{"id":19737,"date":"2024-03-18T18:22:01","date_gmt":"2024-03-18T17:22:01","guid":{"rendered":"https:\/\/djalil.chafai.net\/blog\/?p=19737"},"modified":"2025-04-27T08:24:52","modified_gmt":"2025-04-27T06:24:52","slug":"archimedes-theorem-sphere-cylinder","status":"publish","type":"post","link":"https:\/\/djalil.chafai.net\/blog\/2024\/03\/18\/archimedes-theorem-sphere-cylinder\/","title":{"rendered":"Archimedes theorem on sphere and cylinder"},"content":{"rendered":"
\"\"<\/a>
The Fields Medal and its portrait of Archimedes.<\/figcaption><\/figure>\n

Archimedes theorem.<\/strong> Archimedes (\u1f08\u03c1\u03c7\u03b9\u03bc\u03ae\u03b4\u03b7\u03c2) 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 more generally, if we cut the whole in two pieces by any perpendicular horizontal plane, then this remains for each pieces : the surface of the spherical cap is equal to the surface of the corresponding slice of cylinder. Archimedes was so proud of it that he asked to put the picture of it on his tombstone. This allowed his admirer Marcus Tullius Cicero (-106 \u2013 -43) to identify the tomb, in -75, almost 150 years after the murder of Archimedes by a Roman soldier during the siege of Syracuse.<\/p>\n

\"\"<\/a><\/p>\n

The Archimedes theorem 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 the vertical diameter, gives the uniform distribution on the diameter. Archimedes used antic 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 $1\\leq k\\leq n$,
\n$$\\begin{align*}\\|(X_1,\\ldots,X_k)\\|^2
\n&=X_1^2+\\cdots+X_k^2\\\\
\n&\\overset{\\mathrm{d}}{=}
\n\\frac{Z_1^2+\\cdots+Z_k^2}{Z_1^2+\\cdots+Z_k^2+Z_{k+1}^2+\\cdots+Z_n^2}\\\\
\n&=\\frac{A}{A+B}\\\\
\n&\\sim\\mathrm{Beta}\\Bigr(\\frac{k}{2},\\frac{n-k}{2}\\Bigr),\\end{align*}$$ where the last step comes from the fact that $\\frac{A}{A+B}\\sim\\mathrm{Beta}(\\frac{a}{2},\\frac{b}{2})$ when $$A\\sim\\chi^2(a)=\\Gamma\\Bigr(\\frac{a}{2},\\frac{1}{2}\\Bigr)\\quad\\text{and}\\quad B\\sim\\chi^2(b)=\\Gamma\\Bigr(\\frac{b}{2},\\frac{1}{2}\\Bigr)\\quad\\text{are independent}.$$Recall that the law $\\mathrm{Beta}(\\alpha,\\beta)$ has density proportional to $t\\in[0,1]\\mapsto t^{\\alpha-1}(1-t)^{\\beta-1}$, which is a power when $\\beta=1$. In particular, in the special case where $k=n-2$, we find $$\\mathrm{Beta}\\Bigr(\\frac{k}{2},\\frac{n-k}{2}\\Bigr)=\\mathrm{Beta}\\Bigr(\\frac{n}{2}-1,1\\Bigr),$$and this law has a power density, proportional to $t\\in[0,1]\\mapsto t^{n\/2-2}$. Now, a rotationally invariant random vector $X'$ of $\\mathbb{R}^{n-2}$ is uniformly distributed on the unit ball of $\\mathbb{R}^{n-2}$ iff $\\|X'\\|$ has density proportional to $r\\in[0,1]\\mapsto r^{n-3}$, in other words $\\|X'\\|^2$ has density proportional to $t\\in[0,1]\\mapsto\\sqrt{t}^{n-3}\/\\sqrt{t}=t^{n\/2-2}$, which matches the Beta law above.<\/p>\n

Reverse Archimedes theorem.<\/strong> 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 $$
\n\\frac{(Z_1,\\ldots,Z_n)}{\\sqrt{Z_1^2+\\cdots+Z_n^2+2E}}
\n$$ 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 $$Z_{n+1}^2+Z_{n+2}^2\\sim\\chi^2(2)=\\Gamma\\Bigr(\\frac{2}{2},\\frac{1}{2}\\Bigr)=\\mathrm{Exp}\\Bigr(\\frac{1}{2}\\Bigr)\\sim 2E.$$ The reverse Archimedes principle reveals that the uniform law on the ball concentrates, in high dimension, around the extremal sphere at its edge. Indeed, by the law of large numbers,$$\\frac{\\sqrt{Z_1^2+\\cdots+Z_n^2}}{\\sqrt{Z_1^2+\\cdots+Z_n^2+2E}}=\\frac{1}{\\sqrt{1+O\\bigr(\\frac{1}{n}\\bigr)}}\\underset{n\\to\\infty}{\\longrightarrow}1\\quad\\text{almost surely}.$$This is an instance of the thin-shell phenomenon for convex bodies. From this point of view, in high dimension $n$, the sphere of radius $\\sqrt{n}$ behaves approximately as an isotropic convex body.<\/p>\n

Note.<\/strong> The Archimedes theorem on the sphere and the cylinder is sometimes referred to as the Archimedes principle. But this last term is more classically used for the fact that a body at rest in a fluid is acted upon by a force pushing upward called the buoyant force, equal to the weight of the fluid that the body displaces, related to the famous Eur\u00eaka! These are distinct discoveries. Archimedes was a prolific mind, for instance we still use nowadays Archimedes' screws and Archimedes' parabolic mirrors, among other remarkable and useful elementary things.<\/p>\n

\"\"<\/a>
Opposite side of the Medal, with Archimedes' tomb depicting his theorem on the sphere and the cylinder.<\/figcaption><\/figure>\n

Further reading.<\/strong><\/p>\n