2b:null 2c:["$","title",null,{"children":"Barycenter of the Aperiodic Monotiles | David Hall | Observable"}] 31:T16e7,32:T873,Craig Kaplan's' video ["Aperiodic monotile animation"](https://www.youtube.com/watch?v=W-ECvtIA-5A) shows the whole family of tiles as they morph through the entire spectrum from `Chevron` to `Comma` while maintaining the same number of tiles in the frame. Since the `Chevron` has an area of 4 equilateral triangles and the `Comma` has an area of 8 equilateral triangles, the area of a single tile would double if the edge lengths were the same at the two extremes. This would result in half as many monotiles fitting in the frame. In order to maintain a fixed area, the edge length must scale by a factor of √2 over the whole transition, but I don't think the scaling is a linear function of `t`. Craig's second video, ["Chiral aperiodic monotile animation"](https://www.youtube.com/watch?v=K6wXQvL5KRo), ranges only from the `hat` through the `spectre` to the `turtle`. It does not include the entire range of monotiles. It shows the tiles being resized by maintaining a fixed length for the smaller edge, while increasing the larger edge length to √3. The tiles in the two videos also have an element of "rotation" as they morph. I'm pretty sure that the full spectrum rotates 60 degrees as it morphs. I conjecture that a patch of fixed area tiles will each "rotate" around their own barycenter as they all morph from one extreme to the other. I still need to work out the actual pair of edge lengths as a function of `t` for maintaining a fixed area tile. I'm pretty sure it's not linear. To that end, the barycenter coordinates for all values of `t` are written to the browser's JavaScript console while animation of the slider is enabled. Once I figure out how to calculate the lengths as a function of `t`, I think I can make an animation similar to Craig's first one showing all of the barycenters which will hopefully remain stationary. Whether my conjecture is correct or not, I'd also like to make an animation of the [Voronoi diagram](https://en.wikipedia.org/wiki/Voronoi_diagram) of the barycenters of a large patch of tiles, just to see what sort of polygons they form as the monotile morphs through the whole spectrum as a function of `t`.33:[] 2f:{"id":"dfbfd0aec766916f","slug":null,"title":"Barycenter of the Aperiodic Monotiles","thumbnail":"c10c0ebb75e99256cad50e8aec56b3a50e1107ad43bd0835b60ab72552204988","access_level":"public","create_time":"$D2023-10-23T03:06:47.088Z","update_time":"$D2024-04-20T18:41:48.710Z","view_time":null,"first_public_time":"$D2023-10-23T03:06:47.088Z","has_unpublished_edits":null,"fork":{"id":"e57692300453779b","slug":"interactive-svg","title":"Interactive SVG","owner":{"id":"f35c755083683fe5","login":"observablehq","name":"Observable"}},"stars":0,"starred":false,"owner":{"type":"team","id":"e03a1f35385e7271","login":"david-hall","name":"David Hall","avatar_url":"https://avatars-next.observableusercontent.com/afa38c46229fd6d9c4426dfd17e3359afcc7eb8787d0a9c48fafda7bc081c359","tier":"starter_2024"},"version":2016,"publish_version":null,"role":null,"authored":false,"authors":[{"type":"individual","id":"0d8260f2c3b197fd","login":"david-hall","name":"David Hall","avatar_url":"https://avatars-next.observableusercontent.com/afa38c46229fd6d9c4426dfd17e3359afcc7eb8787d0a9c48fafda7bc081c359","tier":null}],"members":[{"type":"individual","id":"0d8260f2c3b197fd","login":"david-hall","name":"David Hall","avatar_url":"https://avatars-next.observableusercontent.com/afa38c46229fd6d9c4426dfd17e3359afcc7eb8787d0a9c48fafda7bc081c359","tier":null,"role":"editor"}],"comments":[],"custom_thumbnail":false,"trashed":false,"lapsed":null,"subscribed":null,"forks":0,"collection":null,"type":"notebook","body":{"title":"Barycenter of the Aperiodic Monotiles","cells":[{"id":0,"value":"# Barycenter of the Aperiodic Monotiles","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":4,"value":"$31","pinned":false,"hidden":false,"mode":"html","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":970,"value":"Inputs.button([\n [\"Chevron (Red)\", () => set(viewof t, 0)],\n [\"4:1\", () => set(viewof t, 0.2)],\n [\"Hat\", () => set(viewof t, 0.366)],\n [\"Spectre\", () => set(viewof t, 0.5)],\n [\"Turtle\", () => set(viewof t, 0.634)],\n [\"1:4\", () => set(viewof t, 0.8)],\n [\"Comma (Blue)\", () => set(viewof t, 1)]\n])","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":10,"value":"viewof t = Inputs.range([0, 1], {\n value: 0.5,\n step: 0.001,\n text: \"Foo\"\n})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1139,"value":"viewof options = Inputs.checkbox(\n new Map([\n [\"Animation\", \"animation\"],\n [\"Axes\", \"axes\"],\n [\"Barycenter\", \"barycenter\"],\n [\"Bounding Box\", \"boundingbox\"],\n [\"Centroids\", \"centroids\"],\n [\"Edges\", \"edges\"],\n [\"Partitions\", \"partitions\"]\n ]),\n { value: [\"barycenter\", \"edges\", \"partitions\"] }\n)","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1693,"value":"viewof fixedPoint = Inputs.radio(\n new Map([\n [\"Fix Axes\", \"origin\"],\n [\"Fix Barycenter\", \"barycenter\"],\n [\"Fix Bounding Box Center\", \"boundingbox\"]\n ]),\n { value: \"barycenter\" }\n)","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1828,"value":"{\n let step = 0.001;\n if (viewof t.value > 0.5) {\n // motion begins in the direction towards .5\n step *= -1;\n }\n while (options.includes(\"animation\")) {\n if (viewof t.value <= 0 || viewof t.value >= 1) {\n // reverse direction at each limit\n step *= -1;\n }\n viewof t.value += step;\n viewof t.dispatchEvent(new CustomEvent(\"input\"));\n yield Promises.delay(10, viewof t.value);\n }\n yield \"Animation is paused. Select the checkbox above to animate the slider.\";\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1947,"value":"# Abstract\nThe family of aperiodic monotiles is composed of non-convex 14-gons with all edges having one of two lengths, shown here as blue and red. A slider `t` can be adjusted from `0` to `1` controlling the relative lengths of the two edges as a function of `t`.\n\nThe resulting monotile can be decomposed into eight equilateral triangles with an edge length of `blue`, four 120° isosceles triangles with short edge lengths of `red`, and a single 120° parallelogram with edge lengths of `2*blue` and `√3*red`. The areas and centroids of these simple convex polygons can easily be determined. The barycenter [(center of mass)](https://en.wikipedia.org/wiki/Center_of_mass) of the monotile is equal to the vector sum of the individual components' centroids scaled by their individual areas, then dividing their scaled vector sum by the total area of the monotile.","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":1956,"value":"$32","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":1929,"value":"# References","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":1934,"value":"[An aperiodic monotile (PDF)](https://arxiv.org/pdf/2303.10798.pdf)","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":1935,"value":"[A chiral aperiodic monotile (PDF)](https://arxiv.org/pdf/2305.17743.pdf)","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":1931,"value":"[Aperiodic monotile animation (YouTube)](https://www.youtube.com/watch?v=W-ECvtIA-5A)","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":1940,"value":"[Chiral aperiodic monotile animation (YouTube)](https://www.youtube.com/watch?v=K6wXQvL5KRo)","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":1951,"value":"[Centroid by geometric decomposition (Wikipedia)](https://en.wikipedia.org/wiki/Centroid#By_geometric_decomposition)","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":1993,"value":"[Aperiodic monotiles (Observable)](https://observablehq.com/@rreusser/aperiodic-monotile)","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":1994,"value":"[Hexagonal grid (Observable)](https://observablehq.com/@sanderevers/hexagon-tiling-of-an-hexagonal-grid)","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":1997,"value":"[Mathstodon](https://mathstodon.xyz/@Danpiker/110062396666001681)","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":1999,"value":"[Mathigon](https://mathigon.org/polypad/8kVqVH2Mor6JTQ)","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":1634,"value":"# TODO:\n- Add option for fixed area as in Craig's original video.\n- Add option for 60 degree rotation in sync with slider as in Craig's video.\n- Add dynamic scaling to make a fixed area.\n- Add mirroring.\n- Add rotation through all 12 angles.\n- Use a slider with tic marks at presets as in the sandbox below.\n- Adjust margins to match the sides and avoid clipping in Open Graph tests.\n- Remove unused cells and sandbox.\n- Use SVG symbol to facilitate making a supertile with \"key points\" marked.","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":852,"value":"# Data","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":1368,"value":"mutable barycenter = {\n const sum = barycenters.reduce(\n (result, bc) => {\n return vPlus(result, bc);\n },\n { x: 0, y: 0 }\n );\n return vScale(sum, 1 / totalArea);\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1366,"value":"barycenters = barycentersBlue.concat(barycentersRed, barycentersPurple)","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1554,"value":"barycentersBlue = {\n const arr = [];\n const weight = blueTriangleArea;\n const src = centroidsBlue;\n for (let i = 0; i < src.length; i++) {\n const c = vScale(src[i], weight);\n arr.push(c);\n }\n return arr;\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1556,"value":"barycentersRed = {\n const arr = [];\n const weight = redTriangleArea;\n const src = centroidsRed;\n for (let i = 0; i < src.length; i++) {\n const c = vScale(src[i], weight);\n arr.push(c);\n }\n return arr;\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1560,"value":"barycentersPurple = {\n const arr = [];\n const weight = parallelogramArea;\n const src = centroidPurple;\n for (let i = 0; i < src.length; i++) {\n const c = vScale(src[i], weight);\n arr.push(c);\n }\n return arr;\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1332,"value":"centroids = centroidsBlue.concat(centroidsRed, centroidPurple)","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1498,"value":"centroidsBlue = [\n // first 8 are blue triangle centroids\n centroid(0, 1, 15),\n centroid(1, 2, 15),\n centroid(2, 16, 15),\n centroid(3, 16, 2),\n centroid(3, 4, 16),\n centroid(7, 8, 17),\n centroid(10, 11, 18),\n centroid(11, 12, 18)\n]","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1500,"value":"centroidsRed = [\n centroid(4, 5, 14),\n centroid(5, 6, 14),\n centroid(8, 9, 10),\n centroid(12, 13, 0)\n]","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1503,"value":"centroidPurple = [\n //Parallelogram centroid needs just two opposite vertices instead of all four\n centroid(6, 18) // or (14, 17)\n]","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1088,"value":"tileSpecs = {\n const left = Math.sqrt(3) * a;\n const right = 3 * b;\n const width = left + right;\n const top = (3 * a + Math.sqrt(3) * b) / 2;\n const bottom = Math.max(a / 2, (Math.sqrt(3) * b) / 2);\n const height = top + bottom;\n const redBlueRatio = a / b;\n const blueRedRatio = b / a;\n return {\n left,\n right,\n width,\n top,\n bottom,\n height,\n area: totalArea,\n redBlueRatio,\n blueRedRatio\n };\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1922,"value":"tileInfo = {\n const bc = barycenter;\n const s =\n \"{ t: \" +\n t.toFixed(3) +\n \", x: \" +\n bc.x +\n \", y: \" +\n bc.y +\n \", area: \" +\n totalArea +\n \" },\";\n if (options.includes(\"animation\")) {\n console.log(s);\n }\n return s;\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":204,"value":"vectors = {\n const aArray = [];\n const bArray = [];\n for (let i = 0; i < 6; i++) {\n const angle = (2 * Math.PI * i) / 6;\n const c = Math.cos(angle).toFixed(nFixed);\n const s = Math.sin(angle).toFixed(nFixed);\n aArray.push({ x: s * a, y: c * a });\n bArray.push({ x: c * b, y: s * b });\n }\n return { a: aArray, b: bArray };\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":497,"value":"path = [\n vectors.b[5],\n vectors.b[0],\n vectors.b[0],\n vectors.b[1],\n vectors.a[5],\n vectors.a[0],\n vectors.b[3],\n vectors.b[2],\n vectors.a[4],\n vectors.a[3],\n vectors.b[3],\n vectors.b[4],\n vectors.a[2],\n vectors.a[1]\n]","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":527,"value":"p = {\n const arr = [];\n let point = { x: 0, y: 0 };\n for (let i = 0; i < path.length; i++) {\n arr.push(point);\n point = vPlus(point, path[i]);\n }\n // [14] = red internal\n arr.push(vPlus(vPlus(arr[4], vectors.a[4]), vectors.a[5]));\n // The rest are blue internal\n arr.push(vPlus(arr[0], vectors.b[0]));\n arr.push(vPlus(arr[15], vectors.b[0]));\n arr.push(vPlus(arr[8], vectors.b[4]));\n arr.push(vPlus(arr[11], vectors.b[5]));\n\n return arr;\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1812,"value":"slidingScalar = (1 - t) * (Math.sqrt(2) - 1) + 1","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":870,"value":"b = t.toFixed(3) * 1.0","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":54,"value":"a = (1 - b).toFixed(3) * 1.0 // * slidingScalar","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1061,"value":"blueTriangleArea = (b * Math.sqrt(3)) / 4","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1067,"value":"redTriangleArea = (a * Math.sqrt(3)) / 4","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1070,"value":"parallelogramArea = (a * b * 3).toFixed(nFixed) * 1","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1074,"value":"totalArea = 8 * blueTriangleArea + 4 * redTriangleArea + parallelogramArea","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":863,"value":"# Code","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":602,"value":"pathD = (...pathIndices) => pathIndices.map((i) => vStr(p[i]) + \" \")","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":490,"value":"vPlus = (v0, v1) => {\n return {\n x: (v0.x + v1.x).toFixed(nFixed) * 1,\n y: (v0.y + v1.y).toFixed(nFixed) * 1\n };\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1355,"value":"vScale = (v, scalar) => {\n return {\n x: v.x * scalar,\n y: v.y * scalar\n };\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":336,"value":"vStr = (v) => v.x + \" \" + v.y + \" \"","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":967,"value":"// from https://observablehq.com/@observablehq/synchronized-inputs\nfunction set(input, value) {\n input.value = value;\n input.dispatchEvent(new Event(\"input\", { bubbles: true }));\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1164,"value":"visibilityof = (option) => options.includes(option) ? 'visible' : 'hidden'","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1203,"value":"svgViewbox = () => {\n // default values are for fixed origin\n let x = -Math.sqrt(3);\n let y = -1.5;\n let w = 3 - x;\n let h = Math.sqrt(3) / 2 - y;\n switch (fixedPoint) {\n case \"barycenter\":\n // TODO: set specific mathematically exact values rather than adjusting the defaults\n // These are approximated from imperical observation, not ideal.\n // Border gets bigger (tile gets smaller).\n x -= Math.sqrt(3) / 4;\n y += 0.375 / 2;\n // readjust w & h for new x & y values\n w = 3 - x;\n h = Math.sqrt(3) / 2 - y;\n break;\n case \"boundingbox\":\n x = -tileSpecs.left;\n y = -tileSpecs.top;\n w = tileSpecs.width;\n h = tileSpecs.height;\n break;\n case \"origin\":\n // use the defaults from above;\n break;\n default:\n console.log(\"Unexpected fixedPoint:\", fixedPoint);\n break;\n }\n const svgBorder = 0.25;\n x -= svgBorder;\n y -= svgBorder;\n w += svgBorder * 2;\n h += svgBorder * 2;\n return x + \" \" + y + \" \" + w + \" \" + h;\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1263,"value":"centroid = (...pathIndices) => {\n let v = { x: 0, y: 0 };\n pathIndices.forEach((i) => {\n v = vPlus(v, p[i]);\n });\n const n = pathIndices.length;\n if (n > 1) {\n v = { x: v.x / n, y: v.y / n };\n }\n return v;\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":866,"value":"# Settings","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":6,"value":"height = width / 2","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":549,"value":"nFixed = 12","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1034,"value":"# Sandbox","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":1014,"value":"sliderWithPresets.value","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1010,"value":"viewof sliderWithPresets = datalistSlider(0, 1, 0.5, 0.001, presets)","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":981,"value":"function datalistSlider(min, max, value, step, datalist) {\n return form(html``);\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":881,"value":"presets = [\n [\"Chevron\", 0],\n [\"4:1\", 0.2],\n [\"Hat\", 0.366],\n [\"Spectre\", 0.5],\n [\"Turtle\", 0.634],\n [\"1:4\", 0.8],\n [\"Comma\", 1]\n]","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":983,"value":"# Imports","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","output":"","database":"$undefined"},{"id":985,"value":"import { form } from \"@mbostock/form-input\"","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"}],"files":"$W33","stdlib":"1","titling":"auto"}} 30:"$2f:body"