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Archimedes theorem on sphere and cylinder

The Fields Medal and its portrait of Archimedes.

Archimedes theorem. Archimedes (Ἀρχιμήδης) 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 – -43) to identify the tomb, in -75, almost 150 years after the murder of Archimedes by a Roman soldier during the siege of Syracuse.

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 2πr×2r=4πr2. 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 (X1,,Xn) be a random vector of Rn, n3, uniformly distributed on the sphere Sn1. Then its projection (X1,,Xn2) on Rn2 is uniform on the unit ball of Rn2. Indeed, since (X1,,Xn)d=ZZ=(Z1,,Zn)Z21++Z2nwhere Z=(Z1,,Zn)N(0,In), we get, for all 1kn,
(X1,,Xk)2=X21++X2kd=Z21++Z2kZ21++Z2k+Z2k+1++Z2n=AA+BBeta(k2,nk2), where the last step comes from the fact that AA+BBeta(a2,b2) when Aχ2(a)=Γ(a2,12)andBχ2(b)=Γ(b2,12)are independent.Recall that the law Beta(α,β) has density proportional to t[0,1]tα1(1t)β1, which is a power when β=1. In particular, in the special case where k=n2, we find Beta(k2,nk2)=Beta(n21,1),and this law has a power density, proportional to t[0,1]tn/22. Now, a rotationally invariant random vector X of Rn2 is uniformly distributed on the unit ball of Rn2 iff X has density proportional to r[0,1]rn3, in other words X2 has density proportional to t[0,1]tn3/t=tn/22, which matches the Beta law above.

Reverse Archimedes theorem. It states that if Z1,,Zn are i.i.d. N(0,1), n1, and if E is exponential of unit mean independent of Z1,,Zn, then the random vector (Z1,,Zn)Z21++Z2n+2E is uniformly distributed on the unit ball of Rn. To see it, it suffices to use an extended Gaussian sequence (Z1,,Zn+2)N(0,In+2), the Archimedes principle, and the observation that Z2n+1+Z2n+2χ2(2)=Γ(22,12)=Exp(12)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,Z21++Z2nZ21++Z2n+2E=11+O(1n)n1almost 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 n behaves approximately as an isotropic convex body.

Note. 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 Eurêka! 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.

Opposite side of the Medal, with Archimedes' tomb depicting his theorem on the sphere and the cylinder.

Further reading.

Archimedes death, by Édouard Vimont (1846 - 1930). A sphere in a cylinder is drawn on the wall behind him.

Modern times. The reasoning above involving a Beta law is actually a very special aspect of a more general and deeper probabilistic structure involving the Dirichlet law, a generalizatoin of the Euler Beta law, defined using the Euler Gamma law. More precisely, for real parameters a1>0,,an>0, the law Dirichlet(a1,,an) is the law of the random vector
(D1,,Dn):=(G1,,Gn)G1++Gnof Rn, where G1,,Gn are independent with GiGamma(ai,λ), for an arbitrary scaling parameter λ>0. We can safely take without loss of generality λ=1. The support of this law is the simplex of discrete probability distributions with n atoms Δn:={(p1,,pn):p10,,pn0,p1++pn=1}[0,1]n. Its density is
(x1,,xn1)Γ(a1++an)Γ(a1)Γ(an)n1i=1xai1i(1n1i=1xi)an11(x1,,xn)Δn. Moreover, for all 1in, the ith component follows a Beta law :
GiG1++GnBeta(ai,a1++anai). The Dirichlet law structure is stable by summation by blocks. More precisely, for any partitition I1,,Ik of the finite set {1,,n} into non-empty subsets or blocks, we have
(iI1Di,,iIkDi)Dirichlet(iI1ai,,iIkai), and for any non-empty subset I of {1,,n}, iIDiBeta(iIai,iIai).The Dirichlet law plays an important role for spatial points processes, stochastic simulation, as well as in Statistics due to their Bayesian duality with multinomial laws. If U1,,Un are independent uniform random variables on [0,1], and if U(0)U(n) is their non-decreasing reordering, known as the order statistics, then, denoting U(0):=0 and U(n+1):=1, (U(1)U(0),,U(n+1)U(n))d=(E1,,En+1)E1++En+1Dirichlet(1,,1), where E1,,En+1 are independent and identically distributed exponential random variables (with arbitrary parameter). This fact is at the heart of Poisson point processes.

Another important side of what we used is the link between chi-square laws and Gamma laws. In terms of probabilistic culture and structure, there are two basic facts. The first one is χ2(1):=(N(0,1))2=Gamma(12,12) while the second one is the additivity related to the Gamma shape parameter Gamma(a,λ)Gamma(b,λ)=Gamma(a+b,λ). This gives the important fact χ2(n)=(χ2(1))n=Γ(n2,12). In particular the heart of the Box-Muller simulation algorithm involves the special case χ2(2)=Gamma(1,12)=Expo(12)=2log(Uniform[0,1]).

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