2b:null 2c:["$","title",null,{"children":"Babylonian method visualized - computing sqrt | sw1227 | Observable"}] 31:T41f,chart = { const svg = d3.create("svg") .attr("viewBox", [0, 0, width, height]); svg.append("g") .call(xAxis); svg.append("g") .call(yAxis); const drawLine = (datum, color, strokeWidth=1, strokeDashArray="") => { svg.append("path") .datum(datum) .attr("fill", "none") .attr("stroke", color) .attr("stroke-width", strokeWidth) .attr("stroke-dasharray", strokeDashArray) .attr("d", line) }; // y = S / x drawLine(data, "steelblue", 2); // y = x drawLine([{x: 0, y: 0}, {x: S, y: S}], "lightgray", 1, "4 2") // Vertical line drawLine( [{x: initialValue, y: 0}, {x: initialValue, y: S / initialValue}], "red" ); // x(n) => x(n+1) const step = x => (x + S / x) / 2; let x = initialValue; d3.range(10).forEach(_ => { const xNext = step(x); drawLine( [{x: x, y: S / x}, {x: xNext, y: xNext}], "red" ); drawLine( [{x: xNext, y: xNext}, {x: xNext, y: S / xNext}], "red" ); x = xNext; }); return svg.node(); }32:[] 2f:{"id":"890c1eeb0d46ca2d","slug":"babylonian-method-visualized-sqrt","title":"Babylonian method visualized - computing sqrt","thumbnail":"853e47e6806b7025029597ea5e1eeb8357dfeb9715ca4ac0db55a472e8b8bd1d","access_level":"public","create_time":"$D2020-03-15T06:37:57.735Z","update_time":"$D2020-03-15T08:02:07.815Z","view_time":null,"first_public_time":"$D2020-03-15T08:00:16.527Z","fork":false,"stars":1,"starred":false,"owner":{"type":"team","id":"06475ae5027cde36","login":"sw1227","name":"sw1227","avatar_url":"https://avatars-next.observableusercontent.com/3e09ff2c49317e618eedc394f3b33dbb5c952acf0d9654683424e681e7857f8c","tier":"starter_2024"},"version":267,"publish_version":267,"role":null,"authored":false,"authors":[{"type":"individual","id":"4a648b0bd17e2c08","login":"sw1227","name":"sw1227","avatar_url":"https://avatars-next.observableusercontent.com/3e09ff2c49317e618eedc394f3b33dbb5c952acf0d9654683424e681e7857f8c","tier":null}],"members":[{"type":"individual","id":"4a648b0bd17e2c08","login":"sw1227","name":"sw1227","avatar_url":"https://avatars-next.observableusercontent.com/3e09ff2c49317e618eedc394f3b33dbb5c952acf0d9654683424e681e7857f8c","tier":null,"role":"editor"}],"comments":[],"custom_thumbnail":true,"trashed":false,"subscribed":null,"forks":0,"collection":{"id":"f828b02efcb99767","slug":"math","title":"Math","owner":{"id":"06475ae5027cde36","login":"sw1227","name":"sw1227"}},"type":"notebook","body":{"title":"Babylonian method visualized - computing sqrt","cells":[{"id":0,"value":"md`# Babylonian method visualized - computing sqrt`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":169,"value":"md`\nBabylonian method is an algorithm for calculating the square root ${tex`\\sqrt{S}`} of a real number ${tex`S`}.\nIt can be represented as the following iteration:\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":179,"value":"tex.block`\n \\left\\{\n \\begin{array}{l}\n x_0 \\simeq \\sqrt{S} \\:\\:\\:\\: (\\operatorname{e.g.} \\: S/2) \\\\\n x_{n+1} = \\frac{1}{2} \\bigl( x_n + \\frac{S}{x_{n}} \\bigr)\n \\end{array}\n \\right.\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":204,"value":"md`\nI found that this process can be visualized with [cobweb plot](https://en.wikipedia.org/wiki/Cobweb_plot), using the plots of ${tex`y = x`} and ${tex`y = S / x`}.\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":92,"value":"viewof S = {\n const form = html`\n`;\n form.i.oninput = () => form.o.value = `S = ${form.value = form.i.valueAsNumber}`;\n form.i.oninput();\n return form;\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":109,"value":"$31","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":115,"value":"viewof initialValue = {\n const form = html`\n`;\n form.i.oninput = () => form.o.value = `start = ${d3.format(\".1f\")(form.value = form.i.valueAsNumber)}`;\n form.i.oninput();\n return form;\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":34,"value":"data = d3.range(0, S, S/100).map(d => ({x: d+1, y: S/(d+1)}))","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":26,"value":"line = d3.line()\n .x(d => x(d.x))\n .y(d => y(d.y))","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":45,"value":"x = d3.scaleLinear()\n .domain([0, d3.max(data, d => d.x)]).nice()\n .range([margin.left, height - margin.right])","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":28,"value":"y = d3.scaleLinear()\n .domain([0, d3.max(data, d => d.y)]).nice()\n .range([height - margin.bottom, margin.top])","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":47,"value":"xAxis = g => g\n .attr(\"transform\", `translate(0,${height - margin.bottom})`)\n .call(d3.axisBottom(x))","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":49,"value":"yAxis = g => g\n .attr(\"transform\", `translate(${margin.left},0)`)\n .call(d3.axisLeft(y))","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":32,"value":"margin = ({top: 20, right: 20, bottom: 30, left: 30})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":30,"value":"height = Math.min(500, width)","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":15,"value":"d3 = require(\"d3@5\")","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"}],"files":"$W32","stdlib":"1","titling":"auto"}} 30:"$2f:body"