29:null 2a:["$","title",null,{"children":"Moser's Circle Problem | Matt Russell | Observable"}] 2f:T825,function f(N) { const center = { x: 250, y: 250 }; const radius = 200; const PI = Math.PI; //Make an SVG Container const svg = DOM.element("svg:svg", {width:500, height: 500}); const svgContainer = d3.select(svg); function appendHTML(html) { svgContainer.html(svgContainer.html() + html); } //Draw the Ellipse svgContainer.append("ellipse") .attr("cx", center.x) .attr("cy", center.y) .attr("rx", radius) .attr("ry", radius) .attr("fill", "none") .attr("stroke", "black") ; function pointAtRadians(theta) { let x = Math.cos(theta); let y = Math.sin(theta); return { x: x * radius + center.x, y: y * radius + center.y, }; }; var randomPoints = []; for (let i = 0; i < N; i++) { randomPoints.push(2 * PI * Math.random()); } var points = [ // 0, // PI/4 + .1, // PI/2, // PI, // PI + PI/4, // 3 * PI/2, // -.5, // PI/2+.3, // PI/2+.6, // 3 * PI/2 + .15, // 3 * PI/2 + .42, ].concat(randomPoints).map(pointAtRadians).map(p => new Point(p.x, p.y)); var lines = []; for (let i = 0; i < points.length; i++) { for (let j = i+1; j < points.length; j++) { let p1 = points[i]; let p2 = points[j]; let s = new Segment(p1, p2); // let x = svgContainer.append("line"); appendHTML(s.svg({"stroke": "blue"})); // x.html(s.svg({"stroke": "blue"})); lines.push(s); } } var intersections = []; for (let i = 0; i < lines.length; i++) { for (let j = i+1; j < lines.length; j++) { let l1 = lines[i]; let l2 = lines[j]; for (let intersection of l1.intersect(l2)) { intersections.push(intersection); } } } intersections = _.filter(intersections, i => { let distance = i.distanceTo(new Point(center.x, center.y))[0]; return (distance < radius - .1); // - 0.1 is due to floating point stuff }); let num_intersections_in_middle = intersections.length; return { svg: svg, lines: lines.length, intersections: num_intersections_in_middle }; }30:[] 2d:{"id":"640d4df2406e21b3","slug":"mosers-circle-problem","title":"Moser's Circle Problem","thumbnail":"5191e1a686f500ef90e2ba14c2dbc68bc740e86b00de888d4064c69e1299ad07","access_level":"public","create_time":"$D2018-05-27T16:33:03.716Z","update_time":"$D2018-05-29T22:02:42.097Z","view_time":null,"first_public_time":"$D2018-05-29T20:55:21.618Z","has_unpublished_edits":null,"fork":false,"stars":0,"starred":false,"owner":{"type":"team","id":"daa2ceb495af0d02","login":"russelldmatt","name":"Matt Russell","avatar_url":"https://avatars-next.observableusercontent.com/58e4695811bb91f8703e809e4ba2293faf27ca60dded2221c1cb81c04b14a46e","tier":"starter_2024"},"version":169,"publish_version":169,"role":null,"authored":false,"authors":[{"type":"individual","id":"83fe0b7bdc59eef6","login":"russelldmatt","name":"Matt Russell","avatar_url":"https://avatars-next.observableusercontent.com/58e4695811bb91f8703e809e4ba2293faf27ca60dded2221c1cb81c04b14a46e","tier":null}],"members":[{"type":"individual","id":"83fe0b7bdc59eef6","login":"russelldmatt","name":"Matt Russell","avatar_url":"https://avatars-next.observableusercontent.com/58e4695811bb91f8703e809e4ba2293faf27ca60dded2221c1cb81c04b14a46e","tier":null,"role":"editor"}],"comments":[],"custom_thumbnail":false,"trashed":false,"lapsed":null,"subscribed":null,"forks":0,"collection":null,"type":"notebook","body":{"title":"Moser's Circle Problem","cells":[{"id":0,"value":"md`# Moser's Circle Problem`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":163,"value":"md`Draw n points on the circumference of a circle. Connect each pair of points with a straight line segment. What is the maximum number of regions that the circle can be divided into as a function of n?`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":168,"value":"md`Drag the slider around to change the number of points on the circle`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":79,"value":"viewof N = {\n html`

test

`\n return html``\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":123,"value":"html`N = ${N}`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":95,"value":"svg = f(N).svg","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":151,"value":"exact_chart = {\n let fact = n => n <= 0 ? 1 : n * fact(n-1);\n let choose = (n, k) => k > n ? 0 : fact(n)/fact(k)/fact(n-k);\n return makeChart(d3.range(Math.max(15, N + 2)).map(n => { \n return { n : n, lines: choose(n, 2), intersections: choose(n, 4), regions: 1 + choose(n, 2) + choose(n, 4) }\n }))\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":101,"value":"empirical_chart = makeChart(d3.range(15).map(n => { \n let d = f(n); \n return { n : n, lines: d.lines, intersections: d.intersections, regions: 1 + d.lines + d.intersections }\n }))","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":153,"value":"function makeChart(data) { \n function applyBG(color, val) { \n return html`${val}` \n };\n let colorIfN = (val, i) => i === N ? applyBG(\"lightGray\", val) : val;\n return table(data, {\n columns: {\n n: {\n title: \"#points\",\n formatter: colorIfN\n },\n intersections: {\n title: \"#intersections\",\n formatter: colorIfN\n\n },\n lines: {\n title: \"#lines\",\n formatter: colorIfN\n\n },\n regions: {\n title: \"#regions\",\n formatter: colorIfN\n\n },\n }\n }\n )\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":87,"value":"$2f","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":10,"value":"d3 = require('d3')","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":14,"value":"_ = require('lodash')","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":17,"value":"flattenJS = require('flatten-js@0.3.9/dist/flatten.umd.min.js')","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":49,"value":"Point = flattenJS.Point","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":51,"value":"Segment = flattenJS.Segment","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":75,"value":"import {table} from \"@tmcw/tables@444\"","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"}],"files":"$W30","stdlib":"1","titling":"auto"}} 2e:"$2d:body"