31:T628,intro = jl` function orbit_counts( α; # the henon angle parameter sz = 1000, # size of the output canvas n_samples = 100_000, # number of total points we want to record on our canvas max_samples_per_orbit = 50_000, # the maximum number of samples for any individual orbit x_bounds = (-1.0, 1.0), # the x zoom bounds y_bounds = (-1.0, 1.0), # the y zoom bounds fuzz_factor = 0.0 # a fuzz factor to jitter points before plotting ) # create a henon attractor with our given angle h = henon(α) # linear scales, to integer coordinates in A ix, iy = lerp(x_bounds, sz), lerp(y_bounds, sz) # output array of visit counts for each bin counts = zeros(sz, sz) # loop until we collect the desired number of attractor samples count = 0 while count < n_samples # initialize a point p = init(h, x_bounds, y_bounds) for _ in 1:max_samples_per_orbit # iterate the attractor p = h(p) # if the orbit diverges, break out of the loop for this orbit if any(isinf, p) || any(isnan, p) break end # if the point is within bounds, increment the count in the corresponding bin if x_bounds[1] ≤ p[1] ≤ x_bounds[2] && y_bounds[1] ≤ p[2] ≤ y_bounds[2] x = clamp(ix(fuzz(p[1], fuzz_factor)), 1, sz) y = clamp(iy(fuzz(p[2], fuzz_factor)), 1, sz) counts[y, x] += 1 count += 1 end end end # return the array of counts counts end `32:T41a,jl` function visualize(orbits) # normalize each layer of orbit counts to (0, 1), then # concatenate them to create a 3D array of weights weights = cat(normalize.(orbits)..., dims=3) # try visualizing each pixel based on the difference between its # minimum value and maximum value across all of the orbit count layers. # there are other approaches; this is a simple one to start with. A = normalize(maximum(weights, dims=3) .- minimum(weights, dims=3)) # we're going to use the LCH color space. # lightness goes from 0-100 L = 100(1 .- A) # contrast goes from 0-300ish C = 300A # hue cycles with a period of 360 degrees H = 360normalize(sum(weights .^ 3, dims=3)) # create an array of colors by combining our lightness, contrast, and hue colors = RGBA.(LCHuv.(L, C, H), 1) # we want to return a 2D array. By this point our array is # width × height × 1, so we can just drop the third dimension dropdims(colors, dims=3) end # visualize the six orbit count matrices we computed visualize(orbits) 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2f:{"title":"Orbiting the Hénon Attractor","cells":[{"id":2672,"value":"md`# Orbiting the Hénon Attractor`","pinned":false,"hidden":false,"mode":"ojs"},{"id":0,"value":"md`\nThe [Hénon strange attractor](https://mathworld.wolfram.com/HenonMap.html) is a system of two simple equations that gives rise to chaotic behavior. \n\nI first encountered Jared Tarbell's\n[beautiful visualizations](http://www.complexification.net/gallery/machines/henonPhaseDeep/) of the attractor many years ago and recently explored making my own pictures using my favorite programming language at the time, [Julia](https://julialang.org).\n`","pinned":false,"hidden":false,"mode":"ojs"},{"id":2386,"value":"md`Here's a gallery of the most interesting pictures I created along the way. The code for these is described at the bottom of the post.\n\nClick on any picture to see it in full resolution.\n`","pinned":false,"hidden":false,"mode":"ojs"},{"id":2388,"value":"renderRows(\n [\n [\n FileAttachment(\n \"3b10a27c81764d79a7dc06337824b390131556177da972bf163b93996d42eeb5325ee7718215da85108d2137b1506bcbe09e0d4bfa4089579407825ad00cb1fc.png\"\n ),\n FileAttachment(\"out_15.png\")\n ],\n [\n FileAttachment(\n \"339be5e0150598c049b37c90d5502b9c5d6a44cb276502c6c4289db118e5e296f0bb74a832fadf38bb0cf17082e540dddd30066c4ff51da28c8a82076fb6a088.png\"\n ),\n FileAttachment(\"out_13.png\")\n ],\n [\n FileAttachment(\n \"9d4861820ee96418cfb67da96d08790ec7c5f39ed033a55acc5387a1434814169dd2530b7801a9a4288e953f755714100ea922d8a9b385d50403f13a0f781864.png\"\n ),\n FileAttachment(\"out_16.png\")\n ]\n ],\n 'Color variations made by generating several layers of orbits and compositing them together.'\n)","pinned":false,"hidden":false,"mode":"ojs"},{"id":2604,"value":"renderRows(\n [\n [\n FileAttachment(\n \"141cfa3fdf806158fd967932dba5aeccdd1378e88df62bac4e5f56b6504298e5018d4774b39981314c94d3297968a1d743f9b82d06408d46a90d91a27dc5a4bb.png\"\n ),\n FileAttachment(\"out_v3.png\")\n ],\n [\n FileAttachment(\n \"eee7f1d575226626cab4bc577f68add602896affc74e76d58dcb08034c3f57e0fa14fa26910d5486fe620e1a35e31ba2c556d3d86fb430f947d43fbf3d578f78.png\"\n ),\n FileAttachment(\"out_v4.png\")\n ],\n [\n FileAttachment(\n \"0cb760dae761bdf66121e03a2748dc2fb4a4067f81ee73d977047ec7e5d2075a24ef8b7d8aacd1924b19e22a8258ec798e4cd0ee9b08359fdc4de2a08ff433b4.png\"\n ),\n FileAttachment(\"out_v2-1.png\")\n ]\n ],\n html`Variations in blue, including a headless gingerbread man. It turns out there is an unrelated gingerbread man attractor.`\n)","pinned":false,"hidden":false,"mode":"ojs"},{"id":2582,"value":"renderRows(\n [\n [\n FileAttachment(\n \"9bc83b4519cf4c4214a69f355345ba0240a397bcf38c7537b839961f0fcae67f3c3b3d9e00b0e0e8f108adfdb008248bf026f88d7a7803e42c01dffce3ae6872.png\"\n ),\n FileAttachment(\"out_4.png\")\n ],\n [\n FileAttachment(\n \"d237ca2137a937dbf4079ef57454aa37375f49504a848c1fa3907a1e1c7cd491ce07e96e369ddaf31b017ea0e19acbb23ea34a4924bce64d5133239161a45b44.png\"\n ),\n FileAttachment(\"out_5.png\")\n ],\n [\n FileAttachment(\n \"02e80db82e1d077a989074ef9371b05df39ac7c272db90d6586ff5f8b1feb9a16315d68a72b92f54b77748c6591b96a154dfa0c3c2ce463d201703384a0cde23.png\"\n ),\n FileAttachment(\"out_3.png\")\n ]\n ],\n 'Close-up zooms of peaceful swooshy swirls.'\n)","pinned":false,"hidden":false,"mode":"ojs"},{"id":2597,"value":"renderRows(\n [\n [\n FileAttachment(\"861a603f-fbd1-4db5-8764-b2d69611b221.png\"),\n FileAttachment(\"v4.png\")\n ],\n [\n FileAttachment(\n \"4f8f0b9f5f28b71398af891e8d17e105bf2b8849c3be49a2763fc557f036d3a347c7f77fadb21234c769b51c0ff7a9758cffe43b41312493da3974cdf39a428c.png\"\n ),\n FileAttachment(\"out_6.png\")\n ],\n [\n FileAttachment(\n \"d77a0c502d75e3c37bfcaa402e0eeff1dd86d2bb5f1c32fe3553e2961cc54201cf65a40e631f0b474234b29005758348b4c1479e5665dc768a316d2b08718e13.png\"\n ),\n FileAttachment(\"out_20.png\")\n ]\n ],\n html`Exporing a larger zoom of one of the regions in Jared Tarell's Henon Phase Deep.`\n)","pinned":false,"hidden":false,"mode":"ojs"},{"id":2591,"value":"renderRows(\n [\n [\n FileAttachment(\n \"900bb6722957cb991912cbb7198136a0fb0e0886591b65b494ab2e16612d49309082fd561bf5443c556b33ce5335ec49b07a8c4d475bebd10b8d1fe3fa4ca298.png\"\n ),\n FileAttachment(\"out_11.png\")\n ],\n [\n FileAttachment(\n \"982d79c6e627ba00ef21c16dd61b009a65da203d5f416af77e6d8d53f3db63a25e4c48ab781f9892b2fc6bd674f8c2294f17adf6ff38ca0325c2c969703ef098.png\"\n ),\n FileAttachment(\"out_12.png\")\n ],\n [\n FileAttachment(\n \"5285021cea60fbbf3205195321997bf13ce6f4f94486f714dfef60c88e1da4752fcb920850cd1911eafdecacbe7679950ae81274b5d0f2b1077b2a200836c081.png\"\n ),\n FileAttachment(\"out_10.png\")\n ]\n ],\n 'Color variations using palettes from photographs.'\n)","pinned":false,"hidden":false,"mode":"ojs"},{"id":3104,"value":"md`---`","pinned":false,"hidden":false,"mode":"ojs"},{"id":2696,"value":"md`# Behind the Scenes\n\nHere's the [Julia](https://julialang.org) code that I wrote to explore this system.\n\n### 1. Introducing the attractor`","pinned":false,"hidden":false,"mode":"ojs"},{"id":2256,"value":"md`\nTo explore a chaotic attractor we need to define it in code. The Hénon map has an angle parameter ${tex`\\alpha`} so we define \\`henon(α)\\` to return the attractor function for that angle.\n`","pinned":false,"hidden":false,"mode":"ojs"},{"id":2252,"value":"jl`\n# define the attractor\nfunction henon(α)\n c = cos(α)\n s = sin(α)\n attractor((x, y)) = (x*c - (y - x^2)*s, x*s + (y - x^2)*c)\nend\n`","pinned":false,"hidden":false,"mode":"ojs"},{"id":2321,"value":"md`\nWe'll also need some utilities to help us along the way.\n\nFirst, a function to generate random points ${tex`(x, y)`} within our specified bounds, so that we can create detailed zoom images of specific parts of the phase space.\n`","pinned":false,"hidden":false,"mode":"ojs"},{"id":2279,"value":"jl`\n# randomly initialize an point within our bounds, then\n# iterate it a few times to remove initial noisy jumps\nfunction init(h, (xlo, xhi), (ylo, yhi))\n init_x = xlo + (xhi - xlo)rand()\n init_y = ylo + (yhi - ylo)rand()\n p = (init_x, init_y)\n for _ in 1:100\n p = h(p)\n end\n p\nend\n`","pinned":false,"hidden":false,"mode":"ojs"},{"id":2319,"value":"md`Second, a linear scale function, similar to [d3.*scaleLinear*](https://observablehq.com/@d3/d3-scalelinear), which interpolates values in the interval ${tex`(lo, hi)`} to produce integers from ${tex`1`} to ${tex`n`} that we can use for array indexing.`","pinned":false,"hidden":false,"mode":"ojs"},{"id":2304,"value":"jl`\n# map from continuous (lo, hi) to integer 1:n\nfunction lerp((lo, hi), n)\n @assert lo < hi\n x -> 1 + round(Int, (x - lo) / (hi - lo) * (n - 1))\nend`","pinned":false,"hidden":false,"mode":"ojs"},{"id":2327,"value":"md`Finally, a simple \\`fuzz\\` function to randomly jitter points. This technique is responsible for the organic/analog feeling of the images, particularly in deep zooms.`","pinned":false,"hidden":false,"mode":"ojs"},{"id":2308,"value":"jl`fuzz(x, amount) = x + amount * (rand() - 0.5)`","pinned":false,"hidden":false,"mode":"ojs"},{"id":2499,"value":"md`## 2. Computing orbits\n\nNow we get to the heart of the system — it samples Hénon orbits and returns a 2D array of counts indicating how many times each pixel bin was visited by an orbit.\n`","pinned":false,"hidden":false,"mode":"ojs"},{"id":13,"value":"$31","pinned":false,"hidden":false,"mode":"ojs"},{"id":2912,"value":"md`## 3. Making pictures`","pinned":false,"hidden":false,"mode":"ojs"},{"id":2917,"value":"md`\nNext, let's generate six 1,000 × 1,000 arrays of orbit counts to visualize:`","pinned":false,"hidden":false,"mode":"ojs"},{"id":659,"value":"jl`\norbits = [\n orbit_counts(\n 1.5957,\n sz = 1000,\n n_samples = 1_000_000,\n max_samples_per_orbit = 100_000,\n x_bounds = (.45, .55),\n y_bounds = (0.05, .15),\n fuzz_factor = 0.0001\n )\n for _ in 1:6\n]\n`","pinned":false,"hidden":false,"mode":"ojs"},{"id":3233,"value":"md`To make our pictures more interesting we generate multiple \"layers\" of counts and composite them together into the final image. This lets us take advantage of the variation between layers to make our pictures more interesting.`","pinned":false,"hidden":false,"mode":"ojs"},{"id":2739,"value":"md`To talk about colors and color spaces we're going to use the Julia [Images](https://github.com/JuliaImages/Images.jl) package:`","pinned":false,"hidden":false,"mode":"ojs"},{"id":2862,"value":"jl`using Images`","pinned":false,"hidden":false,"mode":"ojs"},{"id":3262,"value":"md`We define one last utility function, after which we're ready to visualize the Hénon attractor:`","pinned":false,"hidden":false,"mode":"ojs"},{"id":3260,"value":"jl`\n# normalize an array of positive numbers to (0, 1), being\n# careful not to introduce NaNs via division by zero.\nnormalize(A) = A ./ max(1e-6, maximum(A))\n`","pinned":false,"hidden":false,"mode":"ojs"},{"id":2978,"value":"md`\nFinally, the pièce de résistance! The \\`visualize\\` function takes a collection of 2D orbit arrays as input and produces an image as output:\n`","pinned":false,"hidden":false,"mode":"ojs"},{"id":740,"value":"$32","pinned":false,"hidden":false,"mode":"ojs"},{"id":2859,"value":"md`All of the pictures in this post were made with variations of this code.\n\n## Interesting directions\n\n- Use contrast normalization techniques, such as [histogram equalization](https://en.wikipedia.org/wiki/Histogram_equalization), which adjusts an image so that the full range of intensities is equally represented. Julia has a a great implementation in the [ImageContrastAdjustment](https://juliaimages.org/ImageContrastAdjustment.jl/stable/) package.\n\n- Try applying [tone mapping](https://en.wikipedia.org/wiki/Tone_mapping) technques, such as [Reinhard tone mapping](https://bruop.github.io/tonemapping/), for which a simple implementation is ${jl`\n# reinhard tone mapping operator\ntonemap(c, c_white) = c * (1 + c / c_white^2) / (1 + c)\n`}\n\n- Experiment with custom [blend modes](https://en.wikipedia.org/wiki/Blend_modes). Since the \\`visualize\\` function controls compositing, we can use standard blending operations in RGB or write our own.\n`","pinned":false,"hidden":false,"mode":"ojs"},{"id":3376,"value":"md`That's it. If you make something cool, [let me know](mailto:yurivish@gmail.com)!\n\nYuri Vishnevsky
July 2020`","pinned":false,"hidden":false,"mode":"ojs"},{"id":3524,"value":"FileAttachment(\"pngout@2.png\").image()","pinned":false,"hidden":false,"mode":"ojs"},{"id":3341,"value":"md`---`","pinned":false,"hidden":false,"mode":"ojs"},{"id":3389,"value":"// Sleep a bit to avoid recomputing with every jl`` evaluation\nPromises.delay(200).then(\n () => md`\n## Appendix: Julia code\n\nHere's the full code from this post.\n\n~~~julia\n${\n // for robustness against evaluation order, which we assume is sequential\n // but may be (and in practice is) reversed:\n (jlCache.findIndex(d => d.startsWith('# define the attractor')) == 0\n ? jlCache\n : Array.from(jlCache).reverse()\n ).join('\\n\\n')\n}\n~~~\n`\n)","pinned":false,"hidden":false,"mode":"ojs"},{"id":3428,"value":"md`---`","pinned":false,"hidden":false,"mode":"ojs"},{"id":3320,"value":"md`## Appendix: Internals\n\n`","pinned":false,"hidden":false,"mode":"ojs"},{"id":2390,"value":"renderRows = async (files, caption) => html`

${await Promise.all(\n files.map(([lowRes, highRes]) =>\n renderImage(lowRes, width / 3, highRes.url())\n )\n)}\n

${caption}

`","pinned":false,"hidden":false,"mode":"ojs"},{"id":2414,"value":"renderImage = async (file, width, url = file.url()) =>\n html`${await file\n .image()\n .then(\n image => (((image.loading = 'lazy'), (image.width = width)), image)\n )}`","pinned":false,"hidden":false,"mode":"ojs"},{"id":2566,"value":"html`