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E28. A balancing cube

A sculpture is in the form of a cube, one meter on a side, balancing unstably on one vertex. Let A be the bottom vertex (on the ground), B one of the three adjacent vertices, and C the vertex at the other end of the space diagonal of the cube from A. Assume the ground is a horizontal plane, and AC is perpendicular to to the ground. Find the distance from B to the ground, with answer in exact form.

Alexander Grothendieck

Alexander Grothendieck passed away last month, one of the giants of twentieth century mathematics. Victor Gutenmacher sent me a link to an article by Pierre Cartier celebrating Grothendieck's life. I am somewhat chagrined that I knew so little about Grothendieck prior to reading the article, which I highly recommend.

Someone once said that there are two types of first-rate mathematicians: problem solvers and theory builders. Grothendieck was one of the latter. He left tens of thousand of pages of work, and directed a brilliant group of researchers at the Institut des Hautes Etudes Scientific who carried forward the program that he developed.

A grand synthesist, Grothendieck worked in fields as seemingly diverse as functional analysis, algebraic geometry, group theory, homological algebra, and Galois theory.

The remembrance of Grothendieck is "A Country Known Only by Name", a title that makes sense after reading the article. The author, Pierre Cartier, was a friend and colleague of Grothendieck and writes about both the man and his work. Grothendieck was principled to the point of eccentricity, and seems to have been very difficult to get along with. After a dispute over military funding in 1970, which offended his pacifism, he retired from involvement in the mathematical community and lived from 1988 in isolation.

This article is at It is lengthy, but in my opinion well worth the time.