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Mathematician’s guide to holidays

Holiday season offers endless opportunities to celebrate, relax, rest, reflect and meditate.  Whether you are enjoying a white Christmas or a palm tree Chanukkah, a mathematician in you might wonder if there is more to the story, a rigorous food for thought, if you will.  So here is a brief guide to the holidays for the mathematically inclined.

1)  Christmas tree lectures

I have my own Christmas tree tradition.  Instead of getting one, I watch new Don Knuth‘s “Christmas tree lecture“.  Here is the most recent one.  But if you have time and enjoy binge-watching here is the archive of past lectures (click on “Computer musings” and select December dates).  If you are one of my Math 206 students, compare how Knuth computed the number of spanning trees in a hypercube (in a 2009 lecture) with the way Bernardi did in his elegant paper.

2)  Algorithmic version of Fermat’s Christmas theorem

Apparently, Fermat’s theorem on sums of two squares first appeared in Fermat’s long letter to Mersenne, written on Christmas Day (December 25, 1640).  For background, see Catalan and French language Wikipedia articles.  Zagier’s “one-sentence proof” is well known and available here.  Long assumed to be mysterious, it was nicely explained by Elsholtz.  Still mysteriously, a related proof also appears in a much earlier paper (in French), by a Russian-American mathematician J. Uspensky (ht. Ustinov).  Can somebody explain to me what’s in that paper?

Interestingly, there is a nice polynomial time algorithm to write a prime  p=1 mod 4  as a sum of two squares, but I could not find a clean version on the web.  If you are curious, start with Cornacchia’s algorithm for more general quadratic Diophantine equations, and read its various proofs (advanced, elementary, short, textbook, in French).  Then figure out why Fermat’s special case can be done in (probabilistic) polynomial time.

3)  Dreidel game analysis

The dreidel is a well known Chanukkah game with simple rules.  Less known is the mathematics behind it.  Start with this paper explaining that it’s unfair, and continue to this paper explaining how to fix it (on average).  Then proceed to this “squared nuts” conjecture by Zeilberger on the expected length of the game (I have a really good joke here which I will suppress).  This conjecture was eventually resolved in this interesting paper, definitely worth $25 promised by Zeilberger.

Now, if you are underwhelmed with the dreidel game, try to prove the festive Star of David Theorem.  When you are done, enjoy this ingenious proof, which is definitely “from the book”.

4)  Santa Claus vs beautiful mathematics

Most readers of this blog are aware of existence of beautiful mathematics.  I can only speculate that a clear majority of them would probably deny the existence of Santa Claus.  However, there are millions of (mostly, very young) people who believe the exact opposite on both counts.  Having grown up in the land of Ded Moroz, we have little to say on the great Santa debate, but we believe it’s worth carefully examining Santa proponent’s views.  Could it be that their arguments can be helpful in our constant struggle to spread the gospel of beautiful mathematics?

We recommend reading “Yes, Virginia, there is Santa Claus column (fully available here), which was originally published by the New York Sun in 1897.  In fact, read it twice, three times, even four times.  I am reluctant to quote from it because it’s short and deserves to be read in full.  But note this passage:  “The most real things in the world are those that neither children nor men can see.”  The new Jewish editor of the Sun reports that the rabbis he consulted think this is “a joyous articulation of faith”.  Maybe.  But to me this evokes some beautiful advanced mathematics.

You see, when mathematicians try to explain that mathematics is beautiful, they tend to give simple visually appealing examples (like here).  But I suggest closing your eyes and imagining beautiful mathematical objects, such as the 600-cell, Poincaré homolgy sphereLie group E8, Monster group, or many other less known higher dimensional constructions such as the associahedron, the Birkhoff polytope, Walz’s flexible cross-polyhedron, etc.  Certainly all of these can be seen by “neither children nor men”.  Yet we can prove that they “are real”.  We can then spend years studying and generalizing them.  This knowledge alone can bring joy to every holiday season…

HAPPY HOLIDAYS EVERYONE!   С НОВЫМ ГОДОМ!

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