{"id":529,"date":"2014-03-01T01:00:57","date_gmt":"2014-03-01T01:00:57","guid":{"rendered":"http:\/\/blogs.ams.org\/visualinsight\/?p=529"},"modified":"2017-11-28T03:58:27","modified_gmt":"2017-11-28T03:58:27","slug":"menger-sponge","status":"publish","type":"post","link":"https:\/\/blogs.ams.org\/visualinsight\/2014\/03\/01\/menger-sponge\/","title":{"rendered":"Menger Sponge"},"content":{"rendered":"
\n
\"Menger<\/a>

Menger Sponge – Niabot<\/p><\/div>\n<\/div>\n

<\/p>\n

Take a cube. Chop it into 3\u00d73\u00d73 = 27 small cubes. Poke holes through it, removing 7 of these small cubes. Repeat this process for each remaining small cube. Do this forever! The result is the Menger sponge<\/a><\/b>.<\/p>\n

What’s the volume of the Menger sponge? At each stage we remove 7\/27 of the volume, so only 20\/27 of the volume is left. As we repeat this forever, the volume drops to zero. So, the final volume is zero!<\/p>\n

What’s the surface area of the Menger sponge? That’s a bit harder to work out, but it’s infinite!<\/p>\n

What’s the dimension of the Menger sponge? That depends on what you mean by ‘dimension’.<\/p>\n

You can compute the Minkowski dimension<\/b><\/a> of a metric space by covering it with balls of radius $1\/n$, counting the minimum number of balls you need, and seeing how this grows as $n \\to \\infty$. If the growth rate is of order $n^d$, the space has Minkowski dimension $d$.<\/p>\n

For a 3-dimensional cube, the number of balls needs to grow as some number times $n^3$. So, we say a cube has Hausdorff dimension 3. Similarly, a smooth curve has Minkowski dimension 1. But the Menger sponge has Minkowski dimension about<\/p>\n

$$ 2.726833… $$<\/p>\n

This number is not an integer. So, the Menger sponge is an example of a fractal<\/a>. <\/p>\n

What is this number? It’s<\/p>\n

$$ \\frac{\\ln 20}{\\ln 3}$$<\/p>\n

The reason is that at each stage of constructing the Menger sponge we subdivide each existing cube into 27 smaller cubes and then remove 7, leaving 20. So, at the $n$th stage of the construction we have covered the Menger spong with $20^n$ cubes having edges of length $1\/3^n$.<\/p>\n

There are also other concepts of dimension. 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. <\/p>\n

The Lebesgue covering dimension of the Menger sponge is also 1. Moreover, Menger showed, back in 1926, that any compact metric space of Lebesgue covering dimension 1 is homeomorphic to a subset of the Menger sponge!<\/p>\n

The image above was created by Niabot<\/a> and licensed under the Creative Commons Attribution 3.0 Unported<\/a> license. It is available on Wikimedia Commons<\/a>.<\/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":"

Take a cube. Chop it into 3\u00d73\u00d73 = 27 small cubes. Poke holes through it, removing 7 of these small cubes. Repeat this process for each remaining small cube. Do this forever! The result is called the Menger sponge<\/a>.<\/p>\n

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