30:T465,### Why so complicated? The formula for the dimension of a McMullen carpet is somewhat more complicated that the box-counting dimension and it's reasonable to ask if there's some fundamental reason why. Fundamentally, it comes down to the fact that fractal dimension of self-affine sets is much more sensitive to the placements of the pieces when compared to self-similar sets - even when those pieces don't overlap at all. We can illustrate this using the example below. The image is generated by an IFS with three functions, all with the same linear part, namely: ${tex.block`A = \begin{pmatrix}1/2 & 0 \\ 0 & 1/5\end{pmatrix}.`} The IFS is then ${tex.block`\begin{aligned} f_1(x,y) &= A\begin{pmatrix}x\\y\end{pmatrix} \\ f_2(x,y) &= A\begin{pmatrix}x\\y\end{pmatrix} + \begin{pmatrix}s\\2/5\end{pmatrix} \\ f_1(x,y) &= A\begin{pmatrix}x\\y\end{pmatrix} + \begin{pmatrix}0\\4/5\end{pmatrix}. \end{aligned}`} Note that ${tex`s`} is a parameter that determines how far the second function shifts to the right. The "Shift" slider below sets the value of ${tex`s`} to be something between ${tex`0`} and ${tex`1/2`}.31:[] 2e:{"title":"McMullen carpets","cells":[{"id":0,"value":"# McMullen carpets\n\nA McMullen carpet is one of the simplest examples of a self-affine set that's not self-similar. To form a McMullen carpet, choose integers ${tex`m`} and ${tex`n`} such that ${tex`10`}, the ${tex`x`}-coordinates of approximating rectangles overlap in such a way that the projection of those rectangles onto the ${tex`x`} axis is a line segment. This is true at all levels of the approximation; thus, the projection of attractor onto the ${tex`x`} axis is a line segment. A projection can only *decrease* the dimension of the set that you're projecting so the dimension of the attractor must be at least 1.\n\nFor ${tex`s=0`} it's pretty easy to see that the attractor is a self-similar Cantor set consisting of 3 copies of itself scaled by the factor 1/5. Thus, the dimension is \n${tex.block`\\frac{\\log 3}{\\log 5}<1.`}\n\nAs a result, no single *continuous* formula can capture the fractal dimension of self-affine sets, even under the assumption of separation of the pieces.","pinned":false,"hidden":false,"mode":"md"},{"id":198,"value":"### Simple code to generate your own\n\nHere's some code that makes it easy to generate your own McMullen carpet!","pinned":false,"hidden":false,"mode":"md"},{"id":190,"value":"{\n // Note how the 0-1 pattern in the matrix below specifies\n // which sub-rectangles are on or off in the image.\n let M = [\n [0, 1, 1],\n [1, 0, 0],\n [0, 1, 0],\n [0, 1, 1]\n ];\n\n // The rest should be automatic.\n let m = M[0].length;\n let n = M.length;\n let A = [\n [1 / m, 0],\n [0, 1 / n]\n ];\n let IFS = [];\n for (let i = 1; i <= m; i++) {\n for (let j = 1; j <= n; j++) {\n if (M[j - 1][i - 1]) {\n IFS.push([A, [(i - 1) / m, 1 - j / n]]);\n }\n }\n }\n return new IteratedFunctionSystem(IFS).render_stochastic({\n n: 60000,\n colors: true,\n axes: true,\n extent: [\n [-0.05, 1.05],\n [-0.05, 1.05]\n ],\n image_width: width < 900 ? width : 900\n });\n}","pinned":true,"hidden":false,"mode":"ojs"},{"id":174,"value":"## Code","pinned":false,"hidden":false,"mode":"md"},{"id":64,"value":"exampleIFS = {\n let m = mn.m;\n let n = mn.n;\n let A = [\n [1 / m, 0],\n [0, 1 / n]\n ];\n let ifs = [];\n let M = 0;\n let N = 0;\n while (N == 0) {\n for (let i = 0; i < m; i++) {\n let Mi = 0;\n for (let j = 0; j < n; j++) {\n if (d3.randomUniform()() < mn.c) {\n N = N + 1;\n Mi = Mi + 1;\n ifs.push([A, [i / m, j / n]]);\n }\n }\n M = M + Math.sign(Mi);\n }\n }\n mn.M = M;\n mn.N = N;\n return new IteratedFunctionSystem(ifs);\n}","pinned":true,"hidden":false,"mode":"ojs"},{"id":112,"value":"mn = {\n redo;\n let c = d3.randomUniform(0.1, 1)();\n let m = d3.randomInt(2, 5)();\n let n = d3.randomInt(m + 1, 7)();\n return { c: c, m: m, n: n };\n}","pinned":true,"hidden":false,"mode":"ojs"},{"id":3,"value":"import { IteratedFunctionSystem } from \"@mcmcclur/iterated-function-systems\"","pinned":true,"hidden":false,"mode":"ojs"}],"files":"$W31","stdlib":"1","titling":"auto"}