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Libres pensées d'un mathématicien ordinaire Posts

Archimedes principle

The Fields Medal and its portrait of Archimedes.

Archimedes theorem or principle. Archimedes of Syracuse (-287 – -212) is one of the greatest minds of all times. One of his discoveries is as follows : if we place a sphere in the tightest cylinder, then the surface of the sphere and of the cylinder are the same, and moreover this remains valid if we cut the whole by a perpendicular plane. Archimedes was so proud of this theorem that he put the picture of it on his tomb. This picture on the stone allowed his admirer Marcus Tullius Cicero (-106 – -43) to identify the tomb, in the year -75, almost one century and a half after the murder of Archimedes by a Roman soldier during the siege of Syracuse.

The Archimedes principle allows to recover the formula for the surface of the sphere : if the sphere has radius $R$, then the surface of the cylinder is $2\pi R\times 2R=4\pi R^2$. The cutting plane part of the Archimedes theorem says that the uniform distribution on the sphere, when projected on a diameter, gives the uniform distribution on the diameter. Archimedes used geometrical methods. Nowadays, with the development of modern analytic and probabilistic tools, we can prove easily an extension of his theorem to arbitrary dimension. More precisely, let $(X_1,\ldots,X_n)$ be a random vector of $\mathbb{R}^n$, $n\geq3$, uniformly distributed on the sphere $\mathbb{S}^{n-1}$. Then its projection $(X_1,\ldots,X_{n-2})$ on $\mathbb{R}^{n-2}$ is uniform on the unit ball of $\mathbb{R}^{n-2}$. Indeed, since $$(X_1,\ldots,X_n)\overset{\mathrm{d}}{=}\frac{Z}{\|Z\|}=\frac{(Z_1,\ldots,Z_n)}{\sqrt{Z_1^2+\cdots+Z_n^2}}$$where $Z=(Z_1,\ldots,Z_n)\sim\mathcal{N}(0,I_n)$, we get, for all $1\leq k\leq n$,
$$\begin{align*}\|(X_1,\ldots,X_k)\|^2
&=X_1^2+\cdots+X_k^2\\
&\overset{\mathrm{d}}{=}
\frac{Z_1^2+\cdots+Z_k^2}{Z_1^2+\cdots+Z_k^2+Z_{k+1}^2+\cdots+Z_n^2}\\
&=\frac{A}{A+B}\\
&\sim\mathrm{Beta}\Bigr(\frac{k}{2},\frac{n-k}{2}\Bigr),\end{align*}$$ since $A/(A+B)\sim\mathrm{Beta}(a/2,b/2)$ when $A\sim\chi^2(a=k)$ and $B\sim\chi^2(b=n-k)$ are independent. In particular, in the special case where $k=n-2$, we find $\mathrm{Beta}(k/2,(n-k)/2)=\mathrm{Beta}(n/2-1,1)$. This law has a power density, proportional to $t\in[0,1]\mapsto t^{n/2-2}$. Now if $X’$ is a random vector of $\mathbb{R}^{n-2}$ uniformly distributed on the unit ball, then $\|X’\|$ has density proportional to $r\in[0,1]\mapsto r^{n-3}$, and thus $\|X’\|^2$ has density proportional to $t\in[0,1]\mapsto\sqrt{t}^{n-3}/\sqrt{t}=t^{n/2-2}$, which matches the Beta law!

Reverse Archimedes principle. It states that if $Z_1,\ldots,Z_n$ are i.i.d. $\mathcal{N}(0,1)$, $n\geq1$, and if $E$ is exponential of unit mean independent of $Z_1,\ldots,Z_n$, then the random vector $$
\frac{(Z_1,\ldots,Z_n)}{\sqrt{Z_1^2+\cdots+Z_n^2+2E}}
$$ is uniformly distributed on the unit ball of $\mathbb{R}^n$. To see it, it suffices to use an extended Gaussian sequence $(Z_1,\ldots,Z_{n+2})\sim\mathcal{N}(0,I_{n+2})$, the Archimedes principle, and the observation that $$Z_{n+1}^2+Z_{n+2}^2\sim\chi^2(2)=\Gamma(2/2,1/2)=\mathrm{Exp}(1/2)\sim 2E.$$ The reverse Archimedes principle combined with the law of large numbers reveal the thin shell phenomenon for the uniform distribution on the ball, and more precisely the concentration of this distribution, in high dimension, on the extremal sphere at its edge. Indeed, thanks to the law of large numbers, almost surely,$$\frac{\sqrt{Z_1^2+\cdots+Z_n^2}}{\sqrt{Z_1^2+\cdots+Z_n^2+2E}}=\frac{1}{\sqrt{1+O(1/n)}}\underset{n\to\infty}{\longrightarrow}1.$$From this point of view, in high dimension $n$, the sphere of radius $\sqrt{n}$ behaves approximately as a normalized isotropic convex body.

Opposite side, with Archimedes tomb illustrating his theorem on the sphere and the cylinder.

Further reading.

Archimedes death, by Édouard Vimont (1846 – 1930)
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