29:null 2a:["$","title",null,{"children":"Fractal tilings | Nicolas Garcia Belmonte | Observable"}] 2f:T6e8,canvas = { const width = 1200; const height = 350; const context = DOM.context2d(width, height); context.fillStyle = "black"; context.fillRect(0, 0, width, height); context.strokeStyle = "#555"; context.lineWidth = 0.2; context.fillStyle = "none"; const rho = 6; let ans = [ isocelesTilings(n, { tiling: 'regular', filter: 'height' }), isocelesTilings(n, { tiling: 'regular', filter: 'cone' }), isocelesTilings(n, { tiling: 'patched', filter: 'height' })]; let dims = [ width / 6, width / 2, 5 * width / 6 ]; ans.forEach((plot, i) => { context.save(); // context.translate((i + 1)*width/4, height/2); context.translate(dims[i], height/2); for (let rate = 0; rate < 4; rate++) { context.rotate(Math.PI/2 * rate); plot.forEach(t => { const queue = [t]; while (queue.length) { let node = queue.shift(); let level = node.depth; let index = level % colors[i].length; let color = colors[i][index]; let [p1, p2, p3] = node.prototile; context.fillStyle = color; context.beginPath(); context.moveTo.apply(context, p1.map(c => c * width/ans.length / rho)); context.lineTo.apply(context, p2.map(c => c * width/ans.length / rho)); context.lineTo.apply(context, p3.map(c => c * width/ans.length / rho)); context.lineTo.apply(context, p1.map(c => c * width/ans.length / rho)); context.closePath(); context.fill(); context.stroke(); queue.push.apply(queue, node.children || []); } }); } context.restore(); }); context.canvas.style.background = "gray"; return context.canvas; }30:Tb2b,canvas1 = { const tile = await FileAttachment("red.jpg").image(); // const tile2 = await FileAttachment("Screen Shot 2022-11-26 at 9@2.46.00 PM.png").image(); // const tile = await FileAttachment("escher_442.png").image(); const width = 500; const height = 500; const context = DOM.context2d(width, height); context.fillStyle = "black"; context.fillRect(0, 0, width, height); context.strokeStyle = "#555"; context.lineWidth = 0.2; context.fillStyle = "none"; const rho = 2.1; let ans = [ isocelesTilings(n, { tiling: 'patched', filter: 'height' })]; ans.forEach((plot, i) => { context.save(); context.translate(width/2, height/2); for (let rate = 0; rate < 4; rate++) { context.save(); context.rotate(Math.PI/2 * rate); plot.forEach(t => { const queue = [t]; while (queue.length) { let node = queue.shift(); let level = node.depth; let index = level % colors[i].length; let color = colors[i][index]; let [p1, p2, p3] = node.prototile.map(p => p.map(c => c * width/2 / rho)); context.save(); context.moveTo.apply(context, p1); context.beginPath(); context.lineTo.apply(context, p2); context.lineTo.apply(context, p3); context.lineTo.apply(context, p1); context.closePath(); context.clip(); context.translate(p1[0], p1[1]); let mid = [(p2[0] + p3[0]) / 2, (p2[1] + p3[1]) / 2]; let vec = [mid[0] - p1[0], mid[1] - p1[1]]; let d = Math.sqrt(vec[0] * vec[0] + vec[1] * vec[1]); let angle = Math.atan2(vec[1], vec[0]); let type; let epsilon = 0.01; if (mid[0] - p1[0] > epsilon) { type = 'right'; } else if (p1[0] - mid[0] > epsilon) { type = 'left'; } else if (mid[1] - p1[1] > epsilon) { type = 'top'; } else if (p1[1] - mid[1] > epsilon) { type = 'bottom'; } let delta = rate % 2? Math.PI/2 : 0; let drawTile = tile; if (node.depth % 2) { // drawTile = tile2; context.scale(1, -1); context.rotate(angle + delta); context.drawImage(drawTile, 0, 0, drawTile.width, drawTile.height, -d, -d, 2 * d, 2 * d); } else { context.rotate(angle + delta); context.drawImage(drawTile, 4, 0, drawTile.width-8, drawTile.height, -d, -d, 2 * d, 2 * d); } // context.globalCompositeOperation = 'xor'; context.restore(); queue.push.apply(queue, node.children || []); } context.restore(); }); } context.restore(); }); context.canvas.style.background = "gray"; return context.canvas; }31:T135a,function isocelesTilings(depth, options) { const { tiling='regular', filter='height' } = options; const sideLength = 1; const angle = Math.PI/4; const prototile = [ [0, 0], [math.cos(angle) * sideLength, -math.sin(angle) * sideLength], [math.cos(angle) * sideLength, math.sin(angle) * sideLength] ]; const maxY = prototile[2][1]; const minY = prototile[1][1]; const getCentroid = (prototile) => { const p0 = prototile[0]; const p1 = prototile[1]; const p2 = prototile[2]; const midp0p1p2 = [(p1[0] + p2[0] + p0[0])/3, (p1[1] + p2[1] + p0[1])/3]; return midp0p1p2; } let accessor; accessor = tilings[tiling]; if (tiling == 'patched') { accessor = tilings['regular']; } const filters = { height: (node, parent, root) => { return node.prototile.every(p => p[1] <= parent.maxY && p[1] >= parent.minY); }, cone: (node, parent, root) => { const centroid = getCentroid(node.prototile); let pNode = node.parent; while(pNode) { let origin = pNode.prototile[0]; let vector = [centroid[0] - origin[0], centroid[1] - origin[1]]; let angle = Math.atan2(vector[1], vector[0]); if (angle > pNode.topCone[1] || angle < pNode.bottomCone[1]) { return false; } pNode = pNode.parent; } return true; } }; const filterApply = filters[filter]; const treeBranch = { prototile, depth: 0, children: [], maxY, minY, topCone: [[0, 0], angle], bottomCone: [[0, 0], -angle], parent: false }; const queue = [treeBranch]; while (queue.length) { let node = queue.shift(); if (node.depth < depth) { if (!node.children || !node.children.length) { node.children = []; let children = accessor.getChildren(node); children.forEach(ch => { if (filterApply(ch, node, treeBranch)) { node.children.push(ch); queue.push(ch); } }); } } } (function removeParent(t) { t.parent = false; if (t.children) { for (const ch of t.children) { removeParent(ch); } } })(treeBranch); const getByLevel = (depth, tree) => { let i = 1; while (tree.depth < depth && tree.children && tree.children.length) { tree = tree.children[i]; i = 1 - i; } let ans = JSON.parse(JSON.stringify(tree, null, 1)); return ans; }; if (tiling == 'patched') { const levels = depth; let rotMat = math.matrix([ [math.cos(Math.PI/2), -math.sin(Math.PI/2)], [math.sin(Math.PI/2), math.cos(Math.PI/2)] ]); let rotMat2 = math.matrix([ [math.cos(-Math.PI/2), -math.sin(-Math.PI/2)], [math.sin(-Math.PI/2), math.cos(-Math.PI/2)] ]); let squares = []; for (let i = 1; i <= Math.floor(levels / 2); i++) { let rootTile = treeBranch.prototile; let scaledTile = rootTile.map(p => { return p.map(c => c / Math.pow(2, i)); }); let rotTile1 = scaledTile.map(p => { return math.multiply(rotMat2, p)._data; }); let rotTile2 = rotTile1.map(p => { return math.multiply(rotMat2, p)._data; }); [rotTile1, rotTile2].forEach(rotTile => { let shift = 0; for (let j = 0; j <= i; j++) { let val = rootTile[2][0] / Math.pow(2, j); if (j !== 0 && j !== i) { val += val; } shift += val; } let t = getByLevel(i*2, treeBranch); let origin = t.prototile[0]; console.log(origin, 'origin'); (function transTree(tree) { tree.prototile = tree.prototile.map(p => { return [p[0] - origin[0] + shift, p[1] - origin[1] + shift]; }); if (tree.children && tree.children.length) { for (let i = 0; i < tree.children.length; i++) { transTree(tree.children[i]); } } })(t); let t2 = getByLevel(i*2, treeBranch); (function transTree2(tree) { tree.prototile = tree.prototile.map(p => { let p1 = [p[0] - origin[0], p[1] - origin[1]]; p1 = math.multiply(rotMat, p1)._data; return [p1[0] + shift, p1[1] + shift]; }); if (tree.children && tree.children.length) { for (let i = 0; i < tree.children.length; i++) { transTree2(tree.children[i]); } } })(t2); let transTile = rotTile.map(p => { return [ p[0] + shift, p[1] + shift ]; }); let nodeTile = { prototile: transTile, depth: i*2, children: [] }; squares.push(nodeTile, t, t2); //squares.push(nodeTile); }); } treeBranch.children.push.apply(treeBranch.children, squares); } return [treeBranch]; }32:T1963,tilings = ({ regular: { getChildren(node) { return [ this.getNodeLeft(node), this.getNodeRight(node) ]; }, getNodeLeft(node) { const angle = Math.PI/4; const protoParent = node.prototile; const p0Left = [(protoParent[2][0] + protoParent[1][0])/2, (protoParent[2][1] + protoParent[1][1])/2]; const p1Left = [p0Left[0] * 2 - protoParent[0][0], p0Left[1] * 2 - protoParent[0][1]]; const p2Left = protoParent[2].slice(); let protoLeft = [p0Left, p1Left, p2Left]; return { prototile: protoLeft, maxY: protoLeft[2][1], minY: protoLeft[1][1], topCone: [protoLeft[0], node.topCone[1] + angle], bottomCone: [protoLeft[0], node.bottomCone[1] + angle], depth: node.depth + 1, left: null, right: null, parent: node }; }, getNodeRight(node) { const angle = Math.PI/4; const protoParent = node.prototile; const p0Right = [(protoParent[2][0] + protoParent[1][0])/2, (protoParent[2][1] + protoParent[1][1])/2]; const p1Right = protoParent[1].slice(); const p2Right = [p0Right[0] * 2 - protoParent[0][0], p0Right[1] * 2 - protoParent[0][1]]; let protoRight = [p0Right, p1Right, p2Right]; return { prototile: protoRight, maxY: protoRight[2][1], minY: protoRight[1][1], topCone: [protoRight[0], node.topCone[1] - angle], bottomCone: [protoRight[0], node.bottomCone[1] - angle], depth: node.depth + 1, left: null, right: null, parent: node }; } }, inverted: { getChildren(node) { return [ this.getNodeLeft2(node), this.getNodeRight(node), this.getNodeCenter(node) ]; }, getNodeLeft2(node) { const protoParent = node.prototile; const p0 = protoParent[0]; const p1 = protoParent[1]; const p2 = protoParent[2]; const mid = [(p1[0] + p2[0])/2, (p1[1] + p2[1])/2]; const dist1 = [mid[0] - p0[0], mid[1] - p0[1]]; const dist1Norm = dist1.map(x => x / math.distance(mid, p0)); const longDist = math.distance(p1, p2); const dist = [p1[0] - p2[0], p1[1] - p2[1]]; const p0Right = p1.slice(); const p1Right = [p0Right[0] + dist[0]/2, p0Right[1] + dist[1]/2]; const p2Right = [p0Right[0] + dist1Norm[0] * longDist/2, p0Right[1] + dist1Norm[1] * longDist/2]; // const p0Right = [ protoParent[2][0] + dist[0], protoParent[2][1] + dist[1]]; // const p1Right = [(protoParent[2][0] + protoParent[1][0])/2 + dist[0], (protoParent[2][1] + protoParent[1][1])/2 + dist[1]]; // const p2Right = [p0Right[0] + dist1[0], p0Right[1] + dist1[1]]; let protoRight = [p0Right, p1Right, p2Right]; return { prototile: protoRight, maxY: protoRight[2][1], minY: protoRight[1][1], topCone: [protoRight[0], node.topCone[1] + this.angle], bottomCone: [protoRight[0], node.bottomCone[1] + this.angle], depth: node.depth + 1, left: null, right: null, parent: node }; }, getNodeLeft(node) { const protoParent = node.prototile; const p0 = protoParent[0]; const p1 = protoParent[1]; const p2 = protoParent[2]; const mid = [(p1[0] + p2[0])/2, (p1[1] + p2[1])/2]; const dist = [mid[0] - p0[0], mid[1] - p0[1]]; const p0Left = protoParent[2].slice(); const p1Left = [(protoParent[2][0] + protoParent[1][0])/2, (protoParent[2][1] + protoParent[1][1])/2]; const p2Left = [p0Left[0] + dist[0], p0Left[1] + dist[1]]; let protoLeft = [p0Left, p1Left, p2Left]; return { prototile: protoLeft, maxY: protoLeft[2][1], minY: protoLeft[1][1], topCone: [protoLeft[0], node.topCone[1] - this.angle], bottomCone: [protoLeft[0], node.bottomCone[1] - this.angle], depth: node.depth + 1, left: null, right: null, parent: node }; }, getNodeRight(node) { const protoParent = node.prototile; const p0 = protoParent[0]; const p1 = protoParent[1]; const p2 = protoParent[2]; const mid = [(p1[0] + p2[0])/2, (p1[1] + p2[1])/2]; const dist = [mid[0] - p0[0], mid[1] - p0[1]]; const p0Right = protoParent[1].slice(); const p1Right = [p0Right[0] + dist[0], p0Right[1] + dist[1]]; const p2Right = [(protoParent[2][0] + protoParent[1][0])/2, (protoParent[2][1] + protoParent[1][1])/2]; let protoRight = [p0Right, p1Right, p2Right]; return { prototile: protoRight, maxY: protoRight[2][1], minY: protoRight[1][1], topCone: [protoRight[0], node.topCone[1] + this.angle], bottomCone: [protoRight[0], node.bottomCone[1] + this.angle], depth: node.depth + 1, left: null, right: null, parent: node }; }, getNodeCenter(node) { const protoParent = node.prototile; const p0 = protoParent[0]; const p1 = protoParent[1]; const p2 = protoParent[2]; const mid = [(p1[0] + p2[0])/2, (p1[1] + p2[1])/2]; const dist1 = [mid[0] - p0[0], mid[1] - p0[1]]; const dist = [p1[0] - p2[0], p1[1] - p2[1]]; const p0Right = p1.slice(); const p1Right = [p0Right[0] + dist[0]/2, p0Right[1] + dist[1]/2]; const p2Right = [protoParent[0][0] + dist[0]/2, protoParent[0][1] + dist[1]/2]; // const p0Right = [ protoParent[2][0] + dist[0], protoParent[2][1] + dist[1]]; // const p1Right = [(protoParent[2][0] + protoParent[1][0])/2 + dist[0], (protoParent[2][1] + protoParent[1][1])/2 + dist[1]]; // const p2Right = [p0Right[0] + dist1[0], p0Right[1] + dist1[1]]; let protoRight = [p0Right, p1Right, p2Right]; console.log(protoRight, dist); return { prototile: protoRight, maxY: protoRight[2][1], minY: protoRight[1][1], topCone: [protoRight[0], node.topCone[1] + this.angle], bottomCone: [protoRight[0], node.bottomCone[1] + this.angle], depth: node.depth + 1, left: null, right: null, parent: node }; } } 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