29:null 2a:["$","title",null,{"children":"Visualizing Juggling: Appendix | Jo Wood | Observable"}] 2f:T873,### Terminology - Units of time are measured in integer _beats_ and we record the state of a juggling system only at these beat markers. - A _throw sequence_ is one or more throws under the vanilla siteswap conditions. - A _catch sequence_ is sequence of balls as they land in either hand over time. For example `🔵🟠🔴🟠🔵` indicates the blue ball lands first, then the yellow ball, then the red etc. - A _pattern_ is a cyclical throw sequence that can be repeated infinitely. - A _prime pattern_ is a cyclical throw sequence that does not involve any sub-cycles within it (a _simple cycle_ in graph theoretic terms), for example **5304**. If a throw sequence does have sub-sequences, it is a _composite pattern_, for example **50244** contains the subsequence **42**) - ${tex`b`} is the number of balls in a pattern or sequence. - ${tex`H`} is the maximum height we are permitted to throw a ball and corresponds to the number of beats it takes for a thrown ball to land. - A _landing schedule_ presents the state of the juggle at a point in time. It comprises a sequence of ${tex`H`} binary digts representing the state of the balls at times ${tex`[t,t+1,t+2,..., t+H-1]`} where 1 indicates a ball in hand, 0 a ball in the air. - ${tex`h`} is the height / number of beats taken to land after throwing a ball. - Throws of all heights are indicated with a single digit with those higher than 9 being symbolised by _a_, _b_, _c_ etc. - A siteswap _word_ can therefore comprise ${tex`n`} digits ${tex`[h_1h_2\dots, h_n]`} from the alphabet ${tex`\{0,1,2,...,H \}`}. It is usually reduced to its _canonical form_ such that the repeating pattern is represented with the smallest number of digits (so **51**, rather than **5151** and **3** rather than **333**) rotated to its highest lexicographic value (highest throw first) starting with the highest throw), so **423** rather than 234 or 342. - Siteswap notation collapses ${tex`C_2`} symmetry. i.e. a throw from left hand to right hand of height ${tex`h`} is considered identical to a throw from right hand to left hand of the same height. - The number of digits in a canonical siteswap is its _period_.30:Tb11,function stave1View(options = {}) { const { w = 640, h = 150, cycles = 2, showLevels = false, minHeight = 0, timeLabel = "", yType = "linear", exp = 1, siteswap = "3", ballFilter = () => true, tFormat = (n) => (d) => (d % (n / 2)) + 1, catchTrajMaxBeat = null, // if number: omit throws/catches whose catch beat > this clrs = {}, extra = [] } = options; const maxHeight = d3.max([...parseSiteswap(siteswap), minHeight]); const pattern = makePatternWithIds({siteswap}); const n = pattern.loopDur; const throws = pattern.throws; const maxX = n * cycles; const beats = d3.range(0, Math.ceil(maxX)); const eps = 1e-9; const data = beats .flatMap((d) => { const thr = throws[d % n]; const ball = thr.ballId; const v = thr.value; if (ball < 0) { return []; } const xCatch = d + v; if (catchTrajMaxBeat != null && xCatch > catchTrajMaxBeat + eps) { return []; } return [ {x: d, y: 0, ball}, {x: (d + d + v) / 2, y: v, ball}, {x: xCatch, y: 0, ball} ]; }) .filter(ballFilter); const catchData = d3 .range(0, n * cycles) .map((d) => ({x: d + throws[d % n].value, ball: throws[d % n].ballId})) .filter((d) => d.ball >= 0) .filter((d) => catchTrajMaxBeat == null || d.x <= catchTrajMaxBeat + eps) .filter(ballFilter); const handData = d3 .range(0, n * cycles) .map((d, i) => ({x: i, y: d % 2 ? maxHeight : 0})); return Plot.plot({ width: w, height: h, marginTop: 10, marginLeft: 20, x: {domain: [0, n * cycles]}, y: { domain: [0, maxHeight], type: yType, exponent: exp, axis: showLevels, ticks: maxHeight, label: null, labelArrow: null, tickSize: 0 }, color: {type: "categorical", ...clrs}, marks: [ Plot.rectY(handData, { x1: (d) => d.x - 0.5, x2: (d) => d.x + 0.5, y: "y", fill: "#eee" }), showLevels ? Plot.gridY() : [], Plot.gridX({ticks: n * cycles}), Plot.gridX({ticks: d3.range(0, n * cycles + 1, n / 2), opacity: 1}), Plot.gridX({ ticks: d3.range(0, n * cycles + 1, n), opacity: 1, strokeWidth: 2 }), Plot.axisX({ ticks: n * cycles, tickSize: 0, tickFormat: tFormat(n), label: timeLabel }), Plot.ruleY([0], {opacity: 0.3}), Plot.lineY(data, { x: "x", y: "y", stroke: "ball", strokeWidth: Math.min(4, w / (6 * catchData.length)), curve: "catmull-rom" }), Plot.dot(catchData, { x: "x", y: 0, fill: "ball", r: Math.min(6, w / (4 * catchData.length)) }), extra ] }); }31:T6e1,function stave2View(options = {}) { const { w = 640, h = 140, cycles = 2, showHands = true, timeLabel = "", siteswap = "3", ballFilter = () => true } = options; const pattern = makePatternWithIds({siteswap}); const n = pattern.loopDur; const throws = pattern.throws; const data = d3 .range(0, n * cycles) .map((d) => ({ x1: d, y1: throws[d % n].fromHand, x2: d + throws[d % n].value, y2: throws[d % n].toHand, ball: throws[d % n].ballId })) .filter(ballFilter); return Plot.plot({ width: w, height: h, x: {domain: [0, n * cycles]}, y: {axis: null}, color: {type: "categorical"}, marks: [ Plot.gridX({ticks: n * cycles}), Plot.gridX({ticks: d3.range(0, n * cycles + 1, n / 2), opacity: 1}), Plot.gridX({ ticks: d3.range(0, n * cycles + 1, n), opacity: 1, strokeWidth: 2 }), Plot.axisX({ ticks: n * cycles, tickSize: 0, tickFormat: (d) => (d % (n / 2)) + 1, label: timeLabel }), Plot.axisX({ ticks: d3.range(0, n * cycles + 1, n / 2), fontSize: 12, fontWeight: "bold", stroke: "white", tickFormat: (d) => (d % (n / 2)) + 1, label: null }), showHands ? Plot.ruleY(["L", "R"], {opacity: 0.3}) : [], Plot.arrow(data, { x1: (d) => (d.y1 === "L" ? d.x1 : d.x2), y1: (d) => (d.y1 === "L" ? d.y1 : d.y2), x2: (d) => (d.y1 === "L" ? d.x2 : d.x1), y2: (d) => (d.y1 === "L" ? d.y2 : d.y1), stroke: "ball", strokeWidth: 4, headLength: 0, bend: -20 }), Plot.dot(data, {x: "x2", y: "y2", fill: "ball", r: 6}) ] }); }32:T1172,function radialStave(options = {}) { const { w = 400, h = w, cycles = 1, showLevels = false, minHeight = 3, timeLabel = "", yType = "linear", exp = 1, siteswap = "3", ballFilter = () => true, catchTrajMaxBeat = null, clrs = {}, extra = [], inner = 0.333, outer = 0.95, rotate = 0, samples = 60 } = options; const {loopDur: n, throws} = makePatternWithIds({siteswap}); const maxX = n * cycles; const maxHeight = d3.max([minHeight, ...throws.map((d) => d.value)]); const scaleMax = Math.max(1, maxHeight); const eps = 1e-9; const rScale = ( yType === "pow" ? d3.scalePow().exponent(exp) : yType === "sqrt" ? d3.scaleSqrt() : d3.scaleLinear() ) .domain([0, scaleMax]) .range([inner, outer]); const point = (beat, height) => { const a = ((rotate + (360 * beat) / maxX) * Math.PI) / 180; const r = rScale(height); return {x: r * Math.sin(a), y: r * Math.cos(a)}; }; const beatData = d3.range(maxX).map((beat) => { const {value, ballId: ball} = throws[beat % n]; return {beat, value, ball, xCatch: beat + value}; }); const keepBall = (ball, x) => ball >= 0 && ballFilter({x, y: 0, ball}); const keepCatch = ({xCatch}) => catchTrajMaxBeat == null || xCatch <= catchTrajMaxBeat + eps; const data = beatData .filter((d) => keepBall(d.ball, d.beat) && keepCatch(d)) .flatMap(({beat, value, ball}) => d3.range(samples + 1).map((i) => { const t = i / samples; return { ...point(beat + value * t, 4 * value * t * (1 - t)), ball, trajectory: beat }; }) ); const catchData = beatData .filter((d) => keepBall(d.ball, d.xCatch) && keepCatch(d)) .map(({xCatch, ball}) => ({...point(xCatch, 0), ball})); const handWedges = d3.range(1, maxX, 2).flatMap((beat) => { const t = d3.range(25).map((i) => i / 24); return [ ...t.map((d) => point(beat - 0.5 + d, scaleMax)), ...t.map((d) => point(beat + 0.5 - d, 0)) ].map((p) => ({...p, beat})); }); const rings = d3.range(Math.floor(maxHeight) + 1).flatMap((level) => d3.range(181).map((i) => ({ ...point((maxX * i) / 180, level), level })) ); const spokes = d3.range(maxX).flatMap((beat) => [ {...point(beat, 0), beat}, {...point(beat, scaleMax), beat} ]); const cycleSpokes = cycles > 1 ? d3.range(0, maxX, n).flatMap((beat) => [ {...point(beat, 0), beat}, {...point(beat, scaleMax), beat} ]) : []; const levelData = showLevels ? d3 .range(1, Math.floor(maxHeight) + 1) .map((level) => ({...point(0, level), level})) : []; return Plot.plot({ width: w, height: h, margin: 10, x: {domain: [-1, 1], axis: null}, y: {domain: [-1, 1], axis: null}, color: {type: "categorical", ...clrs}, marks: [ // Alternating left/right hand background Plot.line(handWedges, { x: "x", y: "y", z: "beat", fill: "#eee", stroke: null }), // Throw-height rings Plot.line(rings, { x: "x", y: "y", z: "level", strokeWidth: 0.1 }), // Beat spokes Plot.line(spokes, { x: "x", y: "y", z: "beat", strokeWidth: 0.2 }), // Cycle boundaries (bar maker) only if we have more than one cycle cycles > 1 ? Plot.line(cycleSpokes, { x: "x", y: "y", z: "beat", strokeOpacity: 0.6, strokeWidth: 2 }) : [], // Throw trajectories Plot.line(data, { x: "x", y: "y", z: "trajectory", stroke: "ball", strokeWidth: Math.min(4, w / (6 * catchData.length)) }), // Catch dots Plot.dot(catchData, { x: "x", y: "y", fill: "ball", r: Math.min(6, w / (4 * catchData.length)) }), // Throw-height labels showLevels ? Plot.text(levelData, { x: "x", y: "y", text: "level", dx: 5, textAnchor: "start" }) : [], // Central label timeLabel ? Plot.text([timeLabel], { x: 0, y: 0, text: (d) => d }) : [], extra ] }); }33:T416,function animate(t, options = {}) { const { dwell = 0.2, handSep = 6, catchOutward = 1, launchInward = 1, pattern = makePatternWithIds({ dwell, handSep, catchOutward, launchInward }), w = 340, h = 400, inset = 40, speed = 10, yDomain = [0, 9], yLabels = true, yType = "linear", exp = 1, extra = [], extraAfter = [] } = options; const pts = positionsAtPhase(((t * speed) % 1) * pattern.loopDur, pattern); return Plot.plot({ width: w, height: h, inset, x: {domain: [-10, 10], axis: null}, y: { ticks: yDomain[1], tickSize: 0, domain: yDomain, grid: yLabels, axis: yLabels, type: yType, exponent: exp }, color: {domain: d3.range(0, 10)}, marks: [ extra, Plot.dot(pts, { x: "x", y: "y", r: 12, fill: "ballId", stroke: "black", strokeOpacity: (d) => (d.state === "hand" ? 1 : 0) }), extraAfter ] }); }34:T938,// All siteswaps with either a maximum or given number of balls and a given period. // This will generate non-canonical siteswap words including rotations and compound patterns // If exactBalls is false, results are grouped by number of balls ([0,1,2,...]) function enumerateSiteswaps(options = {}) { const { maxBalls = 3, period = 3, exactBalls = false, groupByBalls = true } = options; const exact = exactBalls === false ? null : exactBalls; const format = (seq) => seq.map(encodeDigit).join(""); const seqs = []; const perm = Array(period); const used = Array(period).fill(false); const groups = Array.from({length: maxBalls + 1}, () => []); const compareSeqs = (a, b) => { for (let i = 0; i < period; i += 1) { if (a[i] !== b[i]) { return a[i] - b[i]; } } return 0; }; const addSeq = (seq) => { const balls = seq.reduce((sum, h) => sum + h, 0) / period; if (groupByBalls) { groups[balls].push(seq); } else { seqs.push(seq); } }; const emit = (base, extraBallsLeft, i = 0, extras = []) => { if (i === period) { if (extraBallsLeft === 0) { addSeq(base.map((h, j) => h + period * extras[j])); } return; } for (let k = 0; k <= extraBallsLeft; k += 1) { extras.push(k); emit(base, extraBallsLeft - k, i + 1, extras); extras.pop(); } }; const visitPerms = (i = 0) => { if (i === period) { const base = perm.map((t, j) => (t - j + period) % period); const baseBalls = base.reduce((sum, h) => sum + h, 0) / period; if (exact === null) { for ( let extraBalls = 0; extraBalls <= maxBalls - baseBalls; extraBalls += 1 ) { emit(base, extraBalls); } } else { const extraBalls = exact - baseBalls; if (extraBalls >= 0) { emit(base, extraBalls); } } return; } for (let t = 0; t < period; t += 1) { if (!used[t]) { used[t] = true; perm[i] = t; visitPerms(i + 1); used[t] = false; } } }; visitPerms(); if (groupByBalls) { const out = groups.map((group) => group.sort(compareSeqs).map(format)); return exact === null ? out : [out[exact]]; } return seqs.sort(compareSeqs).map(format); }35:T755,function makePatternWithIds(options = {}) { const config = resolveConfig(options); const {siteswap, cycles, startHand} = config; const seq = Array.isArray(siteswap) ? siteswap : parseSiteswap(siteswap); const P = seq.length; const numBalls = seq.reduce((a, v) => a + v, 0) / P; if (!Number.isInteger(numBalls)) { throw new Error("Invalid siteswap: average throw must be an integer."); } const useCycles = cycles === "auto" ? minCycles(seq) : cycles; // Build throws const throws = []; let [hand, beat] = [startHand, 0]; for (let k = 0; k < useCycles; k++) { for (const value of seq) { throws.push(makeThrow(value, beat++, hand, config)); hand = otherHand(hand); // beat++; } } const n = throws.length; // Next indices and ball ids (orbit labels) const nextIdx = throws.map((th, i) => th.isIdle ? null : (i + th.value) % n ); const ballId = new Array(n).fill(-1); let nextBallId = 0; for (let i = 0; i < n; i++) { if (ballId[i] !== -1 || throws[i].isIdle) { continue; } let j = i; while (!throws[j].isIdle && ballId[j] === -1) { ballId[j] = nextBallId; j = nextIdx[j]; } nextBallId += 1; } // Attach ids and modular timings for (let i = 0; i < n; i++) { const th = throws[i]; th.ballId = ballId[i]; th.t0 = mod(th.tLaunchAbs, n); th.t1 = mod(th.tCatchAbs, n); th.nextIdx = nextIdx[i]; } // Hold intervals and endpoints for (const th of throws) { if (th.value === 0) { continue; // no geometry; hand is empty this beat } const nxt = throws[th.nextIdx]; th.h0 = th.t1; th.h1 = nxt.t0; th.holdHand = th.toHand; th.xHoldStart = th.x1; th.yHoldStart = 0; th.xHoldEnd = nxt.x0; th.yHoldEnd = 0; th.carryDir = handDir(th.holdHand); } return {throws, loopDur: n, numBalls}; }36:T89b,const patterns = [ ["3", "cascade"], ["42", "2 in 1 with hold", 15], ["51", "shower", 20], ["60", "three in one", 15], ["423", "the 'W'", 15], ["441", "fake box", 5], ["504", "complete waste of a 5-ball", 7], ["522", "high 3 cascade (practice for 5-ball cascade)", 5], ["531", "reversing half shower", 15], ["603", "high W", 20], ["612", "asynchronous box", 20], ["630", "see-saw high W", 20], ["711", "7-height fake box", 7], ["720", "hot potato cascade", 7], ["801", "high fake box", 7], ["900", "high hot potato cascade", 7], ["4233", "", 15], ["4413", "one-sided fake box", 20], ["4440", "gapped fountain", 20], ["5124", ""], ["5223", ""], ["5241", ""], ["5304", ""], ["5313", "asymmetric reversing high shower"], ["5340", ""], ["5511", ""], ["5520", ""], ["6015", ""], ["6024", ""], ["6051", ""], ["6222", ""], ["6231", ""], ["6312", ""], ["6330", ""], ["6411", ""], ["6420", ""], ["7005", "wonky hot potato cascade", 15], ["7023", ""], ["7041", ""], ["7122", ""], ["7131", "high-low shower"], ["7302", ""], ["7401", ""], ["8004", ""], ["8013", ""], ["8040", "high-low three in one"], ["8112", ""], ["8130", ""], ["8202", ""], ["8400", ""], ["9111", ""], ["9120", ""], ["9201", ""], ["9300", ""], // ["b001", ""], ["42333", "Marden's marvel"], ["42423", "Orinoco flow", 3], ["44133", "frostbite", 3], ["45501", ""], ["50505", "snake"], ["52233", "Rubenstein's revenge"], ["52512", "Boston shuffle"], ["53133", "shuffler's mess"], ["55500", "hot potato five"], ["81411", "super box"], ["423333", ""], ["424233", "", 5], ["517131", "one up, one down"], ["522441", "", 20], ["525501", ""], ["531441", ""], ["531522", ""], ["615060", "", 20], ["615150", "leaky shower", 20], ["622440", "", 20], ["622530", ""], ["631440", ""], ["631530", "", 20], ["712611", ""], ["712620", ""], ["720711", "", 7], ["801711", "", 7], ["801720", "", 7], ["802611", ""], ["802620", ""], ["4233333", "", 7], ["4234233", "", 3], ["4242333", ""], ["4242423", "", 7], ["5224233", "Relf's Rubensteins"], ["33333342", "Icelandic shuffle", 5] ];37:T5e9,function suppressDegree1(g, options = {}) { if (g.nodes().length <= 1) { return g; } const toSeq = (w) => (Array.isArray(w) ? w : [w]); const {combine = (a, b) => [...toSeq(a), ...toSeq(b)]} = options; const inEdges = (n) => typeof g.inEdges === "function" ? g.inEdges(n) : g.inNodes(n).map((u) => ({from: u, weight: g.getEdgeWeight(u, n)})); const outEdges = (n) => typeof g.outEdges === "function" ? g.outEdges(n) : g.adjacent(n).map((v) => ({to: v, weight: g.getEdgeWeight(n, v)})); let changed = true; while (changed) { changed = false; for (const n of g.nodes()) { const [ins, outs] = [inEdges(n), outEdges(n)]; // Do not suppress exterior nodes if (ins.length === 0 || outs.length === 0) { continue; } // Avoid suppressing self-loops if (ins.some(({from}) => from === n) || outs.some(({to}) => to === n)) { continue; } if (ins.length === 1) { const {from: u, weight: wIn} = ins[0]; for (const {to: v, weight: wOut} of outs) { g.addEdge(u, v, combine(wIn, wOut, u, n, v)); } g.removeNode(n); changed = true; break; } if (outs.length === 1) { const {to: v, weight: wOut} = outs[0]; for (const {from: u, weight: wIn} of ins) { g.addEdge(u, v, combine(wIn, wOut, u, n, v)); } g.removeNode(n); changed = true; break; } } } return g; }38:Tdaf,function patternGraph(options = {}) { const {maxPeriod = 3, balls = 3} = options; const keyOf = (a) => a.join(","); const prettyOf = (a) => a.map(encodeDigit).join(""); const canon = (a) => canonicalRotation(primitive(a)); const isValid = (a) => { const p = a.length; const seen = new Array(p).fill(false); for (let i = 0; i < p; i++) { const land = (i + a[i]) % p; if (seen[land]) { return false; } seen[land] = true; } return true; }; // Enumerate all valid siteswaps of exact period p with sum = balls * p const enumeratePeriod = (p) => { const targetSum = balls * p; const used = new Array(p).fill(false); const a = new Array(p).fill(0); const out = []; const dfs = (i, remaining) => { if (i === p) { if (remaining === 0) { out.push([...a]); } return; } for (let t = 0, hi = remaining; t <= hi; t++) { const land = (i + t) % p; if (!used[land]) { used[land] = true; a[i] = t; dfs(i + 1, remaining - t); used[land] = false; } } }; dfs(0, targetSum); return out; }; // Build canonical nodes (primitive + rotation collapsed) const nodes = d3 .range(maxPeriod) .map((i) => i + 1) .flatMap((p) => enumeratePeriod(p)) .reduce((m, raw) => { const c = canon(raw); const k = keyOf(c); if (!m.has(k)) { m.set(k, {arr: c, label: prettyOf(c)}); } return m; }, new Map()); // Adjacent (cyclic) unit transfer on array a at i (j=i+1 mod p). // dir +1 is distancing swap; dir -1 is equalising swap. const transfer = (a, i, dir) => { const p = a.length; if (p === 1) { return null; } const j = (i + 1) % p; const b = [...a]; if (dir === 1) { if (b[j] === 0) { return null; } b[i] += 1; b[j] -= 1; } else { if (b[i] === 0) { return null; } b[i] -= 1; b[j] += 1; } return isValid(b) ? b : null; }; const atExpanded = (prim, k) => prim[k % prim.length]; const graph = new Graph(); const seenEdge = new Set(); // store "lo|hi" with canonical keys (comma form) const idOf = (arr) => arr.map(encodeDigit).join(""); const addEdge = (fromArr, toArr) => { const c1 = canon(fromArr); const c2 = canon(toArr); const k1 = keyOf(c1); const k2 = keyOf(c2); if (k1 === k2) { return; } if (!nodes.has(k1) || !nodes.has(k2)) { return; } const [lo, hi] = k1 < k2 ? [k1, k2] : [k2, k1]; const ek = `${lo}|${hi}`; if (seenEdge.has(ek)) { return; } seenEdge.add(ek); const [n1, n2] = [idOf(nodes.get(lo).arr), idOf(nodes.get(hi).arr)]; graph.addUndirectedEdge(n1, n2); }; // Generate all swaps from any phase of any expansion (up to periodMax) for (const {arr: prim} of nodes.values()) { const p0 = prim.length; for (let p = p0; p <= maxPeriod; p += p0) { // Iterate phases of the expanded pattern without materialising it. for (let phase = 0; phase < p; phase++) { const phased = d3.range(p).map((i) => atExpanded(prim, phase + i)); for (let i = 0; i < p; i++) { for (const dir of [1, -1]) { const moved = transfer(phased, i, dir); if (moved) { addEdge(phased, moved); } } } } } } return graph; }39:T4b7,function drawDistanceGraph(layout) { const maxLen = d3.max(layout.nodes().map((d) => d.dist)); return Plot.plot({ axis: null, width: 900, height: 400, inset: 20, color: {scheme: "Cividis"}, marks: [ Plot.contour(layout.nodes(), { x: "x", y: "y", fill: "dist", interpolate: "random-walk", blur: 10, fillOpacity: 0.25 }), Plot.arrow(layout.edges(), { x1: (d) => d.source.x, y1: (d) => d.source.y, x2: (d) => d.target.x, y2: (d) => d.target.y, headLength: 0, strokeWidth: 0.5 }), Plot.dot(layout.nodes(), { x: "x", y: "y", fill: "white", stroke: "black", r: 18, strokeWidth: (d) => (d.id === "3" ? 9 : 0) }), Plot.dot(layout.nodes(), { x: "x", y: "y", fill: "dist", stroke: "black", r: 18 }), Plot.text(layout.nodes(), { x: "x", y: "y", fill: (d) => (d.dist < maxLen / 2 + 1 ? "white" : "black"), text: "id", fontWeight: "bold", fontSize: (d) => 18 / Math.sqrt(d.id.length) }) ] }); }3a:T51f,function siteswapFSM(nBalls, maxHeight) { const bigP = BigInt(maxHeight); const limit = 1n << bigP; const toLabel = (mask) => { let s = ""; for (let i = 0; i < maxHeight; i++) { s += ((mask >> BigInt(i)) & 1n) === 1n ? "🟤" : "◯"; } return s; }; // Generate bit masks with exactly nBalls bits set. function* masksWithPopcount() { let comb = (1n << BigInt(nBalls)) - 1n; // lowest B bits while (comb < limit) { yield comb; const c = comb & -comb; const r = comb + c; if (r >= limit) { break; } comb = (((r ^ comb) >> 2n) / c) | r; } } // Nodes const nodes = []; const lu = {}; for (const mask of masksWithPopcount()) { const id = mask.toString(); nodes.push({id, label: toLabel(mask)}); lu[id] = toLabel(mask); } const g = new Graph(); g.setLookup(lu); // Edges for (const {id} of nodes) { const mask = BigInt(id); const landsNow = (mask & 1n) === 1n; const shifted = mask >> 1n; if (!landsNow) { g.addEdge(id, shifted.toString(), 0); continue; } for (let t = 1; t <= maxHeight; t++) { const bit = 1n << BigInt(t - 1); if ((shifted & bit) === 0n) { g.addEdge(id, (shifted | bit).toString(), t); } } } return g; }3b:T8d4,function reachable(options = {}) { const { nBalls = 3, maxHeight = 5, maxPeriod = 5, start = undefined, prime = false, g = suppressDegree1(suppressEvenStates(siteswapFSM(nBalls, maxHeight))) } = options; const startState = start ? `${start}` : `${2 ** nBalls - 1}`; const labelOf = (w) => Array.isArray(w) ? w.map((d) => encodeDigit(Number(d))).join("") : typeof w === "number" ? encodeDigit(w) : String(w); const costToStart = g.distancesTo(startState, { wFn: (w) => labelOf(w).length }); const outCache = new Map(); const outsOf = (u) => { const cached = outCache.get(u); if (cached) { return cached; } const outs = g.outEdges(u).map(({to, weight}) => { const lab = labelOf(weight); return {to, lab, labLen: lab.length}; }); outs.sort((a, b) => a.labLen - b.labLen); outCache.set(u, outs); return outs; }; const getCost = (n) => { const c = costToStart.get(n); return Number.isFinite(c) ? c : Infinity; }; const paths = []; const pieces = []; const visited = prime ? new Set([startState]) : null; const traverse = (node, usedLen) => { const remaining = maxPeriod - usedLen; if (remaining <= 0) { return; } if (getCost(node) > remaining) { return; } for (const {to, lab, labLen} of outsOf(node)) { if (labLen > remaining) { break; } const remaining2 = remaining - labLen; if (getCost(to) > remaining2) { continue; } if (to === startState) { pieces.push(lab); paths.push(pieces.join("")); if (!prime && remaining2 > 0) { traverse(startState, usedLen + labLen); } pieces.pop(); continue; } if (prime && visited.has(to)) { continue; } pieces.push(lab); if (prime) { visited.add(to); } traverse(to, usedLen + labLen); if (prime) { visited.delete(to); } pieces.pop(); } }; traverse(startState, 0); return [...new Set(paths.map((d) => primitive(canonicalRotation(d))))].sort( (a, b) => (a.length === b.length ? (a < b ? -1 : 1) : a.length - b.length) ); }3c:Ta45,function hamiltonian(options = {}) { const {nBalls = 3, maxHeight = 5, limit = 22, maxTours = Infinity} = options; const labelOf = (w) => Array.isArray(w) ? w.map(encodeDigit).join("") : typeof w === "string" ? w : typeof w === "number" ? encodeDigit(w) : String(w); const g = suppressDegree1(suppressEvenStates(siteswapFSM(nBalls, maxHeight))); const nodes = g.nodes(); const n = nodes.length; if (n === 0 || n > limit) { return []; } let s = 0; for (let i = 1; i < n; i++) { if (`${nodes[i]}` < `${nodes[s]}`) { s = i; } } const idxOf = new Map(nodes.map((u, i) => [u, i])); const adj = Array.from({length: n}, () => []); for (let ui = 0; ui < n; ui++) { const u = nodes[ui]; for (const {to, weight} of g.outEdges(u)) { const vi = idxOf.get(to); if (vi !== undefined) { const lab = labelOf(weight); adj[ui].push({to: vi, lab, cost: lab.length}); } } // Deterministic ordering adj[ui].sort((a, b) => a.cost !== b.cost ? a.cost - b.cost : a.lab < b.lab ? -1 : a.lab > b.lab ? 1 : 0 ); } // Precompute closing edges back to start const closeToStart = adj.map((es) => es.filter((e) => e.to === s)); const visited = new Uint8Array(n); const path = new Int32Array(n + 1); visited[s] = 1; path[0] = s; const bestByKey = new Map(); let tourCount = 0; const dfs = (pos, u, costSoFar, throwsSoFar) => { if (tourCount >= maxTours) { return; } if (pos === n) { for (const e of closeToStart[u]) { const key = primitive(canonicalRotation(throwsSoFar + e.lab)); const cost = costSoFar + e.cost; path[pos] = s; const tour = Array.from(path, (ix) => nodes[ix]); const prev = bestByKey.get(key); if (!prev || cost < prev.cost) { bestByKey.set(key, {tour, throws: key, cost}); } tourCount++; if (tourCount >= maxTours) { return; } } return; } for (const e of adj[u]) { const v = e.to; if (visited[v]) { continue; } visited[v] = 1; path[pos] = v; dfs(pos + 1, v, costSoFar + e.cost, throwsSoFar + e.lab); visited[v] = 0; if (tourCount >= maxTours) { return; } } }; dfs(1, s, 0, ""); return [...bestByKey.values()].sort((a, b) => a.throws.length === b.throws.length ? a.throws < b.throws ? -1 : 1 : a.throws.length - b.throws.length ); }3d:T7f3,// Find a walk through the FSM graph that matches the given siteswap throw sequence function walkSiteswap(g, siteswap, options = {}) { const {constraints = []} = options; const tokenize = (x) => Array.isArray(x) ? x.map((d) => (typeof d === "number" ? encodeDigit(d) : String(d))) : typeof x === "number" ? [encodeDigit(x)] : [...String(x)]; const rotateLeft = (xs, k) => { if (xs.length === 0) { return []; } const r = ((k % xs.length) + xs.length) % xs.length; return r === 0 ? [...xs] : xs.slice(r).concat(xs.slice(0, r)); }; const throws = tokenize(siteswap); if (throws.length === 0) { return []; } const outIndex = new Map(); for (const u of g.nodes()) { const outs = g.outEdges(u).map(({to, weight}) => ({ to, weight, tokens: tokenize(weight) })); outIndex.set(u, outs); } const matchesAt = (tokens, seq, i) => { if (i + tokens.length > seq.length) { return false; } for (let k = 0; k < tokens.length; k += 1) { if (tokens[k] !== seq[i + k]) { return false; } } return true; }; const traceFrom = (start, seq) => { const walk = []; let [u, i] = [start, 0]; while (i < seq.length) { const matches = outIndex .get(u) .filter((e) => matchesAt(e.tokens, seq, i)); if (matches.length === 0) { return null; } if (matches.length > 1) { throw new Error( `Ambiguous walk from node ${u} at offset ${i}: ` + matches.map((e) => String(e.weight)).join(", ") ); } const e = matches[0]; walk.push({node: u, edge: e.weight}); u = e.to; i += e.tokens.length; } walk.endNode = u; return walk; }; for (let k = 0; k < throws.length; k += 1) { const rotated = rotateLeft(throws, k); for (const u of g.nodes()) { const walk = traceFrom(u, rotated); if (walk) { return walk; } } } return []; }3e:Tc71,function seqToSiteswap(landings, options = {}) { const {empty = " "} = options; const seq = [...landings]; const iMap = new Map(); const throws = Array(seq.length).fill(0); const distinctInOrder = (seq, options = {}) => { const {empty = " "} = options; const seen = new Set(); const out = []; for (const x of seq) { if (x !== empty && !seen.has(x)) { seen.add(x); out.push(x); } } return out; }; const rotateLeft = (xs, k) => { if (xs.length === 0) { return []; } const r = ((k % xs.length) + xs.length) % xs.length; return r === 0 ? [...xs] : xs.slice(r).concat(xs.slice(0, r)); }; const smallestPeriod = (xs) => { const n = xs.length; if (n === 0) { return 0; } const prefix = Array(n).fill(0); for (let i = 1; i < n; i += 1) { let j = prefix[i - 1]; while (j > 0 && xs[i] !== xs[j]) { j = prefix[j - 1]; } if (xs[i] === xs[j]) { j += 1; } prefix[i] = j; } const period = n - prefix[n - 1]; return n % period === 0 ? period : n; }; const canonicalRotationInfo = (digits) => { const n = digits.length; const isGreaterRot = (k, bestK) => { if (digits[k] !== digits[bestK]) { return digits[k] > digits[bestK]; } for (let i = 1; i < n; i += 1) { const a = digits[(k + i) % n]; const b = digits[(bestK + i) % n]; if (a !== b) { return a > b; } } return false; }; let bestK = 0; for (let k = 1; k < n; k += 1) { if (isGreaterRot(k, bestK)) { bestK = k; } } return { shift: bestK, digits: rotateLeft(digits, bestK) }; }; for (const [i, ball] of seq.entries()) { if (ball !== empty) { const ids = iMap.get(ball) ?? []; ids.push(i); iMap.set(ball, ids); } } const heightsByBall = []; for (const [ball, ids] of iMap.entries()) { const counts = new Map(); for (const [k, i] of ids.entries()) { const j = ids[(k + 1) % ids.length]; const h = j > i ? j - i : seq.length - i + j; throws[i] = h; counts.set(h, (counts.get(h) ?? 0) + 1); } for (const [height, freq] of counts) { heightsByBall.push({height, ball, freq}); } } heightsByBall.sort( (a, b) => a.height - b.height || String(a.ball).localeCompare(String(b.ball)) ); const siteswapPeriod = smallestPeriod(throws); const reducedThrows = throws.slice(0, siteswapPeriod); const {shift, digits: canonicalThrows} = canonicalRotationInfo(reducedThrows); const canonicalSeq = rotateLeft(seq, shift); return { seq, throws, siteswap: throws.map(encodeDigit).join(""), period: seq.length, balls: seq.length === 0 ? 0 : throws.reduce((a, b) => a + b, 0) / seq.length, maxHeight: throws.length === 0 ? 0 : Math.max(...throws), heightsByBall, canonicalThrows, canonicalSiteswap: canonicalThrows.map(encodeDigit).join(""), canonicalShift: shift, ballOrder: [...new Set(seq)], canonicalBallOrder: distinctInOrder(canonicalSeq, {empty}) }; }3f:T6ba,// Given a throw sequence, create the graph of all possible edges given the // maximum height and number of balls implied by the sequence. Highlight the // parts that form the walk capable of reproducing the throw sequence. function showWalk(throwSeq, options = {}) { const { g = suppressDegree1( pruneExterior( removeEdgesWith(0, siteswapFSM(throwSeq.balls, throwSeq.maxHeight)) ) ), edgeLabel = null, showLabels = true, onlyWalk = false, scale = 0.6, formatEdgeWeight = (w) => Array.isArray(w) ? w.map((d) => encodeDigit(Number(d))).join("") : typeof w === "number" ? ` ${encodeDigit(w)} ` : String(w) } = options; const walk = walkSiteswap(g, throwSeq.siteswap); const g2 = onlyWalk ? gFromWalk(g, walk) : g; const nStyle = [...new Set(walk.map((d) => d.node))] .map( (d) => `${d} [color=red, penwidth=2, style=filled, fillcolor="#ff000020"];` ) .join(""); const walkedEdges = new Set( walk.map(({node, edge}) => `${node}|${formatEdgeWeight(edge)}`) ); const edgeStyle = ({source, weight}) => { const label = formatEdgeWeight(weight); const isWalkEdge = walkedEdges.has(`${source}|${label}`); return isWalkEdge ? {color: "red", penwidth: 2, fontcolor: "red"} : null; }; return g2.toDot({ compact: true, showDirected: true, showWeights: showLabels, nodeLabels: g2.getLookup(), scale, extra: nStyle, edgeAtts: edgeStyle, edgeLabel: (e, label) => { const formatted = showLabels ? formatEdgeWeight(e.weight) : ""; return typeof edgeLabel === "function" ? edgeLabel(e, formatted) : formatted; } }); }40:T418,// Show the distribution of heights by ball colour. If showFreq is true, // the number of times each ball is thrown to a given height in a single cycle is shown. // If showFreq is false, just the number of distinct balls at each height is shown. function heightPlot(throwSeq, options = {}) { const {showFreq = true, maxHeight, gap = 1, w = 640, h = 300} = options; return Plot.plot({ width: w, height: h, x: {axis: showFreq, label: "frequency", tickSize: 0, tickPadding: 1}, y: { domain: d3.range(1, (maxHeight || throwSeq.maxHeight) + 1), reverse: true, label: "Throw height", tickSize: 0 }, color: { domain: throwSeq.ballOrder, range: throwSeq.ballOrder.map((d) => ballClrs[d]) }, marks: [ Plot.waffleX( throwSeq.heightsByBall, Plot.stackX({ x: (d) => (showFreq ? d.freq : d.freq > 0), y: "height", fill: "ball", stroke: "white", strokeWidth: gap }) ), Plot.ruleX([0]) ] }); }41:T178b,const bib = Refs.create(` @incollection{allouche_ubiquitous_1999, title={The ubiquitous prouhet-thue-morse sequence}, author={Allouche, Jean-Paul and Shallit, Jeffrey}, booktitle={Sequences and their Applications: Proceedings of SETA’98}, pages={1--16}, year={1999}, publisher={Springer}, doi={10.1007/978-1-4471-0551-0_1} } @article{badkobeh_more_2026, title={More characterizations of morphic words}, author={Badkobeh, Golnaz and Ochem, Pascal}, journal={Theoretical Computer Science}, volume={1072}, pages={1-8}, year={2026}, publisher={Elsevier}, doi={10.1016/j.tcs.2026.115875} } @article{badkobeh_infinite_2015, title={Infinite binary words containing repetitions of odd period}, author={Badkobeh, Golnaz and Crochemore, Maxime}, journal={Information Processing Letters}, volume={115}, number={5}, pages={543--547}, year={2015}, publisher={Elsevier}, doi={} } @article{banaian_counting_2016, title={Counting prime juggling patterns}, author={Banaian, Esther and Butler, Steve and Cox, Christopher and Davis, Jeffrey and Landgraf, Jacob and Ponce, Scarlitte}, journal={Graphs and Combinatorics}, volume={32}, number={5}, pages={1675--1688}, year={2016}, publisher={Springer}, url={https://arxiv.org/pdf/1508.05296} } @article{buhler_juggling_1994, author = {Joe Buhler and David Eisenbud and Ron Graham and Colin Wright}, title = {Juggling drops and descents}, journal = {The American Mathematical Monthly}, volume = {101}, number = {6}, pages = {507--519}, year = {1994}, publisher = {Taylor \& Francis}, doi = {10.1080/00029890.1994.11996984} } @book{chung_universal_2007, title={Universal juggling cycles}, author={Chung, Fan and Graham, Ron}, volume={2007}, year={2007}, publisher={de Gruyter, Berlin}, url={https://mathweb.ucsd.edu/~ronspubs/07_03_ujc.pdf} } @article{chung_primitive_2008, title={Primitive juggling sequences}, author={Chung, Fan and Graham, Ron}, journal={The American Mathematical Monthly}, volume={115}, number={3}, pages={185--194}, year={2008}, publisher={Taylor \& Francis}, url={https://www.jstor.org/stable/pdf/27642443.pdf} } @article{currie_properties_2023, title={Properties of a ternary infinite word}, author={Currie, James and Ochem, Pascal and Rampersad, Narad and Shallit, Jeffrey}, journal={RAIRO-Theoretical Informatics and Applications}, volume={57}, pages={1}, year={2023}, publisher={EDP Sciences}, doi={10.1051/ita/2022010} } @article{ehrenborg_juggling_1996, title={Juggling and applications to q-analogues}, author={Ehrenborg, Richard and Readdy, Margaret}, journal={Discrete Mathematics}, volume={157}, number={1-3}, pages={107--125}, year={1996}, publisher={Elsevier}, doi={10.1016/S0012-365X(96)83010-X} } @article{engstrom_geometric_2015, title={Geometric juggling with q-analogues}, author={Engstr{\"o}m, Alexander and Leskel{\"a}, Lasse and Varpanen, Harri}, journal={Discrete Mathematics}, volume={338}, number={7}, pages={1067--1074}, year={2015}, publisher={Elsevier}, doi={10.1016/j.disc.2015.02.004} } @article{fredricksen_gray_1992, title={Gray codes and the Thue-Morse-Hedlund sequence}, author={Fredricksen, H}, journal={J. Combin. Math. Combin. Comput}, volume={11}, pages={3--11}, year={1992}, url={https://combinatorialpress.com/article/jcmcc/Volume%2011/vol-11-paper%201.pdf} } @misc{macauley_braids_2003, title={Braids and juggling patterns}, author={Macauley, Matthew}, year={2003}, url={https://scholarship.claremont.edu/cgi/viewcontent.cgi?article=1154&context=hmc_theses} } @article{magnusson_physics_1989, title={The Physics of Juggling.}, author={Magnusson, Bengt and Tiemann, Bruce}, journal={Physics Teacher}, volume={27}, number={8}, pages={584--88}, year={1989}, publisher={ERIC} } @article{ong_abstract_2023, title={Abstract Art Generated by Thue--Morse Autocorrelation Functions}, author={Ong, Darren C}, journal={Bulletin of the Malaysian Mathematical Sciences Society}, volume={46}, number={5}, pages={153}, year={2023}, publisher={Springer}, url={https://arxiv.org/pdf/2209.11162} } @article{rampersad_correlations_2023, title={Correlations of minimal forbidden factors of the Fibonacci word}, author={Rampersad, Narad and Wiebe, Max}, journal={arXiv preprint arXiv:2309.07070}, url={https://arxiv.org/pdf/2309.07070}, year={2023} } @incollection{shannon_scientific_1993, author = "Claude Shannon", title = "Scientific aspects of juggling", editor = "N. J. A. Sloane and A. D. Wyner", booktitle = "The Collected Papers of Claude E. Shannon", publisher = "IEEE Press", year = 1993, pages = "850--860", url={https://www.jonglage.net/theorie/notation/ladder/refs/Claude%20Shannon%20-%20Scientific%20Aspects%20of%20Juggling.pdf} } @article{simpson_juggling_1986, author = {Charlie Simpson}, title = {Juggling on paper}, journal = {Juggler's World}, volume = {37}, number = {4}, pages = {31--32}, year = {1994}, url = {https://www.jonglage.net/theorie/notation/ladder/refs/Charlie%20Simpson%20-%20Juggling%20on%20paper%20-%20JW-vol37no4-p31.pdf} } @article{spiegelhofer_gaps_2023, title={Gaps in the Thue--Morse word}, author={Spiegelhofer, Lukas}, journal={Journal of the Australian Mathematical Society}, volume={114}, number={1}, pages={110--144}, year={2023}, publisher={Cambridge University Press}, doi={10.1017/S1446788721000380} } @article{stadler_juggling_2002, title={Juggling and vector compositions}, author={Stadler, Jonathan D}, journal={Discrete Mathematics}, volume={258}, number={1-3}, pages={179--191}, year={2002}, publisher={Elsevier}, doi={10.1016/S0012-365X(02)00269-8} } @misc{zinser_juggling_2010, title={Juggling Theory Part I. Siteswap State Diagrams}, author={Zinser, Alexander}, year={2010}, url={https://alexzinser.de/download/jugglingtheory_sssd.pdf} }`);42:[] 2d:{"id":"7340ada34aed4f39","slug":"visualizing-juggling-appendix","title":"Visualizing Juggling: Appendix","thumbnail":"4b147273dd9ea5f40d171452ac1e53be12783205f4870225e64ee6506a4fbdaf","access_level":"public","create_time":"$D2026-02-23T17:12:52.575Z","update_time":"$D2026-09-22T21:33:33.689Z","view_time":null,"first_public_time":"$D2026-09-22T21:16:35.489Z","has_unpublished_edits":null,"fork":{"id":"5afdf94ce77ed6d1","slug":"visualizing-juggling","title":"Visualizing Juggling: Movement in Space and Time","owner":{"id":"9461f58b209074d5","login":"jwolondon","name":"Jo Wood"}},"stars":0,"starred":false,"owner":{"type":"team","id":"9461f58b209074d5","login":"jwolondon","name":"Jo Wood","avatar_url":"https://avatars-next.observableusercontent.com/27f72cecd1ac20853a533421fab228ce5f611cbc0986c1e420f7d3b1eb49ef50","tier":"starter_2024"},"version":7913,"publish_version":7913,"role":null,"authored":false,"authors":[{"type":"individual","id":"e2432cc13dc447d3","login":"jwolondon","name":"Jo Wood","avatar_url":"https://avatars-next.observableusercontent.com/27f72cecd1ac20853a533421fab228ce5f611cbc0986c1e420f7d3b1eb49ef50","tier":null}],"members":[{"type":"individual","id":"e2432cc13dc447d3","login":"jwolondon","name":"Jo Wood","avatar_url":"https://avatars-next.observableusercontent.com/27f72cecd1ac20853a533421fab228ce5f611cbc0986c1e420f7d3b1eb49ef50","tier":null,"role":"editor"},{"type":"team","id":"9461f58b209074d5","login":"jwolondon","name":"Jo Wood","avatar_url":"https://avatars-next.observableusercontent.com/27f72cecd1ac20853a533421fab228ce5f611cbc0986c1e420f7d3b1eb49ef50","tier":"starter_2024","role":"viewer"}],"comments":[],"custom_thumbnail":false,"trashed":false,"lapsed":null,"subscribed":null,"forks":0,"collection":{"id":"990e96f05f9951d9","slug":"juggling","title":"juggling","owner":{"id":"9461f58b209074d5","login":"jwolondon","name":"Jo Wood"}},"type":"notebook","body":{"title":"Visualizing Juggling: Appendix","cells":[{"id":0,"value":"# Visualizing Juggling: Appendix","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":5605,"value":"- [Chapter One: Movement in space and time](https://observablehq.com/@jwolondon/visualizing-juggling)\n- [Chapter Two: Marking time](https://observablehq.com/@jwolondon/visualizing-juggling-marking-time)\n- [Chapter Three: Siteswap](https://observablehq.com/@jwolondon/visualizing-juggling-siteswap)\n- [Chapter Four: Pattern space](https://observablehq.com/@jwolondon/visualizing-juggling-pattern-space)\n- [Chapter Five: Ball state](https://observablehq.com/@jwolondon/visualizing-juggling-ball-state)\n- **Appendix: Theorems, theory and code**\n---","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":6512,"value":"## Appendix A1: Theorems and Theory","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":6514,"value":"### Assumptions\n\nWorking with [vanilla siteswap](https://www.juggle.org/assumptions-in-vanilla-siteswap-paths-to-new-patterns), we assume the following:\n\n- two hands throw alternately at equally spaced time intervals; so no synchronous patterns.\n- a maximum of one ball can be thrown or caught by a hand; so no multiplex throws.\n- both hands throw at the same rhythm.\n","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":6517,"value":"$2f","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":6546,"value":"### Some observations","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":7737,"value":"#### 1. Number of balls can be derived from a valid canonical siteswap word\n\nFor a siteswap pattern with ${tex`h_1h_2\\dots h_n`} it will always be the case that\n\n${tex`b = \\frac{1}{n}\\sum_{i=1}^{n} h_i`}\n\nFor example the following siteswaps all indicate 3 balls\n\n**3** (${tex`3/1 = 3`})\n\n**51** (${tex`(5+1)/2 = 3`})\n\n**423** (${tex`(4+2+3)/3 = 3`})\n\nThe following siteswaps all indicate 4 balls:\n\n**615** (${tex`(6+1+5)/3 = 4`})\n\n**747161514** (${tex`36/9=4`})\n\nAnd the following would not be valid siteswap pattern because it doesn't have an integer average:\n\n**4222** (${tex`10/4 = 2.5`})","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":6529,"value":"---\n\n#### 2. Validating a siteswap word\n\nIn addition to requiring an integer average to maintain the vanilla siteswap constraints, ${tex`[h_1,\\dots,h_n]`} is a valid vanilla siteswap if and only if ${tex`{(h_i + i - 1)\\bmod n : i=1,\\dots,n}`} are unique (i.e. a maximum of one ball lands on any beat). For example,\n\n**531** = ${tex`\\{(5+0) \\bmod 3, (3+1) \\bmod 3, (1+2) \\bmod 3 \\} = \\{2,1,0\\}`} so is valid.\n\n**513** = ${tex`\\{(5+0) \\bmod 3, (1+1) \\bmod 3, (3+2) \\bmod 3 \\} = \\{2,2,2\\}`} so is invalid.","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":6658,"value":"---\n\n#### 3. Number of siteswap words\n\nThe number of sitewaps for ${tex`b`} balls is infinite because composite words can be assembled by concatenating any number of prime patterns. But ${bib.cite(\"buhler_juggling_1994\")} proved that if we constrain the period to ${tex`n`}, the number of possible siteswaps with up to ${tex`b`} balls will be ${tex`(b+1)^n`} including the 'zero-pattern' with 0 balls. This count is for non-canonical siteswaps so include rotation variants and duplicated prime patterns.\n\nFor example, the number of period 3 siteswaps with up to 3 balls is ${tex`4^3 = 64`}:","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":7738,"value":"$${enumerateSiteswaps().map((row,i)=>\"**\"+i+ (i===1?\" ball\":\" balls\")+\":** \"+ row.join(\", \")+\"\\n\\n\")}","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":6811,"value":"If we convert each to its canonical representation, this reduces to 20 unique period-3 patterns:","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":6813,"value":"$${enumerateSiteswaps().map(canonise).map((row,i)=>\"**\"+i+ (i===1?\" ball\":\" balls\")+\":** \"+ row.filter(d=>d.length===3).join(\", \")+\"\\n\\n\")}","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":6915,"value":"$${bib.cite(\"buhler_juggling_1994\")} provide the counting function for canonical siteswaps (Theorem 3, p.518):\n

\n${tex`M(b,n) = \\frac{1}{n}\\sum\\limits_{d \\mid n} \\mu(n/d)((b+1)^d-b^d)`}\n\nwhere ${tex`\\mu`} is the [Möbius function](https://en.wikipedia.org/wiki/Möbius_function) and the ${tex`d`}s are the factors of ${tex`n`}. This allows quick counting of the number of distinct canonical siteswaps for any given pair of values ${tex`n`} and ${tex`b`}. In the table below, marginal references are to the [Online Encyclopedia of Integer Sequences](https://oeis.org) corresponding to the sequence of row or column counts:","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":7011,"value":"canonicalTable()","pinned":false,"hidden":false,"mode":"ts","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":7073,"value":"There is no known closed formula for counting prime patterns ${bib.citep(\"banaian_counting_2016\")}, although they can be computed via graph traversal (see [Chapter Five: Ball state](https://observablehq.com/@jwolondon/visualizing-juggling-ball-state)).","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":6531,"value":"---\n\n#### 4. Binary notation\n\nIf a landing schedule is stored in reverse sequence (so time ${tex`t`} is the least significant bit and ${tex`t+H-1`} is the most significant), it will be even if all balls are in the air at time ${tex`t`}, or odd if there is a ball in hand.","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":6533,"value":"---\n\n#### 5. FSM graph\n\nA Finite State Machine (FSM) representation of all possible states for any given ${tex`H`} will always be _strongly connected_. In other words it is possible to transition from any landing schedule state to any other with a walk across the FSM graph.","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":6535,"value":"---\n\n#### 6. Transition alphabet\n\nFor FSMs for a given ${tex`H`} with ${tex`b`} balls, if ${tex`0 < b < H`}, transitions will comprise the full alphabet ${tex`\\{0,1,2,\\dots,H \\}`}. If ${tex`b=H`}, the alphabet is just ${tex`\\{H\\}`}.","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":6663,"value":"---\n## References\n\n${bib.bibliography()}","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":6216,"value":"---\n\n## Appendix A2: Code","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":406,"value":"### Helper functions","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":396,"value":"_Given one hand (L or R) provide the other (R or L):_","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":258,"value":"function otherHand(h) {\n return h === \"L\" ? \"R\" : \"L\";\n}","pinned":false,"hidden":false,"mode":"ts","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":399,"value":"_Given a hand (L or R) provide its horizontal direction when moving a held ball:_","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":268,"value":"function handDir(h) {\n return h === \"L\" ? +1 : -1;\n}","pinned":false,"hidden":false,"mode":"ts","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":402,"value":"_Coordinates of a hand:_","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":271,"value":"function handX(h, handSep) {\n return h === \"L\" ? -handSep / 2 : handSep / 2;\n}","pinned":false,"hidden":false,"mode":"ts","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1056,"value":"### Static views","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":889,"value":"function trajectoryPlot(data) {\n return Plot.plot({\n width: 340,\n height: 200,\n inset: 40,\n x: {domain: [-10, 10], axis: null},\n y: {domain: [0, 3], axis: null},\n color: {domain: d3.range(0, 10)},\n marks: [\n trajectory(data),\n Plot.dot(data, Plot.selectLast({x: \"x\", y: \"y\", r: 12, fill: \"ball\"}))\n ]\n });\n}","pinned":false,"hidden":false,"mode":"ts","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":5716,"value":"function staticGap(t, options = {}) {\n const {arrows = []} = options;\n return animate(t, {\n w: 280,\n h: 210,\n inset: 30,\n yLabels: false,\n yDomain: [0, 3],\n dwell: 0.1,\n handSep: 7,\n catchOutward: 1.5,\n launchInward: 1.5,\n\n extra: [\n trajectory(\n makeTrajectoryLines([\n {centre: -2.0, spread: 1.125},\n {centre: -5.0, spread: 1.125}\n ]),\n {\n stroke: \"lightgrey\"\n }\n )\n ],\n extraAfter: [\n Plot.dot([{x: 0, y: 2.45}], {\n x: \"x\",\n y: \"y\",\n r: 12,\n strokeDasharray: [2, 2]\n }),\n arrows.map((d) =>\n Plot.arrow([{x1: d.x1, y1: d.y1, x2: d.x2, y2: d.y2}], {\n x1: \"x1\",\n y: \"y1\",\n x2: \"x2\",\n y2: \"y2\",\n bend: d.bend,\n strokeWidth: 2,\n insetEnd: 8\n })\n )\n ]\n });\n}","pinned":false,"hidden":false,"mode":"ts","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1093,"value":"function trajectory(data, options = {}) {\n return Plot.lineY(data, {\n x: \"x\",\n y: \"y\",\n z: \"ball\",\n curve: \"natural\",\n stroke: \"ball\",\n strokeWidth: 1.9,\n strokeDasharray: [5, 6],\n ...options\n });\n}","pinned":false,"hidden":false,"mode":"ts","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1628,"value":"function makeTrajectoryLines(throws) {\n return throws.flatMap((d, i) => {\n const centre = d.centre ? d.centre : 0;\n const spread = d.spread ? d.spread : 1;\n const hand = d.hand ? d.hand : \"L\";\n const thr = d.throw ? d.throw : 3;\n const xs = hand === \"L\" ? d3.range(0, 1.1, 0.1) : d3.range(1, -0.1, -0.1);\n return xs.map((x) => ({\n x: 6 * x * spread + centre,\n y: yParabola(x, thr),\n ball: i\n }));\n });\n}","pinned":false,"hidden":false,"mode":"ts","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":2177,"value":"$30","pinned":false,"hidden":false,"mode":"ts","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1909,"value":"$31","pinned":false,"hidden":false,"mode":"ts","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":7644,"value":"$32","pinned":false,"hidden":false,"mode":"ts","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":854,"value":"### Animated views","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":3192,"value":"$33","pinned":false,"hidden":false,"mode":"ts","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":393,"value":"### General utility functions","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":417,"value":"_Safe modulo:_","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":274,"value":"function mod(a, m) {\n if (m === 0) {\n return 0;\n }\n const r = a % m;\n return r < 0 ? r + m : r;\n}","pinned":false,"hidden":false,"mode":"ts","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":433,"value":"_Is phi betwen t0 and t1?_","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":277,"value":"function inInterval(phi, t0, t1) {\n if (t0 <= t1) {\n return phi >= t0 && phi <= t1;\n } else {\n return phi >= t0 || phi <= t1;\n }\n}","pinned":false,"hidden":false,"mode":"ts","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":435,"value":"_For interpolation:_","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":280,"value":"function progressInInterval(phi, t0, t1, loop) {\n const dur = t0 <= t1 ? t1 - t0 : t1 - t0 + loop;\n const elapsed =\n t0 <= t1 ? phi - t0 : phi >= t0 ? phi - t0 : phi + (loop - t0);\n const safeDur = Math.max(1e-6, dur);\n return {u: elapsed / safeDur, dur: safeDur};\n}","pinned":false,"hidden":false,"mode":"ts","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":283,"value":"function lerp(a, b, u) {\n return a + (b - a) * u;\n}","pinned":false,"hidden":false,"mode":"ts","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":439,"value":"_Parabolic shape, given x, returns y:_","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":289,"value":"function yParabola(u, hPeak) {\n return u < 0 || u > 1 ? 0 : 4 * hPeak * u * (1 - u);\n}","pinned":false,"hidden":false,"mode":"ts","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":441,"value":"_For calculating cycle size:_","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":303,"value":"function lowestCommonMultiple(...args) {\n const lcmPair = (a, b) => {\n return Math.abs(a * b) / highestCommonFactor(a, b);\n };\n return args.reduce((acc, n) => lcmPair(acc, n), 1);\n}","pinned":false,"hidden":false,"mode":"ts","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":385,"value":"function highestCommonFactor(a, b) {\n if (a === 0 && b === 0) {\n return 1;\n }\n while (b != 0) {\n const t = b;\n b = a % b;\n a = t;\n }\n return Math.abs(a);\n}","pinned":false,"hidden":false,"mode":"ts","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":2833,"value":"_For array rotation and max value detection:_","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":2794,"value":"function rotate(arr, k = 0) {\n const n = arr.length;\n const s = ((k % n) + n) % n;\n return arr.slice(s).concat(arr.slice(0, s));\n}","pinned":false,"hidden":false,"mode":"ts","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":2798,"value":"function indexOfMax(xs) {\n return xs.length === 0\n ? -1\n : xs.reduce((best, x, i) => (x > xs[best] ? i : best), 0);\n}","pinned":false,"hidden":false,"mode":"ts","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":6847,"value":"[_Möbius function:_](https://en.wikipedia.org/wiki/Möbius_function)","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":6849,"value":"function mobius(n) {\n if (n === 1) {\n return 1;\n }\n\n let [m, nPrimes] = [n, 0];\n\n for (let p = 2; p * p <= m; p += 1) {\n if (m % p === 0) {\n m /= p;\n nPrimes++;\n\n if (m % p === 0) {\n return 0;\n }\n\n while (m % p === 0) {\n m /= p;\n }\n }\n }\n\n if (m > 1) {\n nPrimes++;\n }\n\n return nPrimes % 2 === 0 ? 1 : -1;\n}","pinned":false,"hidden":false,"mode":"ts","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":6871,"value":"_Ordered list of factors of a positive integer n:_","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":6861,"value":"function factors(n) {\n if (n === 1) {\n return [1];\n }\n const [lower, upper] = [[], []];\n const sqrt = Math.sqrt(n);\n\n for (let i = 1; i <= sqrt; i += 1) {\n if (n % i === 0) {\n lower.push(i);\n if (i !== sqrt) {\n // To avoid adding the square root twice\n upper.push(n / i);\n }\n }\n }\n\n return lower.concat(upper.reverse());\n}","pinned":false,"hidden":false,"mode":"ts","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":7241,"value":"_List intersection:_","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":7243,"value":"function intersection(a, b) {\n return a.filter((x) => b.includes(x));\n}","pinned":false,"hidden":false,"mode":"ts","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":444,"value":"### Siteswap functions","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":3533,"value":"_Siteswap notation:_","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":2996,"value":"function canonicalRotation(digits) {\n const n = digits.length;\n\n const isGreaterRot = (k, bestK) => {\n // Compare rotation by k vs bestK: prefer larger first element, then lexicographically greatest\n const a0 = digits[k];\n const b0 = digits[bestK];\n if (a0 !== b0) {\n return a0 > b0;\n }\n for (let i = 0; i < n; i++) {\n const a = digits[(k + i) % n];\n const b = digits[(bestK + i) % n];\n if (a !== b) {\n return a > b;\n }\n }\n return false;\n };\n\n let bestK = 0;\n for (let k = 1; k < n; k++) {\n if (isGreaterRot(k, bestK)) {\n bestK = k;\n }\n }\n\n // rotate left by bestK (so bestK becomes index 0)\n return bestK === 0\n ? digits\n : digits.slice(bestK).concat(digits.slice(0, bestK));\n}","pinned":false,"hidden":false,"mode":"ts","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":6754,"value":"// Given an array of possibly non-canonical siteswaps reduce it to an array of sorted canonical siteswaps\n// Will reduce the number of distinct patterns if the original list contains rotations of the same pattern\nfunction canonise(siteswaps) {\n return [\n ...new Set(\n siteswaps\n .map((d) => primitive(toSiteswap(canonicalRotation(parseSiteswap(d)))))\n .sort()\n )\n ];\n}","pinned":false,"hidden":false,"mode":"ts","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":4397,"value":"function primitive(a) {\n // Simplifiy a repeating pattern to its single period pattern (e.g. \"3333\"=>\"3\")\n const p = a.length;\n for (let k = 1; k <= p; k++) {\n if (p % k !== 0) {\n continue;\n }\n let ok = true;\n for (let i = 0; i < p; i++) {\n if (a[i] !== a[i % k]) {\n ok = false;\n break;\n }\n }\n if (ok) {\n return a.slice(0, k);\n }\n }\n return a;\n}","pinned":false,"hidden":false,"mode":"ts","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":4389,"value":"function allSitewaps(maxPeriod = 9999, balls = 3) {\n return [\n ...new Set(\n d3\n .range(0, maxPeriod)\n .map((d) => String(d).split(\"\").map(Number))\n .map((d) => toSiteswap(canonicalRotation(primitive(d))))\n .filter((d) => isValidSiteswap(d, balls))\n )\n ].sort((a, b) => Number(a) - Number(b));\n}","pinned":false,"hidden":false,"mode":"ts","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":6682,"value":"$34","pinned":false,"hidden":false,"mode":"ts","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":6873,"value":"function canonicalCount(options = {}) {\n const {b = 3, n = 3} = options;\n return (\n d3.sum(factors(n).map((d) => mobius(n / d) * ((b + 1) ** d - b ** d))) / n\n );\n}","pinned":false,"hidden":false,"mode":"ts","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":7009,"value":"function canonicalTable(options = {}) {\n const {maxB = 9, maxN = 9} = options;\n const oeisCols = [\n \"[A059966](https://oeis.org/A059966)\",\n \"[A065178](https://oeis.org/A065178)\",\n \"[A065179](https://oeis.org/A065179)\",\n \"[A065180](https://oeis.org/A065180)\"\n ];\n\n const oeisRows = [\n ,\n ,\n \"[A002378](https://oeis.org/A002378)\",\n \"[A059270](https://oeis.org/A059270)\",\n \"[A169938](https://oeis.org/A169938)\"\n ];\n const fmt = d3.format(\",\");\n\n const rows = [\n [, ...d3.range(1, maxB + 1).map((b) => `b=${b}`), \"OEISS\"],\n [\"---:\", ...d3.range(1, maxB + 2).map(() => \"---:\")],\n [, ...oeisCols],\n ...d3\n .range(1, maxN + 1)\n .map((n, i) => [\n `**n=${n}**`,\n ...d3.range(1, maxB + 1).map((b) => fmt(canonicalCount({b, n}))),\n oeisRows[i]\n ])\n ];\n\n const content = rows.map((row) => `| ${row.join(\" | \")} |`).join(\"\\n\");\n return md`${content}`;\n}","pinned":false,"hidden":false,"mode":"ts","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":4387,"value":"_Is this a valid siteswap for a given number of balls?_","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":4373,"value":"function isValidSiteswap(siteswap, balls = 3) {\n const digits = parseSiteswap(siteswap);\n const n = digits.length;\n if (d3.sum(digits) !== balls * n) {\n return false;\n }\n\n const seen = new Uint8Array(n);\n for (let i = 0; i < n; i++) {\n const t = (i + digits[i]) % n;\n if (seen[t]) {\n return false;\n }\n seen[t] = 1;\n }\n return true;\n}","pinned":false,"hidden":false,"mode":"ts","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":447,"value":"_Swap between a siteswap string (e.g. \"441\") and array of integers:_","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":306,"value":"function parseSiteswap(str) {\n return [...str].map(decodeThrow);\n}","pinned":false,"hidden":false,"mode":"ts","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":2810,"value":"function toSiteswap(digits) {\n return digits.length === 0\n ? \"\"\n : rotate(digits, -indexOfMax(digits))\n .map((v) => v.toString(36))\n .join(\"\");\n}","pinned":false,"hidden":false,"mode":"ts","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":3027,"value":"function encodeDigit(d) {\n return d < 10 ? String(d) : String.fromCharCode(87 + d);\n}","pinned":false,"hidden":false,"mode":"ts","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":7524,"value":"function decodeThrow(char) {\n return char >= \"0\" && char <= \"9\" ? Number(char) : char.charCodeAt(0) - 87;\n}","pinned":false,"hidden":false,"mode":"ts","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":458,"value":"_How many orbits does a given sequence have?_","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":309,"value":"function countOrbits(seq, cycles) {\n const P = seq.length;\n const N = P * cycles;\n\n const next = new Array(N);\n for (let i = 0; i < N; i++) {\n const v = seq[i % P];\n next[i] = v > 0 ? (i + v) % N : null;\n }\n\n const seen = new Array(N).fill(false);\n let orbits = 0;\n\n for (let i = 0; i < N; i++) {\n if (next[i] === null || seen[i]) {\n continue;\n }\n orbits += 1;\n\n let j = i;\n while (next[j] !== null && !seen[j]) {\n seen[j] = true;\n j = next[j];\n }\n }\n\n return orbits;\n}","pinned":false,"hidden":false,"mode":"ts","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":313,"value":"function minCycles(seq) {\n const P = seq.length;\n const numBalls = seq.reduce((a, v) => a + v, 0) / P;\n const handOK = (K) => (P * K) % 2 === 0;\n const Kmax = Math.max(2, 2 * lowestCommonMultiple(P, numBalls));\n for (let K = 1; K <= Kmax; K++) {\n if (!handOK(K)) {\n continue;\n }\n if (countOrbits(seq, K) === numBalls) {\n return K;\n }\n }\n return P % 2 === 0 ? 1 : 2;\n}","pinned":false,"hidden":false,"mode":"ts","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":478,"value":"_Default geometry settings for juggle animation:_","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":598,"value":"const DEFAULTS = {\n siteswap: \"3\",\n cycles: \"auto\",\n dwell: 0.2,\n handSep: 6,\n catchOutward: 1,\n launchInward: 1,\n startHand: \"L\"\n};\n\nconst resolveConfig = (options = {}) => ({...DEFAULTS, ...options});","pinned":false,"hidden":false,"mode":"ts","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":482,"value":"_Throw a ball and provide updated ball values:_","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":320,"value":"function makeThrow(value, beat, fromHand, config) {\n if (value === 0) {\n // A 0 is an 'idle' beat with no ball present\n return {\n value: 0,\n beat,\n fromHand,\n toHand: fromHand,\n isIdle: true,\n tLaunchAbs: beat,\n tCatchAbs: beat,\n x0: handX(fromHand, config.handSep),\n x1: handX(fromHand, config.handSep)\n };\n }\n\n const toHand = value % 2 ? otherHand(fromHand) : fromHand;\n return {\n value,\n beat,\n fromHand,\n toHand,\n x0:\n handX(fromHand, config.handSep) + handDir(fromHand) * config.launchInward, // inward\n x1: handX(toHand, config.handSep) - handDir(toHand) * config.catchOutward, // outward\n tLaunchAbs: beat + config.dwell,\n tCatchAbs: beat + value - config.dwell\n };\n}","pinned":false,"hidden":false,"mode":"ts","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":485,"value":"_Generate the patterns of balls across cycles:_","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":323,"value":"$35","pinned":false,"hidden":false,"mode":"ts","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":488,"value":"_Calculate the geometry of balls at a given phase and pattern:_","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":326,"value":"function airPositionsAtPhase(phi, pattern) {\n const {throws, loopDur} = pattern;\n const pts = [];\n for (const th of throws) {\n if (inInterval(phi, th.t0, th.t1)) {\n const {u} = progressInInterval(phi, th.t0, th.t1, loopDur);\n const [x, y] = [lerp(th.x0, th.x1, u), yParabola(u, th.value)];\n pts.push({x, y, ballId: th.ballId, state: \"air\"});\n }\n }\n return pts;\n}","pinned":false,"hidden":false,"mode":"ts","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":330,"value":"function holdsAtPhase(\n phi,\n pattern,\n carry = {liftY: 0.0, ampX: 0.1, dipY: 0.3}\n) {\n const {throws, loopDur} = pattern;\n const held = [];\n for (const th of throws) {\n if (inInterval(phi, th.h0, th.h1)) {\n const {u} = progressInInterval(phi, th.h0, th.h1, loopDur);\n const bump = Math.sin(Math.PI * u);\n const offX = th.carryDir * (carry.ampX ?? 0.1) * bump;\n const offY = -(carry.dipY ?? 0.1) * bump;\n const x = lerp(th.xHoldStart, th.xHoldEnd, u) + offX;\n const y = lerp(th.yHoldStart, th.yHoldEnd, u) + offY;\n held.push({x, y, ballId: th.ballId, hand: th.holdHand, state: \"hand\"});\n }\n }\n return held;\n}","pinned":false,"hidden":false,"mode":"ts","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":334,"value":"function positionsAtPhase(phi, pattern) {\n const air = airPositionsAtPhase(phi, pattern);\n const hand = holdsAtPhase(phi, pattern);\n return [...air, ...hand];\n}","pinned":false,"hidden":false,"mode":"ts","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":3827,"value":"$36","pinned":false,"hidden":false,"mode":"ts","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":3410,"value":"### Graph utilities","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":3412,"value":"import {Graph} from \"observable:@jwolondon/gds-graph\";","pinned":false,"hidden":false,"mode":"ts","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":7305,"value":"$37","pinned":false,"hidden":false,"mode":"ts","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":7311,"value":"function pruneExterior(g) {\n // Create a new graph that is the given one with any non-circuit nodes/edges removed\n const h = new Graph(g.serialize());\n let changed = true;\n while (changed) {\n changed = false;\n for (const n of h.nodes()) {\n if (h.indegree(n) === 0 || h.outdegree(n) === 0) {\n h.removeNode(n);\n changed = true;\n }\n }\n }\n return h;\n}","pinned":false,"hidden":false,"mode":"ts","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":3396,"value":"$38","pinned":false,"hidden":false,"mode":"ts","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":3643,"value":"$39","pinned":false,"hidden":false,"mode":"ts","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":3672,"value":"function nodesAndEdges(options = {}) {\n const {maxPeriod = 3, root = \"3\"} = options;\n const g = patternGraph({maxPeriod}).component(root);\n const {nodes, links} = g.serialize();\n return {\n nodes: nodes.map((d) => ({\n id: d.id,\n dist: g.shortestPath(root, d.id).length\n })),\n edges: links\n };\n}","pinned":false,"hidden":false,"mode":"ts","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":6448,"value":"function lineDigraph(g) {\n const h = new Graph();\n const lu = {};\n for (const from of g.nodes()) {\n for (const edge1 of g.outEdges(from)) {\n const n1 = `${edge1.weight}`;\n const n1Label = `${from}_${edge1.to}_${edge1.weight}`;\n lu[n1Label] = `${edge1.weight}`;\n\n for (const edge2 of g.outEdges(edge1.to)) {\n const n2 = `${edge2.weight}`;\n const n2Label = `${edge1.to}_${edge2.to}_${edge2.weight}`;\n lu[n2Label] = `${edge2.weight}`;\n h.addEdge(n1Label, n2Label);\n }\n }\n }\n h.setLookup(lu);\n return h;\n}","pinned":false,"hidden":false,"mode":"ts","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":7204,"value":"// Allow edge labels to be concatenations of single letters\nfunction formatEdgeWeight(w) {\n return Array.isArray(w)\n ? w.map((d) => encodeDigit(Number(d))).join(\"\")\n : typeof w === \"number\"\n ? encodeDigit(w)\n : String(w);\n}","pinned":false,"hidden":false,"mode":"ts","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":7394,"value":"// For creating and displaying a line graph from a graph\n// Edges become nodes and pairs of adjacent edges become edges\nfunction showLineGraph(g, options = {}) {\n const {\n nodeFill = \"#98D1C3\",\n nodeStroke = \"#356A66\",\n compact = true,\n showDirected = true,\n showWeights = true,\n scale = 1\n } = options;\n\n const edgeWeight = ({source, target, index, weight}) =>\n weight ?? g.getEdgeWeights(source, target)[index];\n\n const g2 = g.lineGraph({\n idFn: (e) =>\n `${formatEdgeWeight(edgeWeight(e))}|${e.source}|${e.target}|${e.index}`,\n edgeWeight: ({from, to}) =>\n `${formatEdgeWeight(from.weight)}${formatEdgeWeight(to.weight)}`\n });\n\n const nodeLabels = Object.fromEntries(\n g2.nodes().map((n) => [n, n.split(\"|\")[0]])\n );\n\n return g2.toDot({\n compact,\n showDirected,\n showWeights,\n nodeLabels,\n edgeLabel: (e, label) => ` ${label} `,\n extra: `node [style=filled, fillcolor=\"${nodeFill}\", color=\"${nodeStroke}\"];`,\n scale\n });\n}","pinned":false,"hidden":false,"mode":"ts","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":4826,"value":"### Finite State Machine","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":4715,"value":"$3a","pinned":false,"hidden":false,"mode":"ts","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":5151,"value":"function suppressEvenStates(g, options = {}) {\n const {\n keepNode = () => false,\n isEvenState = (id) => (BigInt(id) & 1n) === 0n\n } = options;\n\n const toSeq = (w) => (Array.isArray(w) ? w : [w]);\n const combine = options.combine ?? ((a, b) => [...toSeq(a), ...toSeq(b)]);\n\n while (true) {\n const n = g.nodes().find((id) => !keepNode(id) && isEvenState(id));\n if (!n) {\n break;\n }\n\n const [ins, outs] = [g.inEdges(n), g.outEdges(n)];\n for (const {from, weight: wIn} of ins) {\n for (const {to, weight: wOut} of outs) {\n g.addEdge(from, to, combine(wIn, wOut, from, n, to));\n }\n }\n g.removeNode(n);\n }\n return g;\n}","pinned":false,"hidden":false,"mode":"ts","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":7195,"value":"function removeEdgesWith(test, g) {\n const seen = new Set();\n\n const pred =\n typeof test === \"function\"\n ? test\n : (w) => (Array.isArray(w) ? w.includes(test) : w === test);\n\n for (const u of g.nodes()) {\n for (const v of g.adjacent(u)) {\n const key = `${u}|${v}`;\n if (seen.has(key)) {\n continue;\n }\n seen.add(key);\n\n const kept = g.getEdgeWeights(u, v).filter((w) => !pred(w));\n g.removeEdge(u, v);\n\n for (const w of kept) {\n g.addEdge(u, v, w);\n }\n }\n }\n return g;\n}","pinned":false,"hidden":false,"mode":"ts","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":7400,"value":"// Given a walk over a subset of a graph, return a graph of just the walked part\nfunction gFromWalk(g, walk) {\n const same = (a, b) =>\n Array.isArray(a) === Array.isArray(b) &&\n (Array.isArray(a)\n ? a.length === b.length && a.every((x, i) => x === b[i])\n : a === b);\n\n const weightKey = (w) => (Array.isArray(w) ? w.join(\"\") : String(w));\n const edgeKey = (u, v, w) => `${u}|${v}|${weightKey(w)}`;\n\n const out = new Graph();\n const seen = new Set();\n\n if (\n typeof g.getLookup === \"function\" &&\n typeof out.setLookup === \"function\"\n ) {\n out.setLookup(g.getLookup());\n }\n\n for (const {node, edge} of walk) {\n const e = g.outEdges(node).find(({weight}) => same(weight, edge));\n\n if (!e) {\n out.addNode(node);\n } else {\n const k = edgeKey(node, e.to, e.weight);\n\n if (!seen.has(k)) {\n seen.add(k);\n out.addEdge(node, e.to, e.weight);\n }\n }\n }\n\n return out;\n}","pinned":false,"hidden":false,"mode":"ts","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":7403,"value":"// Given a throw sequence, create a graph of a walk that produces it.\nfunction walkedGraph(throwSeq, options = {}) {\n const {\n g = suppressDegree1(\n pruneExterior(\n removeEdgesWith(0, siteswapFSM(throwSeq.balls, throwSeq.maxHeight))\n )\n )\n } = options;\n\n return gFromWalk(g, walkSiteswap(g, throwSeq.siteswap));\n}","pinned":false,"hidden":false,"mode":"ts","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":6337,"value":"$3b","pinned":false,"hidden":false,"mode":"ts","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":6345,"value":"function siteswaps(options = {}) {\n const {nBalls = 3, maxHeight = 5, maxPeriod = 5, prime = false} = options;\n const g = suppressDegree1(suppressEvenStates(siteswapFSM(nBalls, maxHeight)));\n\n const seen = new Set();\n for (const node of g.nodes()) {\n for (const s of reachable({g, start: node, maxPeriod, prime})) {\n seen.add(s);\n }\n }\n return [...seen].sort((a, b) =>\n a.length === b.length ? (a < b ? -1 : 1) : a.length - b.length\n );\n}","pinned":false,"hidden":false,"mode":"ts","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":7296,"value":"function primeTable(options = {}) {\n const {nBalls = 3, maxHeight = 5, maxPeriod = 9, ss = undefined} = options;\n const ss2 =\n ss === undefined\n ? siteswaps({nBalls, maxHeight, maxPeriod, prime: true})\n : ss;\n\n // Group by period, preserving siteswaps order.\n const byP = new Map();\n for (const s of ss2) {\n const p = s.length;\n const a = byP.get(p);\n if (a) {\n a.push(s);\n } else {\n byP.set(p, [s]);\n }\n }\n\n const ps = [...byP.keys()].sort((a, b) => a - b);\n const maxRows = ps.reduce((m, p) => Math.max(m, byP.get(p).length), 0);\n\n const rows = [];\n for (let i = 0; i < maxRows; i++) {\n const row = {};\n for (const p of ps) {\n const s = byP.get(p)[i];\n if (s !== undefined) {\n row[`p${p}`] = s;\n }\n }\n if (Object.keys(row).length) {\n rows.push(row);\n }\n }\n\n return rows;\n}","pinned":false,"hidden":false,"mode":"ts","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":6351,"value":"$3c","pinned":false,"hidden":false,"mode":"ts","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":7198,"value":"$3d","pinned":false,"hidden":false,"mode":"ts","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":7166,"value":"### Sequence processing and visualization","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":7161,"value":"const alphabetToBalls = [\"🟠\", \"🔵\", \"🔴\", \"🟢\", \"🟣\", \"🟡\", \"🟤\", \"⚪\"];\nconst ballClrs = {\n \"🟠\": \"rgb(230,180,66)\",\n \"🔵\": \"rgb(75,104,201)\",\n \"🔴\": \"rgb(238,122,100\",\n \"🟢\": \"rgb(130,195,177)\",\n \"🟣\": \"rgb(174,88,242)\",\n \"🟡\": \"rgb(250,223,77)\",\n \"🟤\": \"rgb(158,124,95)\",\n \"⚪\": \"rgb(210,210,210)\"\n};","pinned":false,"hidden":false,"mode":"ts","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":7171,"value":"$3e","pinned":false,"hidden":false,"mode":"ts","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":7223,"value":"$3f","pinned":false,"hidden":false,"mode":"ts","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":7192,"value":"$40","pinned":false,"hidden":false,"mode":"ts","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":7447,"value":"function mFromImages(name, options = {}) {\n const {images = {0: \"0\", 1: \"1\"}, k = 5, notes = \"\"} = options;\n const alphabet = Object.keys(images).join(\"\");\n return {\n name,\n alphabet,\n images,\n f: (w) =>\n [...w]\n .map((x) => {\n if (!(x in images)) {\n throw new RangeError(\n `Symbol ${JSON.stringify(x)} is not in alphabet ${alphabet}`\n );\n }\n return images[x];\n })\n .join(\"\"),\n k,\n notes\n };\n}","pinned":false,"hidden":false,"mode":"ts","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":7229,"value":"const morphisms = {\n b2: mFromImages(\"Thue-Morse b₂\", {images: {0: \"01\", 1: \"10\"}}),\n b3: mFromImages(\"Hall-Thue b₃\", {images: {0: \"012\", 1: \"02\", 2: \"1\"}}),\n b4: mFromImages(\"b₄\", {images: {0: \"01\", 1: \"21\", 2: \"03\", 3: \"23\"}}),\n b5: mFromImages(\"b₅\", {\n images: {0: \"01\", 1: \"23\", 2: \"4\", 3: \"21\", 4: \"0\"}\n }),\n fib: mFromImages(\"Fibonacci (fib)\", {images: {0: \"01\", 1: \"0\"}, k: 14}),\n p: mFromImages(\"Currie ternary (p)\", {\n images: {0: \"01\", 1: \"21\", 2: \"0\"},\n k: 11\n }),\n pd: mFromImages(\"Period doubling (pd)\", {\n images: {0: \"01\", 1: \"00\"},\n k: 12\n }),\n metallicSturmian: (a, options = {}) => {\n const {k = 6} = options;\n return mFromImages(`Metallic Sturmian, a=${a}`, {\n images: {\n 0: `${\"0\".repeat(a)}1`,\n 1: \"0\"\n },\n k\n });\n },\n w4: mFromImages(\"w₄\", {\n images: {0: \"0123\", 1: \"1032\", 2: \"1032\", 3: \"0123\"},\n k: 5\n }),\n fib2: mFromImages(\"Ternary fib\", {\n images: {0: \"01\", 1: \"2\", 2: \"10\"},\n k: 11\n })\n};","pinned":false,"hidden":false,"mode":"ts","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":7226,"value":"// Generate a word from the given morphism by iterating some number of steps\nfunction iterateMorphism(morph, options = {}) {\n if (morph.constant) {\n // For constant words, we don't need to iterate.\n return {word: morph.constant};\n }\n const {seed = \"0\", steps = 5} = options;\n let word = seed;\n for (let i = 0; i < steps; i++) {\n word = morph.f(word);\n }\n return word;\n}","pinned":false,"hidden":false,"mode":"ts","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":7434,"value":"// Like iterateMorphism, but we specify the length of the finite word, not the number of iteration steps\nfunction iterateMorphismLen(morph, options = {}) {\n if (morph.constant) {\n // For constant words, we don't need to iterate.\n return {word: morph.constant};\n }\n const {minLength = 33, maxSteps = 50} = options;\n let [word, steps] = [\"\", 0];\n while (steps < maxSteps && word.length < minLength) {\n steps++;\n word = iterateMorphism(morph, {steps});\n }\n return {word, steps};\n}","pinned":false,"hidden":false,"mode":"ts","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":7272,"value":"// Iterates the morphism keeping track of the sequence at each iteration.\nfunction scanMorphism(f, n = 6) {\n const words = [\"0\"];\n for (let i = 1; i < n; i++) {\n words.push(f(words.at(-1)));\n }\n const balls = {\n \"0\": \"🟠\",\n \"1\": \"🔵\",\n \"2\": \"🔴\",\n \"3\": \"🟢\",\n \"4\": \"🟣\",\n \"5\": \"🟡\",\n \"6\": \"🟤\",\n \"7\": \"⚪\"\n };\n const catches = words.map((d) => [...d].map((b) => balls[b]).join(\"\"));\n return catches.map((d) => ({catch: d, throw: seqToSiteswap(d).siteswap}));\n}","pinned":false,"hidden":false,"mode":"ts","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":7735,"value":"function invGap(seq) {\n return \"0\" + [...seq].map((gap) => \"1\".repeat(gap) + \"0\").join(\"\");\n}","pinned":false,"hidden":false,"mode":"ts","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":6493,"value":"### References","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":6495,"value":"import {Refs} from \"observable:@gicentre/references-class\";","pinned":false,"hidden":false,"mode":"ts","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":6498,"value":"$41","pinned":false,"hidden":false,"mode":"ts","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":4311,"value":"### Page style","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":4314,"value":"import {defaultStyle} from \"observable:@jwolondon/utilities-cdt-2026\";","pinned":false,"hidden":false,"mode":"ts","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":4317,"value":"defaultStyle()","pinned":false,"hidden":false,"mode":"ts","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"}],"files":"$W42","stdlib":"2","titling":"auto"}} 2e:"$2d:body"