaginary part between -1 and +1 could give a discrete group, while values with imaginary part more than +2 or less than -2 would always be discrete.*","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":2069,"value":"$3d","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":179,"value":"md`## Imports`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":158,"value":"import {slider} from \"@jashkenas/inputs\"","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":1438,"value":"import {select} from \"@jashkenas/inputs\"\n","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":260,"value":"mathjs = require('mathjs');","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":271,"value":"math = mathjs.create(mathjs.all, {});","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":16,"value":"d3 = require(\"d3@5\")","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"}],"files":"$W3e","stdlib":"1","titling":"auto"}} 2e:"$2d:body"