因为前一段时间看了一些近代，也了解了一些数论的皮毛，发现其中有个娱乐性的乌拉姆螺旋看起来颇有观赏性，于是自己做了一些实验，这里记录一下，并且给出了简单实现的源代码。
什么是乌拉姆螺旋(Ulam Spiral)
质数螺旋（国外叫作 Ulam spiral — “乌拉姆螺旋”）是在1963年被美籍波兰数学家斯塔尼斯拉夫·乌拉姆（Stanislaw Ulam）（1909年 － 1984年）发现。 他在会议无聊中在一张草稿纸上画了一个简单的整数螺旋。
(from Wikipedia)The Ulam spiral, or prime spiral (in other languages also called the Ulam Cloth) is a simple method of visualizing the prime numbers that reveals the apparent tendency of certain quadratic polynomials to generate unusually large numbers of primes. It was discovered by the mathematician Stanislaw Ulam in 1963, while he was doodling during the presentation of a “long and very boring paper” at a scientific meeting. Shortly afterwards, in an early application of computer graphics, Ulam with collaborators Myron Stein and Mark Wells used MANIAC II at Los Alamos Scientific Laboratory to produce pictures of the spiral for numbers up to 65,000. In March of the following year, Martin Gardner wrote about the Ulam spiral in his Mathematical Games column; the Ulam spiral featured on the front cover of the issue of Scientific American in which the column appeared.
简单的Ulam螺旋
Ulam螺旋的染色图
比较大的螺旋
扭曲的螺旋
生成基本Ulam Spiral的代码(HTML+JS)
<!DOCTYPE html><html><head><meta charset="utf-8"><title>Ulam Spiral</title><meta name="Keywords" content=""><meta name="Description" content=""><style type="text/css">body, h1{margin:0;}canvas{margin: 20px; }</style></head><body onload="draw()"><canvas id="canvas" width=1000 height=1000 style="border: 0px;"></canvas><script>function isPrime(n) {if (isNaN(n) || !isFinite(n) || n%1 || n<2) return false; if (n%2==0) return (n==2);if (n%3==0) return (n==3);var m=Math.sqrt(n);for (var i=5;i<=m;i+=6) {if (n%i==0)return false;if (n%(i+2)==0) return false;}return true;}var prime_colors = ["#444444", "#007FFF", "#cccccc", "green", "#ff3300"];function dot(context, x, y, r, num) {context.beginPath();context.arc(x, y, r, 0, Math.PI * 2, true);if (isPrime(num)) {context.fillStyle = prime_colors[(num-1)%10/2];} else {context.fillStyle = "#f3f3f3";}context.fill();if (1) {if (isPrime(num)) {context.fillStyle = "white";} else {context.fillStyle = "#aaaaaa";}context.textAlign='center';context.textBaseline='middle';context.font = (r*0.9)+"px Courier New";context.fillText(num, x, y);}}function draw() {var a = 40;var r = a/2;var border_length = 22;var cx = r+1000/2;var cy = r+1000/2;var canvas=document.getElementById('canvas');var context=canvas.getContext('2d');var num = 1;var direction = 0;var stride = 1;for (k=0; k<border_length; k++) {for (var i=0; i<stride; i++) {dot(context, cx, cy, r, num);if (direction == 0) cx += a;if (direction == 1) cx -= a;num+=1;}for (var i=0; i<stride; i++) {dot(context, cx, cy, r, num);if (direction == 0) cy -= a;if (direction == 1) cy += a;num+=1;}direction = ++direction%2;stride++;}}</script></body></html>
/wiki/Ulam_spiral/link?url=3-xkN-IcKPcc0qLrmMBpF9chgvk7el5E8VgUP427Dfvfa5TjZtOTX_R_6l6HxodRX0l_Vif5ontvjSQCbM9EKaGardner, M. (March 1964), “Mathematical Games: The Remarkable Lore of the Prime Number”, Scientific American 210: 120–128, doi:10.1038/scientificamerican0364-120
