{"id":952,"date":"2014-07-01T01:00:12","date_gmt":"2014-07-01T01:00:12","guid":{"rendered":"http:\/\/blogs.ams.org\/visualinsight\/?p=952"},"modified":"2017-12-01T17:42:27","modified_gmt":"2017-12-01T17:42:27","slug":"sierpinski-carpet","status":"publish","type":"post","link":"https:\/\/blogs.ams.org\/visualinsight\/2014\/07\/01\/sierpinski-carpet\/","title":{"rendered":"Sierpinski Carpet"},"content":{"rendered":"
\n
\"Sierpinski<\/a>

Sierpinski Carpet – Noon Silk<\/p><\/div>\n<\/div>\n

To build the Sierpinski carpet<\/a><\/b> you take a square, cut it into 9 equal-sized smaller squares, and remove the central smaller square. Then you apply the same procedure to the remaining 8 subsquares, and repeat this ad infinitum<\/i>. This image by Noon Silk<\/a> shows the first six stages of the procedure.<\/p>\n

The Sierpinski carpet is the set of points in the unit square whose coordinates written in base three do not both have a digit ‘1’ in the same position. It is thus a 2-dimensional analogue of the Cantor set. <\/p>\n

The Lebesgue covering dimension<\/b><\/a> of a topological space $X$ is the least natural number $n$ such that every finite open cover of $X$ admits a refinement to a finite open cover in which no point of $X$ is included in more than $n+1$ sets. For example, a smooth curve embedded in $\\mathbb{R}^n$ has Lebesgue covering dimension 1. More generally, any compact metric space with Lebesgue dimension 1 is called a curve<\/b>. <\/p>\n

The Sierpinski carpet is a plane curve<\/b>: that is, a curve homeomorphic to a subspace of the plane. In fact, in 1916 Sierpinski showed that his carpet is a universal plane curve<\/b>: any plane curve is homeomorphic to a subspaces of the Sierpinksi carpet!<\/p>\n

The Menger sponge is also a curve, but not a plane curve:<\/p>\n

• Menger sponge<\/a>.<\/p>\n

Menger showed in 1926 that his sponge is a universal curve<\/b>: any curve is homeomorphic to a subspace of the Menger sponge. For example, the Sierpinski carpet is clearly a subspace of the Menger sponge, since every face of that sponge is a Sierpinski carpet.<\/p>\n

The ‘universality properties’ of the Sierpinski carpet and Menger sponge are not universal properties in the sense of category theory: they do not uniquely characterize these spaces up to homeomorphism. For example, the disjoint union of a Sierpinski carpet and a circle is also a universal plane curve. <\/p>\n

However, in 1958 Gordon Whyburn<\/a> uniquely characterized the Sierpinski carpet as follows: any plane curve that is locally connected<\/a> and has no ‘local cut points’ is homeomorphic to the Sierpinski carpet. Here a local cut point<\/b> is a point $p$ for which some connected neighborhood $U$ of $p$ has the property that $U – \\{p\\}$ is not connected. So, for example, any point of the circle is a local cut point.<\/p>\n

Whyburn also gave another nice characterization of the Sierpinski carpet. A continuum<\/a><\/b> is a nonempty connected compact metric space<\/b><\/a>. Suppose $X$ is a continuum embedded in the plane. Suppose the complement $\\mathbb{R}^2 – X$ has countably many connected components $C_1, C_2, C_3, \\dots$, and suppose:<\/p>\n

• the diameter of $C_i$ goes to zero as $i \\to \\infty$;<\/p>\n

• the boundary of $C_i$ and the boundary of $C_j$ are disjoint if $i \\ne j$; <\/p>\n

• the boundary of $C_i$ is a simple closed curve for each $i$; <\/p>\n

• the union of the boundaries of $C_i$ is dense in $X$.<\/p>\n

Then $X$ is homeomorphic to the Sierpinski carpet! <\/p>\n

For more properties of Sierpinski carpet see:<\/p>\n

• Janusz J. Charatonik, Pawel Krupski and Pavel Pyrih, Sierpinski carpet<\/a>.<\/p>\n

I had a little contest to create this image, and besides Noon Silk<\/a>, also Nathan Reed<\/a>, Chris Greene<\/a>, Geoffrey Irving<\/a> created images; Noon Silk merely got the job done a few minutes earlier. Silk used Mathematica code roughly like this:<\/p>\n

\r\ncarpet[n_] :=\r\n  Nest[ArrayFlatten[{{#, #, #} , {#, 0, #} , {#, #, #}}] & 1, n];\r\ndata = Table[\r\n   ArrayPlot[carpet[k],\r\n   ColorRules -> {0 -> White, 1 -> RGBColor[66\/255, 0, 130\/255]} \r\n    ], {k, 1, 6, 1}\r\n   ];\r\nExport[\"anim.gif\", data, \"DisplayDurations\" -> 1,\r\n  ImageSize -> {750, 750}];\r\n<\/pre>\n
Greene used Javascript<\/a>, and Irving used Python<\/a>, creating a version that goes to the 10th stage.  Nathan Reed used Python<\/a> to create a ‘negative’ version where the holes are purple.<\/p>\n
\n
Visual Insight<\/i> is a place to share striking images that help explain advanced topics in mathematics. I\u2019m always looking for truly beautiful images, so if you know about one, please drop a comment here<\/a> and let me know!<\/p>\n
<\/div>","protected":false},"excerpt":{"rendered":"
To build the Sierpinski carpet<\/a><\/b> you take a square, cut it into 9 equal-sized smaller squares, and remove the central smaller square. Then you apply the same procedure to the remaining 8 subsquares, and repeat this ad infinitum<\/i>.  This image by Noon Silk<\/a> shows the first six stages of the procedure.<\/p>\n
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