<story><title>High-dimensional spheres are &quot;spikey&quot;</title><url>http://www.penzba.co.uk/cgi-bin/PvsNP.py?SpikeySpheres#HN2</url></story><parent_chain><item><author>xyzzyz</author><text>Another way of looking at this phenomenon is to inscribe a high dimensional sphere in a hypercube. As the dimension grows, the sphere/cube volume ratio gets arbitrarily small, although sphere touches the cube at every side. It seems that most of the mass of the cube was focused near vertices. This is because the side of a cube is a whole lot closer than the vertex from the center of the cube -- for instance, if we take a million-dimensional cube with a side of length 2, then the distance from the center of the cube to the side is 1, but the distance from the center to the vertex is 1000. It&apos;s not the spheres that are &quot;spikey&quot;, the cubes are.</text></item></parent_chain><comment><author>tel</author><text>One should also take a close look at the formulas for volumes of &lt;i&gt;unit-radius&lt;/i&gt; n-balls. For even numbers of dimensions it&apos;s&lt;p&gt;&lt;pre&gt;&lt;code&gt;    pi^(n/2)/fact(n/2)
&lt;/code&gt;&lt;/pre&gt;
which behaves very interestingly[1]. It&apos;s equal to pi in two dimensions, peaks at the 4-dimensional ball (about 1.6 pi), then is driven to effectively zero by the time you get to 20 dimensions or so (value of ~8/1000 pi).&lt;p&gt;Again, however, the radius isn&apos;t changing. I don&apos;t think of unit balls as being spiky--they&apos;re still rotationally invariant in all d rotational degrees of freedom, so calling them spiky doesn&apos;t sit well with me.&lt;p&gt;Instead, imagine you lived on a line, only walking up and back along it forever. Then, one day, someone introduced you to the plane and then a volume. These are vastly larger than the space you were afforded by the line: exponential blowup.&lt;p&gt;It&apos;s also driven home by how the Lesbegue measure in d-dimensions assigns zero measure to all (d-1)-dimensional (or lesser) objects. Adding a dimension just makes space immensely larger.&lt;p&gt;And so pegging the size of our unit ball to its 1-dimensional parameter, the radius, causes its volume to vanish.&lt;p&gt;[1] Wolfram Alpha chart showing the volume of the n-sphere in units of pi &lt;a href=&quot;http://www.wolframalpha.com/input/?i=pi%5E%28n%2F2-1%29%2FGamma%5Bn%2F2%2B1%5D+from+0+to+30&quot; rel=&quot;nofollow&quot;&gt;http://www.wolframalpha.com/input/?i=pi%5E%28n%2F2-1%29%2FGa...&lt;/a&gt;</text></comment>
<story><title>High-dimensional spheres are &quot;spikey&quot;</title><url>http://www.penzba.co.uk/cgi-bin/PvsNP.py?SpikeySpheres#HN2</url></story><parent_chain><item><author>xyzzyz</author><text>Another way of looking at this phenomenon is to inscribe a high dimensional sphere in a hypercube. As the dimension grows, the sphere/cube volume ratio gets arbitrarily small, although sphere touches the cube at every side. It seems that most of the mass of the cube was focused near vertices. This is because the side of a cube is a whole lot closer than the vertex from the center of the cube -- for instance, if we take a million-dimensional cube with a side of length 2, then the distance from the center of the cube to the side is 1, but the distance from the center to the vertex is 1000. It&apos;s not the spheres that are &quot;spikey&quot;, the cubes are.</text></item></parent_chain><comment><author>Jimmie</author><text>This made it click for me. Thanks for explaining.&lt;p&gt;Just to expand a bit on the vertex thing, in 2 dimensions length is sqrt(x^2 + y^2). For a square of side length 2 the distance from the center to each vertex is sqrt(1^2 + 1^2) or sqrt(2 * 1). For 3 dimensions it&apos;s sqrt(x^2 + y^2 + z^2) or sqrt(3 * 1). For one million dimensions the distance to each vertex is sqrt(1,000,000) or 1,000 while the distance to each side is still 1.</text></comment>