2c:null 2d:["$","title",null,{"children":"Periodic Orbits of the Modular Flow | Christopher Henn | Observable"}] 32:T425,// This algorithm is due to Mostovoy latticeToExpS1 = m => { // Compute the Voronoi cell of a lattice centered at the origin const cell = voronoiCellForLattice(m); // Pick the top most vertex of the cell const vertex = _.max(cell, ([x, y]) => y); // Generate some points of the lattice near the origin const points = clippedLatticePoints(m); // Lattices either have a rectangular or hexagonal Voronoi cell const isRectangular = cell.length === 4; // Find the 2 or 3 lattice points nearest to our selected vertex const nearest = _.sortBy(points, p => distance(p, vertex)).slice(0, isRectangular ? 2 : 3); // Translate the nearest points so the selected vertex is at the origin. These 2 or 3 nearest points define three lines through the origin, i.e. they define 3 points in RP^1 const translatedNearest = nearest.map(n => sub2(n, vertex)); // Go from RP^1 to S^1 (they are homeomorphic) const latticeInExpS1 = translatedNearest.map(([x, y]) => 2 * (y >= 0 ? cArg([x, y]) : cArg([-x, -y]))) return latticeInExpS1; }33:T579,pathsInFViz = paths => { const size = 400; const domainSize = 4; const xScale = d3.scaleLinear().domain([-domainSize / 2, domainSize / 2]).range([0, size]); const yScale = d3.scaleLinear().domain([0, domainSize]).range([size, 0]); const ctx = DOM.context2d(size, size); // Draw y-axis ctx.strokeStyle = "lightgray"; ctx.lineWidth = 1; ctx.beginPath(); ctx.moveTo(xScale(0), 0); ctx.lineTo(xScale(0), size); ctx.stroke(); // Draw x-axis ctx.beginPath(); ctx.moveTo(0, size); ctx.lineTo(size, size); ctx.stroke(); // Draw the boundary of the fundamental region const originX = xScale(0); const originY = yScale(0); const unitRadius = xScale(1) - xScale(0); // Yucky assumptions about aspect ratio here ctx.strokeStyle = "black"; ctx.lineWidth = 1.5; ctx.beginPath(); ctx.arc(originX, originY, unitRadius, Math.PI / 3 + Math.PI, 2 * Math.PI / 3 + Math.PI); ctx.stroke(); ctx.beginPath(); ctx.moveTo(xScale(- 1 / 2), yScale(Math.sqrt(3) / 2)); ctx.lineTo(xScale(- 1 / 2), 0); ctx.moveTo(xScale(1 / 2), yScale(Math.sqrt(3) / 2)); ctx.lineTo(xScale(1 / 2), 0); ctx.stroke(); // Draw paths for (let path of paths) { ctx.beginPath(); ctx.moveTo(xScale(path[0]), xScale(path[0])); for (let [x, y] of path) { ctx.lineTo(xScale(x), yScale(y)); } ctx.stroke(); } return ctx.canvas; }34:[] 30:{"id":"850437833e1193e0","slug":"the-modular-flow-in-tex-mathcal-l-and-tex-mathbb-h","title":"Periodic Orbits of the Modular Flow","thumbnail":"f868e97b41ef464f2a4fc244471ec9559111c22d27ec7de8a9959edf20229602","access_level":"public","create_time":"$D2018-04-05T20:38:20.729Z","update_time":"$D2021-04-07T03:07:48.244Z","view_time":null,"first_public_time":"$D2018-04-06T18:27:45.565Z","fork":false,"stars":4,"starred":false,"owner":{"type":"team","id":"f3428374116cf9e8","login":"chnn","name":"Christopher Henn","avatar_url":"https://avatars-next.observableusercontent.com/78bb870dc1a41660b957b0e3e7c72e7a1393fc8fc056e370a3d26079dff4b2c9","tier":"starter_2024"},"version":2157,"publish_version":2157,"role":null,"authored":false,"authors":[{"type":"individual","id":"c1d31c67ff33b8b2","login":"chnn","name":"Christopher Henn","avatar_url":"https://avatars-next.observableusercontent.com/78bb870dc1a41660b957b0e3e7c72e7a1393fc8fc056e370a3d26079dff4b2c9","tier":null}],"members":[{"type":"individual","id":"c1d31c67ff33b8b2","login":"chnn","name":"Christopher Henn","avatar_url":"https://avatars-next.observableusercontent.com/78bb870dc1a41660b957b0e3e7c72e7a1393fc8fc056e370a3d26079dff4b2c9","tier":null,"role":"editor"}],"comments":[],"custom_thumbnail":false,"trashed":false,"subscribed":null,"forks":2,"collection":null,"type":"notebook","body":{"title":"Periodic Orbits of the Modular Flow","cells":[{"id":0,"value":"md`# Periodic Orbits of the Modular Flow`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":215,"value":"md`We visualize a [periodic orbit of the modular flow](https://golem.ph.utexas.edu/category/2014/04/the_modular_flow_on_the_space.html) in the space of lattices for ${tex.block`A = \\begin{pmatrix}${a[0][0]} & ${a[0][1]} \\\\ ${a[1][0]} & ${a[1][1]}\\end{pmatrix} \\in \\mathrm{SL}_2(\\mathbb{Z})`}`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":195,"value":"latticeViz(orbitForA[i])","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":1593,"value":"{\n if (!this) {\n const input = html``;\n \n input.oninput = () => mutable time = input.valueAsNumber;\n \n return html`${input} ${tex`\\text{time} = ${time.toFixed(2)}`}`; \n }\n \n const input = this.querySelector(\"input[type='range']\");\n\n input.max = orbitTime(a) - 0.01;\n input.value = time;\n \n this.replaceChild(tex`\\text{time} = ${time.toFixed(2)}`, this.querySelector(\"span\"));\n \n return this;\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":1074,"value":"viewof animate = html``","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":1116,"value":"a = [[15, -8],\n [2, -1]]","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":2094,"value":"md`## In ${tex`\\mathrm{exp}_{\\{2,3\\}} S^1`}`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":1755,"value":"multiCanvas([expS1CircleViz(orbitForAInExpS1[i]), expS1FlatViz(orbitForAInExpS1, i)])","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":2007,"value":"{\n if (!this) {\n const input = html``;\n \n input.oninput = () => mutable time = input.valueAsNumber;\n \n return html`${input} ${tex`\\text{time} = ${time.toFixed(2)}`}`; \n }\n \n const input = this.querySelector(\"input[type='range']\");\n\n input.max = orbitTime(a) - 0.01;\n input.value = time;\n \n this.replaceChild(tex`\\text{time} = ${time.toFixed(2)}`, this.querySelector(\"span\"));\n \n return this;\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":916,"value":"md`## In the Upper Half Plane and Modular Surface`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":721,"value":"pathsInFViz(pathInF(orbitForAInH))","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":2153,"value":"{\n if (!this) {\n const input = html``;\n \n input.oninput = () => mutable time = input.valueAsNumber;\n \n return html`${input} ${tex`\\text{time} = ${time.toFixed(2)}`}`; \n }\n \n const input = this.querySelector(\"input[type='range']\");\n\n input.max = orbitTime(a) - 0.01;\n input.value = time;\n \n this.replaceChild(tex`\\text{time} = ${time.toFixed(2)}`, this.querySelector(\"span\"));\n \n return this;\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":2131,"value":"md`

\n\n## Appendix: Visualization Code`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":898,"value":"orbitTime = m => Math.log(Math.abs(eigenvalues(m)[0]))","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":901,"value":"// Gives a single lattice in the orbit corresponding to m\nlatticeInOrbit = m => inverse(transpose(eigenvectors(m)))","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":910,"value":"deltaT = t => [[Math.exp(t), 0],\n [0, Math.exp(-t)]]","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":913,"value":"// How many points in the orbit to compute\norbitResolution = 500","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":904,"value":"orbit = m => {\n const p = latticeInOrbit(m);\n const t = orbitTime(m);\n\n return d3\n .range(0, t, t / orbitResolution)\n .map(tt => mult22(deltaT(tt), p));\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":930,"value":"// Returns boolean indicating whether a vector is in F\ninF = ([x, y]) => x >= - 1 / 2 && x <= 1 / 2 && norm([x, y]) >= 1","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":933,"value":"tau = m => {\n // Rotate generators by pi if they are in lower half plane\n const [w1, w2] = transpose(m).map(orientGenerator);\n \n // Compute tau, depending on the angle between generators\n return cArg(w1) - cArg(w2) > 0 ? cDiv(w1, w2) : cDiv(w2, w1);\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":936,"value":"// Inverse mobius transform. This formula is brought to you by Mathematica's `ComplexExpand` and Satan\ninvMobius = ([[a, b], [c, d]], [x, y]) => {\n const denom = (a - c * x) ** 2 + c ** 2 * y ** 2;\n \n return [\n (-a * b + b * c * x + a * d * x - c * d * x * x - c * d * y * y) / denom,\n (a * d * y - b * c * y) / denom\n ];\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":939,"value":"// Find the unique mobius transform that sends `z` to F\ntranslateFromF = z => {\n const s = [[0, -1], [1, 0]];\n const t = [[1, 1], [0, 1]];\n let [a, b] = z;\n let gamma = [[1, 0], [0, 1]]; // Identity matrix\n \n while (!inF([a, b])) {\n let next;\n \n if (a < -0.5) {\n next = inverse(t);\n } else if (a > 0.5) {\n next = t;\n } else {\n next = s;\n }\n \n gamma = mult22(gamma, next);\n [a, b] = invMobius(gamma, z);\n }\n \n return gamma;\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":942,"value":"pathInF = path => {\n const translates = path.map(translateFromF);\n const segments = [];\n let segment = [];\n let i = 0;\n \n while (i < path.length) {\n segment.push(invMobius(translates[i], path[i]));\n \n if (i === path.length - 1 || !equal22(translates[i], translates[i + 1])) {\n segments.push(segment);\n segment = [];\n }\n \n i++;\n }\n \n return segments;\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":1381,"value":"voronoiCellForLattice = m => {\n const clip = latticeOriginClip(m);\n const ps = clippedLatticePoints(m, clip);\n const voronoi = d3.voronoi().extent([[-clip, -clip], [clip, clip]])(ps);\n const originPolygon = voronoi.polygons().find(({data: [x, y]}) => x === 0 && y === 0);\n\n return originPolygon;\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":1428,"value":"$32","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":29,"value":"transpose = ([[a, b], [c, d]]) => [[a, c], [b, d]]","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":40,"value":"det = ([[a, b], [c, d]]) => a * d - b * c","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":44,"value":"inverse = ([[a, b], [c, d]]) => {\n const denom = (- b) * c + a * d;\n \n return [[d / denom, - b / denom], [- c / denom, a / denom]];\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":671,"value":"eigendata = ([[a, b], [c, d]]) => {\n const sq = Math.sqrt(a * a + 4 * b * c - 2 * a * d + d * d);\n const eigendata = [\n [(a + d - sq) / 2, [-((-a + d + sq) / (2 * c)), 1]],\n [(a + d + sq) / 2, [-((-a + d - sq) / (2 * c)), 1]],\n ];\n \n return _.sortBy(eigendata, d => 1 / d[0]);\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":52,"value":"eigenvalues = m => eigendata(m).map(d => d[0])","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":696,"value":"eigenvectors = m => eigendata(m).map(d => d[1])","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":423,"value":"// Multiply two 2x2 matrices together\nmult22 = ([[a, b], [c, d]], [[x, y], [w, z]]) => [[b * w + a * x, a * y + b * z], [d * w + c * x, c * y + d * z]]","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":802,"value":"equal22 = ([[a, b], [c, d]], [[w, x], [y, z]]) => a === w && b === x && c === y && d === z","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":426,"value":"add2 = ([a, b], [c, d]) => [a + c, b + d]","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":1457,"value":"sub2 = ([a, b], [c, d]) => [a - c, b - d]","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":1486,"value":"radToR2 = r => [Math.cos(r), Math.sin(r)]","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":1443,"value":"distance = ([x0, y0], [x1, y1]) => Math.sqrt((y1 - y0) ** 2 + (x1 - x0) ** 2)","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":799,"value":"norm = ([x, y]) => distance([0, 0], [x, y])","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":790,"value":"cArg = ([a, b]) => {\n if (a > 0) {\n return Math.atan(b / a);\n } else if (a < 0 && b >= 0) {\n return Math.atan(b / a) + Math.PI;\n } else if (a < 0 && b < 0) {\n return Math.atan(b / a) - Math.PI;\n } else if (a === 0 && b > 0) {\n return Math.PI / 2;\n } else if (a === 0 ** b < 0) {\n return 0 - Math.PI / 2;\n } else if (a === 0 && b === 0) {\n throw new Error('arg not defined');\n }\n\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":793,"value":"cMult = ([a, b], [c, d]) => [(a * c) - (b * d), (b * c) + (a * d)]","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":796,"value":"cDiv = ([a, b], [c, d]) => {\n const denom = c * c + d * d;\n\n return [(a * c + b * d) / denom, (b * c - a * d) / denom];\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":1488,"value":"cExp = ([a, b]) => cMult([Math.exp(a), 0], [Math.cos(b), Math.sin(b)])","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":1962,"value":"// If necessary, reflect vector `[a, b]` across x- and y-axis into the upper half plane\norientGenerator = ([a, b]) => b < 0 ? [-a, -b] : [a, b]","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":107,"value":"latticePoints = ([[a, b], [c, d]], size = 10) => {\n const range = d3.range(- size, size);\n const ijs = d3.cross(range, range);\n \n return ijs.map(([i, j]) => add2([a * i, c * i], [b * j, d * j]));\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":827,"value":"d3 = require(\"d3@5\")","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":832,"value":"_ = require(\"underscore\")","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":1603,"value":"mutable time = 0.1","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":990,"value":"animationTicker = {\n const t1 = orbitTime(a);\n\n while (animate) { \n yield Promises.delay(20);\n mutable time = +((time + 0.01) % t1).toFixed(2);\n }\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":1606,"value":"i = Math.round((time / orbitTime(a)) * orbitForA.length)","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":1625,"value":"// timeControl = () => {\n// const input = this\n// ? this.querySelector(\"input\")\n// : html``;\n\n// input.value = time;\n// input.oninput = () => mutable time = input.valueAsNumber;\n \n// if (animate) {\n// input.disabled = true;\n// }\n \n// return html`${input} ${tex`\\text{time} = ${time}`}`;\n// }","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":1640,"value":"latticeOriginClip = ([[a, b], [c, d]]) => Math.max(...[a, b, c, d].map(Math.abs)) * 3","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":1643,"value":"clippedLatticePoints = (m, clip = latticeOriginClip(m)) => {\n const inClip = ([x, y]) => x >= -clip && x <= clip && y >= -clip && y <= clip;\n \n return latticePoints(m, 20).filter(inClip);\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":1779,"value":"multiCanvas = (cs, spacing = 30) => html`\n \n\n

\n ${cs}\n

\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":838,"value":"orbitForA = orbit(a)","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":865,"value":"latticeViz = m => {\n const domain = [-2, 2];\n const width = 600;\n const height = 400;\n const xScale = d3.scaleLinear().domain(domain).range([0, width]);\n const yScale = d3.scaleLinear().domain(domain).range([0, height]);\n const points = latticePoints(m, 50);\n const ctx = DOM.context2d(width, height);\n const r = 2.5;\n \n ctx.fillStyle = 'black';\n \n for (let [x, y] of points) {\n ctx.beginPath();\n ctx.arc(xScale(x), yScale(y), r, 0, 2 * Math.PI);\n ctx.fill();\n }\n \n return ctx.canvas;\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":868,"value":"orbitForAInH = orbitForA.slice(0, i + 1).map(tau)","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":1686,"value":"orbitForAInExpS1 = orbitForA.map(latticeToExpS1)","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":2118,"value":"pathInHViz = path => {\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":472,"value":"$33","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":1655,"value":"expS1CircleViz = ps => {\n const size = 300;\n const ctx = DOM.context2d(size, size);\n const domain = [-1.1, 1.1];\n const xScale = d3.scaleLinear().domain(domain).range([0, size]);\n const yScale = d3.scaleLinear().domain(domain).range([size, 0]);\n const originX = xScale(0);\n const originY = yScale(0);\n const unitRadius = xScale(1) - xScale(0);\n \n ctx.beginPath();\n ctx.arc(originX, originY, unitRadius, 0, 2 * Math.PI);\n ctx.stroke();\n \n ctx.beginPath();\n for (let p of ps) {\n const [x0, y0] = radToR2(p);\n const [x, y] = [xScale(x0), yScale(y0)];\n ctx.moveTo(x, y);\n ctx.arc(x, y, 4, 0, 2 * Math.PI);\n }\n ctx.fill();\n \n return ctx.canvas;\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":1673,"value":"expS1FlatViz = (path, i = path.length - 1, width = 200, height = 300) => {\n const xScale = d3.scaleLinear().domain([0, 2 * Math.PI]).range([0, width]);\n const yScale = d3.scaleLinear().domain([0, path.length - 1]).range([0, height]);\n const points = [];\n \n let j = 0;\n \n while (j <= i) {\n const r = yScale(j);\n \n for (let rad of path[j]) {\n points.push([xScale(rad % (2 * Math.PI)), r]);\n }\n \n j++;\n }\n \n const ctx = DOM.context2d(width, height);\n \n ctx.beginPath();\n for (let [x, y] of points) {\n ctx.moveTo(x, y);\n ctx.arc(x, y, 0.5, 0, 2 * Math.PI);\n }\n ctx.fill();\n \n ctx.lineWidth = 2;\n ctx.strokeRect(0, 0, width, height);\n \n return ctx.canvas;\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"}],"files":"$W34","stdlib":"1","titling":"auto"}} 31:"$30:body"