speice.io/assets/js/f8d03b4b.436bc33b.js

"use strict";(self.webpackChunkspeice_io=self.webpackChunkspeice_io||[]).push([["6602"],{30218:function(s,e,a){a.r(e),a.d(e,{shiftData:()=>_,default:()=>T,assets:()=>M,coefs:()=>L,metadata:()=>n,frontMatter:()=>R,toData:()=>S,contentTitle:()=>z,toc:()=>A,simpleData:()=>F});var n=a("37144"),l=a("85893"),i=a("50065"),t=a("32042"),m=a("13772"),r=a("57536"),c=a("45342"),h=a("1618"),d=a("71004"),x=a("67"),p=a("86861"),j=a("40653"),o=a("87320"),N=a("67294");function g(s){let{f:e}=s,{width:a,height:n,setPainter:i}=(0,N.useContext)(o.wn);return(0,N.useEffect)(()=>{a&&n&&i(e({width:a,height:n}))},[a,n]),(0,l.jsx)(l.Fragment,{})}function u(s,e,a){var n,l,i,t,m;let[r,c]=(n=s,l=e,[Math.floor(n*(i=a.width)),Math.floor(l*i)]);if(r<0||r>=a.width||c<0||c>=a.height)return;let h=(t=r,m=c,4*a.width*m+4*t);a.data[h]=0,a.data[h+1]=0,a.data[h+2]=0,a.data[h+3]=255}var y=a("42974");let v={Gasket:function(s){let{f:e}=s;return(0,l.jsx)(o.ke,{name:"gasket",children:(0,l.jsx)(g,{f:e})})},plot:u,randomBiUnit:y.J,randomInteger:function(s,e){return Math.floor(Math.random()*(e-s))+s}};var f=a("37955"),w=a("22237"),b=a("57037");function k(){let[s,e]=(0,N.useState)(1),[a,n]=(0,N.useState)(1),[i,t]=(0,N.useState)(1),m=(s,e)=>[s/2,e/2],r=(s,e)=>[(s+1)/2,e/2],c=(s,e)=>[s/2,(e+1)/2],{width:h,height:d,setPainter:x}=(0,N.useContext)(o.wn);(0,N.useEffect)(()=>{x(function*(s){let{width:e,height:a,transforms:n}=s,l=new ImageData(e,a),[i,t]=[(0,y.J)(),(0,y.J)()],m=e*a*.5;for(let s=0;s<m;s++){let[e,a]=(0,f.m)(n);[i,t]=a(i,t),s>20&&u(i,t,l),s%1e3==0&&(yield l)}yield l}({width:h,height:d,transforms:[[s,m],[a,r],[i,c]]}))},[s,a,i]);let p=(s,e,a)=>(0,l.jsx)(l.Fragment,{children:(0,l.jsxs)("div",{className:b.Z.inputElement,children:[(0,l.jsxs)("p",{children:[(0,l.jsx)(w.Z,{children:s}),": ",e]}),(0,l.jsx)("input",{type:"range",min:0,max:1,step:.01,value:e,onInput:s=>a(Number(s.currentTarget.value))})]})});return(0,l.jsx)(l.Fragment,{children:(0,l.jsxs)("div",{className:b.Z.inputGroup,style:{display:"grid",gridTemplateColumns:"1fr 1fr 1fr"},children:[p("F_0",s,e),p("F_1",a,n),p("F_2",i,t)]})})}let R={slug:"2024/11/playing-with-fire",title:"Playing with fire: The fractal flame algorithm",date:new Date("2024-12-16T21:30:00.000Z"),authors:["bspeice"],tags:[]},z=void 0,M={authorsImageUrls:[void 0]},F=[{x:0,y:0},{x:1,y:1},{x:2,y:2}],L={a:1,b:0,c:.5,d:0,e:1,f:1.5},S=([s,e])=>({x:s,y:e}),_=F.map(({x:s,y:e})=>S((0,p.z)(s,e,L))),A=[{value:"Iterated function systems",id:"iterated-function-systems",level:2},{value:"Solution set",id:"solution-set",level:3},{value:"Transform functions",id:"transform-functions",level:3},{value:"Fixed set",id:"fixed-set",level:3},{value:"Sierpinski&#39;s gasket",id:"sierpinskis-gasket",level:2},{value:"The chaos game",id:"the-chaos-game",level:3},{value:"Plotting",id:"plotting",level:3},{value:"Weights",id:"weights",level:3},{value:"Summary",id:"summary",level:2}];function I(s){let e={a:"a",admonition:"admonition",annotation:"annotation",blockquote:"blockquote",code:"code",em:"em",h2:"h2",h3:"h3",hr:"hr",li:"li",math:"math",mfrac:"mfrac",mi:"mi",mn:"mn",mo:"mo",mrow:"mrow",mspace:"mspace",mstyle:"mstyle",msub:"msub",msup:"msup",mtable:"mtable",mtd:"mtd",mtext:"mtext",mtr:"mtr",munderover:"munderover",p:"p",semantics:"semantics",span:"span",strong:"strong",ul:"ul",...(0,i.a)(),...s.components},{Details:a}=e;return!a&&function(s,e){throw Error("Expected "+(e?"component":"object")+" `"+s+"` to be defined: you likely forgot to import, pass, or provide it.")}("Details",!0),(0,l.jsxs)(l.Fragment,{children:[(0,l.jsxs)(e.p,{children:["Wikipedia describes ",(0,l.jsx)(e.a,{href:"https://en.wikipedia.org/wiki/Fractal_flame",children:"fractal flames"})," as:"]}),"\n",(0,l.jsxs)(e.blockquote,{children:["\n",(0,l.jsx)(e.p,{children:"a member of the iterated function system class of fractals"}),"\n"]}),"\n",(0,l.jsx)(e.p,{children:"It's tedious, but technically correct. I choose to think of them a different way: beauty in mathematics."}),"\n","\n",(0,l.jsx)("center",{children:(0,l.jsx)("img",{src:m.Z,style:{filter:(0,t.Z)()?"":"invert(1)"}})}),"\n",(0,l.jsx)(e.p,{children:"I don't remember when exactly I first learned about fractal flames, but I do remember being entranced by the images they created.\nI also remember their unique appeal to my young engineering mind; this was an art form I could participate in."}),"\n",(0,l.jsxs)(e.p,{children:["The ",(0,l.jsx)(e.a,{href:"https://flam3.com/flame_draves.pdf",children:"Fractal Flame Algorithm paper"})," describing their structure was too much\nfor me to handle at the time (I was ~12 years old), so I was content to play around and enjoy the pictures.\nBut the desire to understand it stuck around. Now, with a graduate degree under my belt, I wanted to revisit it."]}),"\n",(0,l.jsx)(e.p,{children:"This guide is my attempt to explain how fractal flames work so that younger me \u2014 and others interested in the art \u2014\ncan understand without too much prior knowledge."}),"\n",(0,l.jsx)(e.hr,{}),"\n",(0,l.jsx)(e.h2,{id:"iterated-function-systems",children:"Iterated function systems"}),"\n",(0,l.jsx)(e.admonition,{type:"note",children:(0,l.jsx)(e.p,{children:"This post covers section 2 of the Fractal Flame Algorithm paper"})}),"\n",(0,l.jsxs)(e.p,{children:['As mentioned, fractal flames are a type of "',(0,l.jsx)(e.a,{href:"https://en.wikipedia.org/wiki/Iterated_function_system",children:"iterated function system"}),',"\nor IFS. The formula for an IFS is short, but takes some time to work 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",(0,l.jsxs)(e.span,{className:"katex",children:[(0,l.jsx)(e.span,{className:"katex-mathml",children:(0,l.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,l.jsxs)(e.semantics,{children:[(0,l.jsx)(e.mrow,{children:(0,l.jsx)(e.mi,{children:"S"})}),(0,l.jsx)(e.annotation,{encoding:"application/x-tex",children:"S"})]})})}),(0,l.jsx)(e.span,{className:"katex-html","aria-hidden":"true",children:(0,l.jsxs)(e.span,{className:"base",children:[(0,l.jsx)(e.span,{className:"strut",style:{height:"0.6833em"}}),(0,l.jsx)(e.span,{className:"mord mathnormal",style:{marginRight:"0.05764em"},children:"S"})]})})]})," is the set of points in two dimensions (in math terms, ",(0,l.jsxs)(e.span,{className:"katex",children:[(0,l.jsx)(e.span,{className:"katex-mathml",children:(0,l.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,l.jsxs)(e.semantics,{children:[(0,l.jsxs)(e.mrow,{children:[(0,l.jsx)(e.mi,{children:"S"}),(0,l.jsx)(e.mo,{children:"\u2208"}),(0,l.jsxs)(e.msup,{children:[(0,l.jsx)(e.mi,{mathvariant:"double-struck",children:"R"}),(0,l.jsx)(e.mn,{children:"2"})]})]}),(0,l.jsx)(e.annotation,{encoding:"application/x-tex",children:"S \\in \\mathbb{R}^2"})]})})}),(0,l.jsxs)(e.span,{className:"katex-html","aria-hidden":"true",children:[(0,l.jsxs)(e.span,{className:"base",children:[(0,l.jsx)(e.span,{className:"strut",style:{height:"0.7224em",verticalAlign:"-0.0391em"}}),(0,l.jsx)(e.span,{className:"mord 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"solution" of some kind to our equation.\nOur goal is to find all the points in ',(0,l.jsxs)(e.span,{className:"katex",children:[(0,l.jsx)(e.span,{className:"katex-mathml",children:(0,l.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,l.jsxs)(e.semantics,{children:[(0,l.jsx)(e.mrow,{children:(0,l.jsx)(e.mi,{children:"S"})}),(0,l.jsx)(e.annotation,{encoding:"application/x-tex",children:"S"})]})})}),(0,l.jsx)(e.span,{className:"katex-html","aria-hidden":"true",children:(0,l.jsxs)(e.span,{className:"base",children:[(0,l.jsx)(e.span,{className:"strut",style:{height:"0.6833em"}}),(0,l.jsx)(e.span,{className:"mord mathnormal",style:{marginRight:"0.05764em"},children:"S"})]})})]}),", plot them, and display that image."]}),"\n",(0,l.jsxs)(e.p,{children:["For example, if we say 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time."}),"\n",(0,l.jsxs)(e.p,{children:['To start, we need to generate some random numbers. The "bi-unit square" is the range ',(0,l.jsxs)(e.span,{className:"katex",children:[(0,l.jsx)(e.span,{className:"katex-mathml",children:(0,l.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,l.jsxs)(e.semantics,{children:[(0,l.jsxs)(e.mrow,{children:[(0,l.jsx)(e.mo,{stretchy:"false",children:"["}),(0,l.jsx)(e.mo,{children:"\u2212"}),(0,l.jsx)(e.mn,{children:"1"}),(0,l.jsx)(e.mo,{separator:"true",children:","}),(0,l.jsx)(e.mn,{children:"1"}),(0,l.jsx)(e.mo,{stretchy:"false",children:"]"})]}),(0,l.jsx)(e.annotation,{encoding:"application/x-tex",children:"[-1, 1]"})]})})}),(0,l.jsx)(e.span,{className:"katex-html","aria-hidden":"true",children:(0,l.jsxs)(e.span,{className:"base",children:[(0,l.jsx)(e.span,{className:"strut",style:{height:"1em",verticalAlign:"-0.25em"}}),(0,l.jsx)(e.span,{className:"mopen",children:"["}),(0,l.jsx)(e.span,{className:"mord",children:"\u2212"}),(0,l.jsx)(e.span,{className:"mord",children:"1"}),(0,l.jsx)(e.span,{className:"mpunct",children:","}),(0,l.jsx)(e.span,{className:"mspace",style:{marginRight:"0.1667em"}}),(0,l.jsx)(e.span,{className:"mord",children:"1"}),(0,l.jsx)(e.span,{className:"mclose",children:"]"})]})})]}),",\nand we can do this using an existing API:"]}),"\n","\n",(0,l.jsx)(x.Z,{language:"typescript",children:"export function randomBiUnit() {\n    return Math.random() * 2 - 1;\n}"}),"\n",(0,l.jsxs)(e.p,{children:["Next, we need to choose a random integer from ",(0,l.jsxs)(e.span,{className:"katex",children:[(0,l.jsx)(e.span,{className:"katex-mathml",children:(0,l.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,l.jsxs)(e.semantics,{children:[(0,l.jsx)(e.mrow,{children:(0,l.jsx)(e.mn,{children:"0"})}),(0,l.jsx)(e.annotation,{encoding:"application/x-tex",children:"0"})]})})}),(0,l.jsx)(e.span,{className:"katex-html","aria-hidden":"true",children:(0,l.jsxs)(e.span,{className:"base",children:[(0,l.jsx)(e.span,{className:"strut",style:{height:"0.6444em"}}),(0,l.jsx)(e.span,{className:"mord",children:"0"})]})})]})," to ",(0,l.jsxs)(e.span,{className:"katex",children:[(0,l.jsx)(e.span,{className:"katex-mathml",children:(0,l.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,l.jsxs)(e.semantics,{children:[(0,l.jsxs)(e.mrow,{children:[(0,l.jsx)(e.mi,{children:"n"}),(0,l.jsx)(e.mo,{children:"\u2212"}),(0,l.jsx)(e.mn,{children:"1"})]}),(0,l.jsx)(e.annotation,{encoding:"application/x-tex",children:"n - 1"})]})})}),(0,l.jsxs)(e.span,{className:"katex-html","aria-hidden":"true",children:[(0,l.jsxs)(e.span,{className:"base",children:[(0,l.jsx)(e.span,{className:"strut",style:{height:"0.6667em",verticalAlign:"-0.0833em"}}),(0,l.jsx)(e.span,{className:"mord mathnormal",children:"n"}),(0,l.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2222em"}}),(0,l.jsx)(e.span,{className:"mbin",children:"\u2212"}),(0,l.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2222em"}})]}),(0,l.jsxs)(e.span,{className:"base",children:[(0,l.jsx)(e.span,{className:"strut",style:{height:"0.6444em"}}),(0,l.jsx)(e.span,{className:"mord",children:"1"})]})]})]}),":"]}),"\n","\n",(0,l.jsx)(x.Z,{language:"typescript",children:"export function randomInteger(\n    min: number,\n    max: number\n) {\n    let v = Math.random() * (max - min);\n    return Math.floor(v) + min;\n}"}),"\n",(0,l.jsx)(e.h3,{id:"plotting",children:"Plotting"}),"\n",(0,l.jsxs)(e.p,{children:["Finally, implementing the ",(0,l.jsx)(e.code,{children:"plot"})," function. This blog series is interactive,\nso everything displays directly in the browser. As an alternative,\nsoftware like ",(0,l.jsx)(e.code,{children:"flam3"}),' and Apophysis can "plot" by saving an image to disk.']}),"\n",(0,l.jsxs)(e.p,{children:["To see the results, we'll use the ",(0,l.jsx)(e.a,{href:"https://developer.mozilla.org/en-US/docs/Web/API/Canvas_API",children:"Canvas API"}),".\nThis allows us to manipulate individual pixels in an image and show it on screen."]}),"\n",(0,l.jsxs)(e.p,{children:["First, we need to convert from fractal flame coordinates to pixel coordinates.\nTo simplify things, we'll assume that we're plotting a square image\nwith range ",(0,l.jsxs)(e.span,{className:"katex",children:[(0,l.jsx)(e.span,{className:"katex-mathml",children:(0,l.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,l.jsxs)(e.semantics,{children:[(0,l.jsxs)(e.mrow,{children:[(0,l.jsx)(e.mo,{stretchy:"false",children:"["}),(0,l.jsx)(e.mn,{children:"0"}),(0,l.jsx)(e.mo,{separator:"true",children:","}),(0,l.jsx)(e.mn,{children:"1"}),(0,l.jsx)(e.mo,{stretchy:"false",children:"]"})]}),(0,l.jsx)(e.annotation,{encoding:"application/x-tex",children:"[0, 1]"})]})})}),(0,l.jsx)(e.span,{className:"katex-html","aria-hidden":"true",children:(0,l.jsxs)(e.span,{className:"base",children:[(0,l.jsx)(e.span,{className:"strut",style:{height:"1em",verticalAlign:"-0.25em"}}),(0,l.jsx)(e.span,{className:"mopen",children:"["}),(0,l.jsx)(e.span,{className:"mord",children:"0"}),(0,l.jsx)(e.span,{className:"mpunct",children:","}),(0,l.jsx)(e.span,{className:"mspace",style:{marginRight:"0.1667em"}}),(0,l.jsx)(e.span,{className:"mord",children:"1"}),(0,l.jsx)(e.span,{className:"mclose",children:"]"})]})})]})," for both ",(0,l.jsxs)(e.span,{className:"katex",children:[(0,l.jsx)(e.span,{className:"katex-mathml",children:(0,l.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,l.jsxs)(e.semantics,{children:[(0,l.jsx)(e.mrow,{children:(0,l.jsx)(e.mi,{children:"x"})}),(0,l.jsx)(e.annotation,{encoding:"application/x-tex",children:"x"})]})})}),(0,l.jsx)(e.span,{className:"katex-html","aria-hidden":"true",children:(0,l.jsxs)(e.span,{className:"base",children:[(0,l.jsx)(e.span,{className:"strut",style:{height:"0.4306em"}}),(0,l.jsx)(e.span,{className:"mord mathnormal",children:"x"})]})})]})," and ",(0,l.jsxs)(e.span,{className:"katex",children:[(0,l.jsx)(e.span,{className:"katex-mathml",children:(0,l.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,l.jsxs)(e.semantics,{children:[(0,l.jsx)(e.mrow,{children:(0,l.jsx)(e.mi,{children:"y"})}),(0,l.jsx)(e.annotation,{encoding:"application/x-tex",children:"y"})]})})}),(0,l.jsx)(e.span,{className:"katex-html","aria-hidden":"true",children:(0,l.jsxs)(e.span,{className:"base",children:[(0,l.jsx)(e.span,{className:"strut",style:{height:"0.625em",verticalAlign:"-0.1944em"}}),(0,l.jsx)(e.span,{className:"mord mathnormal",style:{marginRight:"0.03588em"},children:"y"})]})})]}),":"]}),"\n","\n",(0,l.jsx)(x.Z,{language:"typescript",children:"export function camera(\n  x: number,\n  y: number,\n  size: number\n): [number, number] {\n  return [\n    Math.floor(x * size),\n    Math.floor(y * size)\n  ];\n}"}),"\n",(0,l.jsxs)(e.p,{children:["Next, we'll store the pixel data in an ",(0,l.jsxs)(e.a,{href:"https://developer.mozilla.org/en-US/docs/Web/API/ImageData",children:[(0,l.jsx)(e.code,{children:"ImageData"})," object"]}),".\nEach pixel on screen has a corresponding index in the ",(0,l.jsx)(e.code,{children:"data"})," array.\nTo plot a point, we set that pixel to be black:"]}),"\n","\n",(0,l.jsx)(x.Z,{language:"typescript",children:'// hidden-start\nimport { camera } from "./cameraGasket";\n\n// hidden-end\nfunction imageIndex(\n  x: number,\n  y: number,\n  width: number\n) {\n  return y * (width * 4) + x * 4;\n}\n\nexport function plot(\n  x: number,\n  y: number,\n  img: ImageData\n) {\n  let [pixelX, pixelY] =\n    camera(x, y, img.width);\n\n  // Skip coordinates outside the display\n  if (\n    pixelX < 0 ||\n    pixelX >= img.width ||\n    pixelY < 0 ||\n    pixelY >= img.height\n  )\n    return;\n\n  const i = imageIndex(\n    pixelX,\n    pixelY,\n    img.width\n  );\n\n  // Set the pixel to black by setting\n  // the first three elements to 0\n  // (red, green, and blue, respectively),\n  // and 255 to the last element (alpha)\n  img.data[i] = 0;\n  img.data[i + 1] = 0;\n  img.data[i + 2] = 0;\n  img.data[i + 3] = 0xff;\n}'}),"\n",(0,l.jsx)(e.p,{children:"Putting it all together, we have our first image:"}),"\n","\n","\n",(0,l.jsx)(j.Z,{scope:v,noInline:!0,children:"// Hint: try changing the iteration count\nconst iterations = 100000;\n\n// Hint: negating `x` and `y` creates some cool images\nconst xforms = [\n  (x, y) => [x / 2, y / 2],\n  (x, y) => [(x + 1) / 2, y / 2],\n  (x, y) => [x / 2, (y + 1) / 2]\n];\n\nfunction* chaosGame({ width, height }) {\n  let img =\n    new ImageData(width, height);\n  let [x, y] = [\n    randomBiUnit(),\n    randomBiUnit()\n  ];\n\n  for (let i = 0; i < iterations; i++) {\n    const index =\n      randomInteger(0, xforms.length);\n    [x, y] = xforms[index](x, y);\n\n    if (i > 20)\n      plot(x, y, img);\n\n    if (i % 1000 === 0)\n      yield img;\n  }\n\n  yield img;\n}\n\nrender(<Gasket f={chaosGame} />);\n"}),"\n",(0,l.jsx)("hr",{}),"\n",(0,l.jsx)("small",{children:(0,l.jsxs)(e.p,{children:["The image here is slightly different than in the paper.\nI think the paper has an error, so I'm plotting the image\nlike the ",(0,l.jsx)(e.a,{href:"https://github.com/scottdraves/flam3/blob/7fb50c82e90e051f00efcc3123d0e06de26594b2/rect.c#L440-L441",children:"reference implementation"}),"."]})}),"\n",(0,l.jsx)(e.h3,{id:"weights",children:"Weights"}),"\n",(0,l.jsxs)(e.p,{children:['There\'s one last step before we finish the introduction. So far, each transform has\nthe same chance of being picked in the chaos game.\nWe can change that by giving them a "weight" (',(0,l.jsxs)(e.span,{className:"katex",children:[(0,l.jsx)(e.span,{className:"katex-mathml",children:(0,l.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,l.jsxs)(e.semantics,{children:[(0,l.jsx)(e.mrow,{children:(0,l.jsxs)(e.msub,{children:[(0,l.jsx)(e.mi,{children:"w"}),(0,l.jsx)(e.mi,{children:"i"})]})}),(0,l.jsx)(e.annotation,{encoding:"application/x-tex",children:"w_i"})]})})}),(0,l.jsx)(e.span,{className:"katex-html","aria-hidden":"true",children:(0,l.jsxs)(e.span,{className:"base",children:[(0,l.jsx)(e.span,{className:"strut",style:{height:"0.5806em",verticalAlign:"-0.15em"}}),(0,l.jsxs)(e.span,{className:"mord",children:[(0,l.jsx)(e.span,{className:"mord mathnormal",style:{marginRight:"0.02691em"},children:"w"}),(0,l.jsx)(e.span,{className:"msupsub",children:(0,l.jsxs)(e.span,{className:"vlist-t 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