3f:null 40:["$","title",null,{"children":"Barnsley-Harrington Set, Barnsley-Hardin Set, and more! | Silvas Workspapce | Observable"}] 46:T154f,## Ramblings > *The following statements are simply observations and may or may not be true* 1. There is ***persistance*** of some values of ${tex`c\in BH(\tau)`} for ${tex`|c|\leq2^{-0.5}`} and all ${tex`\tau`}. 2. For each ${tex`c\in\mathbb{D}`} there is an ***arithmetic sequence*** of ${tex`\tau`} for which the attractor ${tex`{\rm A}_c`} is the same. For example, the attractor ${tex`{\rm A}_c`} does not change when - ${tex`c=\dfrac{1}{\sqrt2}{\rm e}^{{\rm i}\pi/4}`} and either ${tex`\tau\equiv0\mod2`} (*the twin dragon*) or ${tex`\tau\equiv1\mod4`} (*the Levy's curve*) or ${tex`\tau\equiv3\mod4`} (*the reflected Heighway dragon*) or ${tex`\tau=\frac{2n+1}{2^m}+4t`} for fixed ${tex`n,m\in\mathbb{N}`} and varying ${tex`t\in\mathbb{Z}`}; - ${tex`c=\dfrac{1}{\sqrt2}{\rm e}^{{\rm i}3\pi/4}`} and either ${tex`\tau\equiv0\mod2`} (*the reflected twin dragon*) or ${tex`\tau\equiv1\mod4`} (*a solid equilateral triangle*) or ${tex`\tau\equiv3\mod4`} (*the Heighway dragon*) or ${tex`\tau=4t+\frac{2n+1}{2^m}`} for fixed ${tex`n,m\in\mathbb{N}`} and varying ${tex`t\in\mathbb{Z}`}; - ${tex`c=\pm\dfrac{1}{\sqrt2}{\rm e}^{{\rm i}\pi/2}`} and ${tex`\tau=2t+\frac{2^m+1}{2^m}`} for fixed ${tex`m\in\mathbb{N}`} and varying ${tex`t\in\mathbb{Z}`} (*a rectangle*; obvious?); - ${tex`c=+0.5`} and ${tex`\tau\in\mathbb{R}`} (*the segment ${tex`[-2,2]`}*; this is obvious); - ${tex`c=\dfrac{1}{\sqrt2}{\rm e}^{{\rm i}\pi/7}`} and ${tex`\tau=\frac{7}{2}t`} for ${tex`t\in\mathbb{Z}`}; - ${tex`c=\dfrac{1}{4}(1+\sqrt{7}{\rm i})=\dfrac{1}{\sqrt2}{\rm e}^{{\rm i}\arctan(\sqrt{7})}`} and ${tex`\tau=\frac{\pi}{2\arctan(\sqrt{7})}n`} for ${tex`n\in\mathbb{Z}`} (*the Tame Twin Dragon*); 3. If we let ${tex`\tau\to\infty`}, then ${tex`BH(\tau)`} seems to be "approaching" (in Hausdorff sense?) to ${tex`BH(1)`}.

**[Sketch] Proof of (2.)** Let ${tex`\theta=t\pi`} for ${tex`t\in\mathbb{R}`} then let ${tex`\tau=\frac{1}{2t}n`} for ${tex`n\in\mathbb{Z}`}, then ${tex.block`b={\rm e}^{-{\rm i}2\tau\theta}c=\vert c\vert{\rm e}^{{\rm i}(1-\frac{1}{t}n)t\pi}=(-1)^{n}c.`} The IFS is ${tex`\{-1-c z, 1+c z\}`} if ${tex`n`} is odd, and ${tex`\{-1+c z, 1+c z\}`} if ${tex`n`} is even. Denote by ${tex`\hat{\rm A}_c`} the attractor of the former IFS, and let ${tex`{\rm A}_c`} be the attractor of the latter. By definition we have ${tex.block`\hat{\rm A}_c=(-1-c\hat{\rm A}_c)\cup(1+c\hat{\rm A}_c)\text{ and }{\rm A}_c=(-1+c{\rm A}_c)\cup(1+c{\rm A}_c).`} Observe that the attractor ${tex`{\rm A}_c`} is symmetric with respect to the involution ${tex`s(z)=-z`}: set ${tex`f(z)=1+cz`} and ${tex`g(z)=-1+cz`}, then one checks easily that ${tex`f=s\circ g\circ s`} and ${tex`g=s\circ f\circ s`} proving that ${tex`s({\rm A}_c)={\rm A}_c`}. Now, note that ${tex`g(s(z))=-1-cz`}, thus ${tex.block`{\rm A}_c=f({\rm A}_c)\cup g({\rm A}_c)=f({\rm A}_c)\cup g(s({\rm A}_c))=\hat{\rm A}_c`} where the last equality is given by the fact that the attractor of an IFS is unique. Let ${tex`\theta=t\pi`} for ${tex`t\in\mathbb{R}`} then let ${tex`\tau=\frac{1}{t}n`} for ${tex`n\in\mathbb{Z}`}, then ${tex.block`b=\vert c\vert{\rm e}^{{\rm i}(1-\frac{2}{t}n)t\pi}=c.`} Then for each integer ${tex`n`}, the IFS is ${tex`\{-1+cz, 1+cz\}`} and thus only one attractor. Let ${tex`\theta=t\pi`} for ${tex`t\in\mathbb{R}`} then let ${tex`\tau=\frac{1}{t}n+k`} for ${tex`n\in\mathbb{Z}`} and for a fixed ${tex`k\in\mathbb{R}`}, then ${tex.block`b=\vert c\vert{\rm e}^{{\rm i}(1-\frac{2}{t}n-2k)t\pi}=\vert c\vert{\rm e}^{{\rm i}(1-2k)\theta}.`} Then for each integer ${tex`n`}, the IFS is ${tex`\{-1+bz, 1+cz\}`} and thus only one attractor. ☐

**Examples from (2.)** - Let ${tex`\theta=\frac{\pi}{4}`} and ${tex`\tau\equiv0\mod2`}. Since ${tex`\tau=2n`} for integers ${tex`n`}, then ${tex.block`b={\rm e}^{-{\rm i}2\tau\theta}c=\vert c\vert{\rm e}^{{\rm i}(1-4n)\frac{\pi}{4}}=(-1)^{n}c.`} The IFS is ${tex`\{-1-c z, 1+c z\}`} if ${tex`n`} is odd, and ${tex`\{-1+c z, 1+c z\}`} if ${tex`n`} is even. However, the two IFS give the same attractor by the argument above. - Let ${tex`\theta=\frac{\pi}{4}`} and ${tex`\tau\equiv1\mod4`}. Since ${tex`\tau=4n+1`} for integers ${tex`n`}, then ${tex.block`b=\vert c\vert{\rm e}^{{\rm i}(1-2(4n+1))\frac{\pi}{4}}=\overline{c}.`} Then for each integer ${tex`n`}, the IFS is ${tex`\{-1+\overline{c}z, 1+c z\}`} and thus only one attractor. - Let ${tex`\theta=\frac{\pi}{4}`} and ${tex`\tau\equiv3\mod4`}. Since ${tex`\tau=4n+3`} for integers ${tex`n`}, then ${tex.block`b=\vert c\vert{\rm e}^{{\rm i}(1-2(4n+3))\frac{\pi}{4}}=-\overline{c}.`} Then for each integer ${tex`n`}, the IFS is ${tex`\{-1-\overline{c}z, 1+c z\}`} and thus only one attractor. - Let ${tex`\theta=\frac{\pi}{7}`} and ${tex`\tau=\frac{7}{2}n`} for integers ${tex`n`}. Then ${tex.block`b=\vert c\vert{\rm e}^{{\rm i}(1-7n)\frac{\pi}{7}}=(-1)^{n}c.`} The IFS is ${tex`\{-1-c z, 1+c z\}`} if ${tex`n`} is odd, and ${tex`\{-1+c z, 1+c z\}`} if ${tex`n`} is even. However, the two IFS give the same attractor by the argument above. - Let ${tex`\theta=\arctan(\sqrt{7})`} and ${tex`\tau=\frac{\pi}{2\arctan(\sqrt{7})}n`} for integers ${tex`n`}. Then ${tex.block`b=\vert c\vert{\rm e}^{{\rm i}(1-\frac{\pi}{\arctan(\sqrt{7})}n)\arctan(\sqrt{7})}=(-1)^{n}c.`} The IFS is ${tex`\{-1-c z, 1+c z\}`} if ${tex`n`} is odd, and ${tex`\{-1+c z, 1+c z\}`} if ${tex`n`} is even. However, the two IFS give the same attractor by the argument above.47:T5ba,// A canvas to draw the Parametr Plane set BH_canvas = { const minRe = -0.707; const maxRe = 0.707; const minIm = -0.707; const maxIm = 0.707; const w = 0.5 * width; const canvas = d3 .create('canvas') .attr('width', w) .attr('height', w); drawBarnsleySet(canvas.node(), minRe,maxRe,minIm,maxIm); const context = canvas.node().getContext("2d"); context.fillStyle = "#ccc";//"#ff3b3b"; const c = Lambda.attr('c').split(',').map(parseFloat); const ij = cplx_to_pxl(new Complex(c[0],c[1]),{x:minRe,y:minIm},{x:maxRe,y:maxIm},w,w); context.fillRect(ij[0] + 1, ij[1] - 1, 5, 5); const render = function(ij) { const xy = pxl_to_cplx(ij, minRe,maxRe,minIm,maxIm, w); // The lazy programmer's way to cover up the previous clicked point drawBarnsleySet(canvas.node(), minRe,maxRe,minIm,maxIm); context.fillRect(ij[0] + 1, ij[1] - 1, 5, 5); d3.select(Attractor_svg) .select('image') .attr('xlink:href', drawAttractor( new Complex(xy[0], xy[1]) )); Lambda.attr('c', pxl_to_cplx(ij, minRe,maxRe,minIm,maxIm, w)); Lambda.html(`c = ${d3.format('.2f')(xy[0])} + ${d3.format('.2f')(xy[1])} i`); }; let clicked = false; canvas.on('mousedown', function() { clicked = true; render(d3.mouse(this)); }); canvas.on('mousemove', function() { if (clicked) render(d3.mouse(this)); }); canvas.on('mouseup', function() { clicked = false; }); return canvas; }48:T4f9,// A function to draw the Barnsley Set function drawBarnsleySet(canvas,minRe,maxRe,minIm,maxIm){ const w = width / 2; const BHS_context = canvas.getContext("2d"); const canvasData = BHS_context.createImageData(canvas.width, canvas.height); const clrBarnsley = PALETTE[3]; const clrOutside = PALETTE[2]; for (let i = 0; i < canvas.width; i++) { for (let j = 0; j < canvas.height; j++) { const xy = pxl_to_cplx([i, j], minRe, maxRe, minIm, maxIm, w); const current_c = new Complex(xy[0],xy[1]); let col; if (current_c.abs>=1.0){//we know the fate already col = clrOutside; } else if (current_c.abs>0.707){//we know the fate as well col = clrOutside;//clrBarnsley; } else if (current_c.abs>0.5){//interesting region let isInside = disconnected_general(current_c, MAXDepth); col = isInside<0 ? clrOutside : clrBarnsley; } else {//not in the set col = clrOutside; } const idx = (i + j * canvas.width) * 4; canvasData.data[idx + 0] = col[0]; canvasData.data[idx + 1] = col[1]; canvasData.data[idx + 2] = col[2]; canvasData.data[idx + 3] = 255; } } BHS_context.putImageData(canvasData, 0, 0); }49:T4eb,// Generate the dataURL that holds an image of the Attractor. function drawAttractor(lambda){ const w = Math.floor(width / 2); const context = DOM.context2d(w, w); context.canvas.width = w; context.canvas.height = w; const lowerLeft = {x:-3.0, y:-3.0}; const upperRight = {x:3.0, y:3.0}; const canvasData = context.createImageData(w, w); //set every pixel to background color for (let i = 0; i < w; i = i + 1) { for (let j = 0; j < w; j = j + 1) { const idx = (i + j * w) * 4; canvasData.data[idx + 0] = PALETTE[2][0]; canvasData.data[idx + 1] = PALETTE[2][1]; canvasData.data[idx + 2] = PALETTE[2][2]; canvasData.data[idx + 3] = 255; } } const finalPoints = generate_attractor(lambda.abs, lambda.arg, isRANDOM); //update the pixel color for the points in the attractor finalPoints.forEach( pt => { const pix = cplx_to_pxl(pt.val,lowerLeft, upperRight, w, w); if (pix) { const idx = (pix[0] + pix[1] * w) * 4; canvasData.data[idx + 0] = pt.col[0]; canvasData.data[idx + 1] = pt.col[1]; canvasData.data[idx + 2] = pt.col[2]; canvasData.data[idx + 3] = 255; } }); context.putImageData(canvasData, 0, 0); return context.canvas.toDataURL(); }4a:T425,class Complex { constructor(a,b,polar=false){ if (polar) { this.real = a*Math.cos(b); this.imag = a*Math.sin(b); this.arg = b; this.abs = a; } else { this.real = a; this.imag = b; this.arg = Math.atan2(this.imag,this.real); this.abs = Math.sqrt(Math.pow(this.real,2)+Math.pow(this.imag,2)); } } toString(polar=false){ if (polar){ return (" |λ| = "+this.abs+"; θ = π ("+this.arg/Math.PI+")"); } else{ if (this.imag<0) { return (" λ = "+this.real+""+this.imag+" i"); } else { return (" λ = "+this.real+"+"+this.imag+" i"); } } } print(polar=false){ if (polar){ console.log(this.abs+"*exp[PI*i*("+this.arg/Math.PI+")]"); } else{ if (this.imag<0) { console.log(this.real+""+this.imag+"*i"); } else { console.log(this.real+"+"+this.imag+"*i"); } } } norm_sq(){ return Math.pow(this.real,2)+Math.pow(this.imag,2); } }4b:T46e,class Ball { constructor(center,rad,word='',ref=''){ this.center = center; this.radius = rad; this.word = word; this.freq = {'f':0,'g':0}; this.ref = (ref === '' ? new Complex(1,0) : ref); this.calc_letters(); } calc_letters(){ if (this.word==='') return; for (let c of this.word) { this.freq[c] +=1; } } is_disjoint(ball){ return sum(this.center,times(ball.center,-1)).abs > (this.radius + ball.radius); } image_under(fcn,curr_c) { const new_center = (fcn == 'g' ? sum(this.center,times(this.ref,-1)) : sum(this.center,this.ref) ); const new_rad = curr_c.abs * this.radius; const g_scale = times(new Complex(1,-TAU*2*curr_c.arg,true),curr_c); const ref_g = times(raise(curr_c,this.freq.f),raise(g_scale,this.freq.g+1)); const ref_f = times(raise(curr_c,this.freq.f+1),raise(g_scale,this.freq.g)); const new_ref = (fcn == 'g' ? ref_g : ref_f); const new_word = this.word + fcn; return new Ball(new_center, new_rad, new_word, new_ref); } does_not_contain(pnt){ return (sum(pnt,times(-1,this.center)).abs > this.radius); } }4c:[] 44:{"id":"80f7deb0263f6120","slug":"bh-sets","title":"Barnsley-Harrington Set, Barnsley-Hardin Set, and more!","thumbnail":"30098056c66f9b7ddb0393b77232b36888f3d3769f6574a3c2ed55359ff0706c","access_level":"public","create_time":"$D2024-04-06T12:58:07.071Z","update_time":"$D2024-05-01T14:50:15.066Z","view_time":null,"first_public_time":"$D2024-04-07T14:21:02.768Z","has_unpublished_edits":null,"fork":false,"stars":1,"starred":false,"owner":{"type":"team","id":"768fc3a9171c17e2","login":"silvasworkspace","name":"Silvas Workspapce","avatar_url":"https://avatars-next.observableusercontent.com/ae8c2ad703961bf12e8816260fa3fb8249cd96e422b39ee5615250633bb459c0","tier":"starter_2024","bio":"","following":false},"version":879,"publish_version":null,"role":null,"authored":false,"authors":[{"type":"individual","id":"26e328310b657089","login":"silvestristefano","name":"Stefano Silvestri","avatar_url":"https://avatars-next.observableusercontent.com/ae8c2ad703961bf12e8816260fa3fb8249cd96e422b39ee5615250633bb459c0","tier":null,"bio":"","following":false}],"members":[{"type":"individual","id":"26e328310b657089","login":"silvestristefano","name":"Stefano Silvestri","avatar_url":"https://avatars-next.observableusercontent.com/ae8c2ad703961bf12e8816260fa3fb8249cd96e422b39ee5615250633bb459c0","tier":null,"role":"editor"},{"type":"team","id":"768fc3a9171c17e2","login":"silvasworkspace","name":"Silvas Workspapce","avatar_url":"https://avatars-next.observableusercontent.com/ae8c2ad703961bf12e8816260fa3fb8249cd96e422b39ee5615250633bb459c0","tier":"starter_2024","role":"viewer"}],"comments":[],"custom_thumbnail":false,"trashed":false,"lapsed":null,"subscribed":null,"forks":0,"collection":null,"type":"notebook","body":{"titling":"auto","title":"Barnsley-Harrington Set, Barnsley-Hardin Set, and more!","cells":[{"id":298,"value":"# Barnsley-Harrington Set, Barnsley-Hardin Set, and more!","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"","database":"$undefined"},{"id":271,"value":"widget = put_it_all_together()","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":330,"value":" ${tex`\\tau`}: change it to see what happens!","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"","database":"$undefined"},{"id":286,"value":"viewof TAU = Inputs.range([-10,10],{value:0,step:1});","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":328,"value":"Ckeck the box if you wish to calculate the Attractor in a stochastic way.","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"","database":"$undefined"},{"id":93,"value":"// isRANDOM = false; //algorithm to implement to generate the attractor\nviewof isRANDOM = Inputs.toggle({ label: \"Stochastic:\" })","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":726,"value":"## Introduction","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"","database":"$undefined"},{"id":425,"value":"Let ${tex`c`} be a non-zero value of the unit disk with ${tex`\\theta={\\rm Arg}(c)`}. Set ${tex`\\tau`} to be a real value.\n\nThe colored sets on the left picture form the Attractor ${tex`\\mathrm{A}_c`} of the Iterated Function System (IFS) ${tex.block`\\{-1+{\\rm e}^{-2\\tau\\theta{\\rm i}}c z, 1+c z\\}`}\n\nThe colored set on the right picture is the parameter plane ${tex`BH(\\tau)`} defined as: ${tex.block`BH(\\tau)=\\{c~|~\\mathrm{A}_c\\text{ is connected }\\}`}\n\nMore precisely, it is depicted only the intersection of ${tex`BH(\\tau)`} with the disk of radius ${tex`2^{-0.5}`}, since it is easy to show that the annulus ${tex`\\{z~|~2^{-0.5}\\leq |z| \\leq 1\\}`} is completely contained in ${tex`BH(\\tau)`} for any ${tex`\\tau`}. ","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"","database":"$undefined"},{"id":626,"value":"## Notes","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"","database":"$undefined"},{"id":458,"value":"The set ${tex`BH(0)`} was first introduced by Barnsley and Harrington in 1985.\n> Barnsley, M. F., and Harrington, A. N. *A Mandelbrot set for pairs of linear maps*, Physica **15D** (1985), 421--432. [DOI](https://doi.org/10.1016/S0167-2789%2885%2980008-7)\n\nThe set ${tex`BH(1)`} was first introduced by Barnsley and Hardin in 1989.\n> Barnsley, M. F., and D. P. Hardin. *A Mandelbrot Set Whose Boundary Is Piecewise Smooth*, Transactions of the American Mathematical Society, vol. **315**, no. 2 (1989), 641-–659. [DOI](https://doi.org/10.2307/2001299)\n\n","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"","database":"$undefined"},{"id":628,"value":"$46","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"","database":"$undefined"},{"id":309,"value":"## The code\nAdapted from [original](https://editor.p5js.org/silvas/full/mY17XsIbn)","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"","database":"$undefined"},{"id":324,"value":"### The maximum number of iterations for the BH set.","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"","database":"$undefined"},{"id":215,"value":"MAXDepth = 9; //where: B-Set, use: depends on width/height","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":326,"value":"### The maximum number of iterations for the Attractor.","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"","database":"$undefined"},{"id":76,"value":"MAXIterations = isRANDOM ? 1e4 : 15;\n// viewof MAXIterations = Inputs.range(isRANDOM ? [1e4,1e6] : [1, 15], isRANDOM ? { value: 30000, step: 1e4 } : { value: 15, step: 1 });","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":601,"value":"### Color Palette","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"","database":"$undefined"},{"id":39,"value":"PALETTE = [\n [240, 139, 0], //right part of attractor\n [73, 114, 183], //left part of attractor\n [250, 250, 250], // background\n [Math.abs(31-(1-TAU)*(150-31))%200,\n 119,\n Math.abs(180-(1-TAU)*(250-180))%250] // BHS\n];","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":239,"value":"function put_it_all_together() {\n let container = d3\n .create('div')\n .style('width', width + 'px')\n .style('height', 0.54 * width + 'px');\n container.append(() => BH_canvas.node()).style('float', 'left');\n container.append(() => Attractor_svg);\n container.append(() => Lambda.node());\n return container.node();\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":197,"value":"### Parameter plane code","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"","database":"$undefined"},{"id":211,"value":"$47","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":202,"value":"$48","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":199,"value":"function disconnected_general(curr_c,depth){\n // adjusted from https://www.aldenwalker.org/schottky.html\n const initial_ball = new Ball(new Complex(0,0), 1/(1-curr_c.abs));\n\n let stack = [ [initial_ball.image_under('f',curr_c), initial_ball.image_under('g',curr_c)] ]\n let steps = 0;\n while (stack.length > 0 ) {\n let b = stack.pop();\n \n if (b[0].is_disjoint(b[1])) continue;\n \n if (b[0].word.length == depth) return steps;\n \n let ff = b[0].image_under('f',curr_c);\n let fg = b[0].image_under('g',curr_c);\n let gf = b[1].image_under('f',curr_c);\n let gg = b[1].image_under('g',curr_c);\n //there are 4 children: [ff,gg],[ff,gf],[fg,gg],[fg,gf], but\n //[fg,gg] and [ff,gf] have the same displacement, so forget [ff,gf]\n stack.push([ff,gg]);\n stack.push([ff,gf]);\n stack.push([fg,gg]);\n stack.push([fg,gf]);\n \n steps++;\n }\n return -1;\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":66,"value":"### Dynamical Plane","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"","database":"$undefined"},{"id":82,"value":"// A place to draw the Attractor\n// The picture of the Attractor is drawn to a canvas which becomes the background of an SVG.\nAttractor_svg = {\n let w = width / 2;\n let svg = d3\n .create(\"svg\")\n .attr('width', w)\n .attr('height', w);\n \n // Set up the SVG background\n const mapbg = DOM.uid('mapbg');\n\n const c = Lambda.attr('c').split(',').map(parseFloat);\n const c_start = new Complex(c[0],c[1]);\n\n svg\n .append(\"defs\")\n .append('pattern')\n .attr('id', mapbg.id)\n .attr('patternUnits', 'userSpaceOnUse')\n .attr('width', w)\n .attr('height', w)\n .append(\"image\")\n .attr(\"xlink:href\", drawAttractor(c_start))\n .attr('width', w)\n .attr('height', w);\n\n svg\n .append(\"rect\")\n .attr(\"x\", 0)\n .attr(\"y\", 0)\n .attr(\"width\", w)\n .attr(\"height\", w)\n .attr(\"fill\", mapbg);\n\n return svg.node();\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":84,"value":"$49","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":68,"value":"function generate_attractor(rad, ang, ranMethod=false){\n const lambda = new Complex(rad, ang, true);\n const rot = new Complex(1,-TAU*2*ang,true);\n const fcnsList = [\n new CplxLinearMap(lambda,new Complex(1,0)),\n new CplxLinearMap(times(lambda,rot),new Complex(-1,0))\n ];\n \n const startPoint = new Complex(0,0);\n const ifs = new IFS(fcnsList);\n return ifs.iterate([{val:startPoint,col:[255,255,255]}],MAXIterations,ranMethod);\n}\n","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":70,"value":"### Starting point","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"","database":"$undefined"},{"id":72,"value":"// A place to display lambda\nLambda = d3\n .create('div')\n .style('width', width + 'px')\n .style('height', 0.033 * width + 'px')\n .style('text-align', 'center')\n .attr('c',\"-0.5,0.5\")\n .text(\"c = -0.5+0.5 i\")","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":61,"value":"### Utility classes and functions","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"","database":"$undefined"},{"id":0,"value":"$4a","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":12,"value":"function conj(z){\n z = (z instanceof Complex) ? z : new Complex(z,0);\n return new Complex(z.real,-z.imag);\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":15,"value":"function sum(z,w){\n z = (z instanceof Complex) ? z : new Complex(z,0);\n w = (w instanceof Complex) ? w : new Complex(w,0);\n \n return new Complex(z.real+w.real,z.imag+w.imag);\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":18,"value":"function times(z,w){\n z = (z instanceof Complex) ? z : new Complex(z,0);\n w = (w instanceof Complex) ? w : new Complex(w,0);\n \n let a = z.abs*w.abs;\n let t = z.arg+w.arg;\n return new Complex(a,t,true);\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":23,"value":"function raise(z,n){\n z = (z instanceof Complex) ? z : new Complex(z,0);\n \n let a = z.abs**n;\n let t = n*z.arg;\n return new Complex(a,t,true); \n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":26,"value":"function divide(z,w){\n z = (z instanceof Complex) ? z : new Complex(z,0);\n w = (w instanceof Complex) ? w : new Complex(w,0);\n \n let a = z.abs/w.abs;\n let t = z.arg-w.arg;\n return new Complex(a,t,true); \n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":29,"value":"class CplxLinearMap {\n constructor(scale,translate){\n this.scale = scale;\n this.trans = translate;\n }\n appliedTo(z){\n return sum(times(this.scale,z),this.trans);\n }\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":36,"value":"class IFS {\n constructor(fcnsList){\n // fcnsList = [CplxLinearMaps]\n this.fcns = fcnsList;\n this.numFcns = fcnsList.length;\n }\n apply_deterministic(zList){\n let output = [] \n this.fcns.forEach( (func,i)=>{\n zList.forEach( z =>{\n output.push({val:func.appliedTo(z.val),col:PALETTE[i]})\n })\n })\n return output; \n }\n apply_isRANDOM(zList){\n const randInd = Math.floor(Math.random()*this.numFcns);\n zList.push({val:this.fcns[randInd].appliedTo(zList[zList.length-1].val),col:PALETTE[randInd]});\n return zList; \n }\n iterate(pts, maxIter,rand=false){\n let iteration = 0;\n while (iterationimg_w || y<0 || y>img_h) ? null : [Math.floor(x),Math.floor(y)];\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":58,"value":"d3 = require('d3@5')","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"}],"files":"$W4c","stdlib":"1"}} 45:"$44:body"