Ham sandwich theorem finally solved

Ham sandwich theorem finally solved

The Ham Sandwich Theorem has been a treat and a spur to mathematicians for more than half a century. There was a bit of a kerfuffle about who invented it, but that question did get settled.

The Ham Sandwich Theorem cropped up in a branch of mathematics called algebraic topology.

The theorem describes a particular truth about certain shapes. Most published papers on the topic make a hash of explaining it to anyone who is not an algebraic topologist. But the authors of a 2001 paper called Leftovers from the Ham Sandwich Theorem wrapped up an important little leftover - they put the idea into clear language.

The Ham Sandwich Theorem, they wrote, "rescues the careless sandwich maker by guaranteeing that it is always possible to slice the sandwich with one cut so that the ham and both slices of bread are each divided into equal halves, no matter how haphazardly the ingredients are arranged".

For a while, most ham sandwich theorising dealt with simple cases. A paper called Computing a Ham-Sandwich Cut in Two Dimensions, published in 1986 in the Journal of Symbolic Computation, is typical. It considered only ham sandwiches that had been flattened flatter than even the chintziest cook would dare to devise. Mathematicians often do things this way, first considering the extreme cases, digesting those thoroughly, and only then moving on to more substantial versions. Indeed, the Computing a Ham-Sandwich Cut in Two Dimensions paper itself contains a section called Getting Rid of Degenerate Cases.

People did solve the mystery of slicing a thick ham sandwich. And, inevitably, they developed a hunger for more substantial problems. In 1990, Yugoslavian theorists were writing in the Bulletin of the London Mathematical Society about An Extension of the Ham Sandwich Theorem. Two years later a theorist at Yaroslav State University in Russia published a paper called A Generalization of the Ham Sandwich Theorem.

That same year, a team of hungry American, Czech, and German mathematicians assembled a master collection of recipes for slicing ham sandwiches. Mathematicians almost never use the word "recipe", so they called their paper Algorithms for Ham-Sandwich Cuts. You'll find it in the December 1994 issue of the journal, Discrete and Computational Geometry.

Research then moved on to exotic, distantly related questions, exemplified by a 1998 monograph called Green Eggs and Ham.

And who started this? A 2004 paper called The Early History of the Ham Sandwich Theorem took care of a lingering leftover: it identified the inventor. Mathematico-historians WA Beyer and Andrew Zardecki of Los Alamos National Laboratory in New Mexico say that it was a Jewish theorist who introduced the ham sandwich into mathematical theory. Beyer and Zardecki trace the theorem back to a 1945 paper by the Polish mathematician Hugo Steinhaus that "represents work Steinhaus did in Poland on the ham sandwich problem in World War II while hiding out with a Polish farm family".

(Thanks to Stanley Eigen for bringing the ham sandwich research to my attention.)

Article by Marc Abrahams, editor of the bimonthly Annals of Improbable Research and organiser of the Ig Nobel prize. You can read the original article in the Guardian here.