What does it look like if we extend the idea behind Ulam's spiral and look at prime lengths of the Archimedean spiral?
Sadly, not like much.
Created
December 6, 2016 18:44
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Archimedean Ulam Spiral
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| <!DOCTYPE html> | |
| <meta charset="utf-8"> | |
| <style> | |
| .line { | |
| fill: none; | |
| stroke: red; | |
| stroke-width: 1.5px; | |
| } | |
| .primePoint { | |
| fill: blue; | |
| stroke: none; | |
| } | |
| .compositePoint { | |
| fill: none; | |
| stroke: none; | |
| } | |
| </style> | |
| <svg width="960" height="600"></svg> | |
| <script src="https://d3js.org/d3.v4.min.js"></script> | |
| <script> | |
| var width = 960, | |
| height = 500, | |
| radius = Math.min(width, height) / 2 - 30; | |
| var nTurns = 25, | |
| scaleFactor = 1; | |
| var data = d3.range(0, 2 * nTurns * Math.PI, .01).map(function(t) { | |
| return [t, t]; | |
| }) | |
| var r = d3.scaleLinear() | |
| .domain([0, nTurns * 2 * Math.PI]) | |
| .range([0, radius]) | |
| var line = d3.radialLine() | |
| .radius(function(d) { return r(d[1]); }) | |
| .angle(function(d) { return -d[0] + Math.PI / 2; }); | |
| var svg = d3.select("svg") | |
| .attr("width", width) | |
| .attr("height", height) | |
| .append("g") | |
| .attr("transform", "translate(" + width * .5 + "," + height * .5 + ")"); | |
| svg.append("path") | |
| .datum(data) | |
| .attr("class", "line") | |
| .attr("d", line); | |
| var spiralArcLength = function(theta) { | |
| return 0.5 * (theta * Math.sqrt(1 + theta**2) + Math.log(theta + Math.sqrt(1 + theta**2))); | |
| } | |
| var spiralArcLengthDerivative = function(theta) { | |
| return (1 + theta**2) / Math.sqrt(1 + theta**2); | |
| } | |
| var spiralInverseArcLength = function(length) { | |
| length = length * scaleFactor; | |
| var x = Math.sqrt(8*length + 5) * 0.5 - 0.5, | |
| delta = Infinity, | |
| precision = 0.0001; | |
| while (Math.abs(delta) > precision) { | |
| var newX = x - (spiralArcLength(x) - length)/spiralArcLengthDerivative(x); | |
| delta = newX - x; | |
| x = newX; | |
| } | |
| return x; | |
| } | |
| function isPrime(value) { | |
| for(var i = 2; i < value; i++) { | |
| if(value % i === 0) { | |
| return false; | |
| } | |
| } | |
| return value > 1; | |
| } | |
| var totalSpiralLength = spiralArcLength(nTurns * Math.PI * 2); | |
| points = d3.range(1, totalSpiralLength / scaleFactor, 1).map(function(t) { | |
| var theta = spiralInverseArcLength(t); | |
| return {"theta":theta, "radius": r(theta), "x":Math.cos(theta) * r(theta), "y": - Math.sin(theta) * r(theta), "n": t }; | |
| }) | |
| var circles = svg.selectAll("circle").data(points); | |
| circles.enter().append("circle") | |
| .attr("class", function(d) { if (isPrime(d.n)) { return "primePoint"; } else { return "compositePoint"; } }) | |
| .attr("cx", function(d) { return d.x; }) | |
| .attr("cy", function(d) { return d.y; }) | |
| .attr("r", r(Math.log(nTurns) - 1)); | |
| </script> |
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