Four colors will do

     As I work with Gizem Karaali, an editor of the Journal of Humanistic Mathematics, to plan a reading of mathematical poetry at the JMM (Joint Mathematics Meetings) in Boston on 6 January 2012, my thoughts return to a poetry reading that I helped to organize at JMM in Baltimore in 1992. One of the participants was a friend and former colleague, Frank Bernhart, whose work is guided by the rhythm pattern of a well-known song.
     Bernhart is an expert on the Four-Color Theorem and his poem celebrates its history -- including consideration of its proof (in 1976) by Kenneth Appel and Wolfgang Haken. (The theorem asserts that any map drawn on a flat surface or on a sphere requires only 4 colors to ensure that no regions sharing a boundary segment have the same color.)


Cartographic Claustropobia       by Frank Bernhart
 (A Four Color History Tour)         

          DIRECTIONS:  read like "Higgledy-piggledy, My Black Hen," or
          "The Wonderful One-Hoss Shay," contrasting smooth and staccato phrases.

One color, TWO color, THREE-and-FOUR,
In coloring maps, do WE-need-more?
The map of England, Guthrie tried,
With EXACTLY four he's SATIS-fied.
So he told DeMorgan, who THEN told Cayley,
And plenty of others -- the news spread daily.
Til Kempe, a lawyer, made demonstration,
Earned himself a nice ovation.

PLANE maps or SPHERE maps -- it don't matter,
Makes no difference, 'cept in patter.
But torus and Klein sack, please refuse,
All higher surfaces you MUST NOT use.
Abstain from topology, draw simple lines,
Let the map be described by finite signs.
And using the Kuratowski planarity rule,
You may trade the map for its planar dual.

Two color, THREE color, FOUR-and-Five,
said Percy John Heawood, not now alive,
are always enough, and I'll attemp'
to repair the gap I found in Kempe.
For years he labored, with great agility
To make four suffice, with high probability,
Accompanied by false proofs, dare we mention?
The authors could hold a crackpot convention.

Now, finally, thankfully, the answer's FOUR,
in colorful cartography -- no NEED for more.
And what team broke the century-long spell?
None other than Haken and his FRIEND Appel.
The easiest direction, discharging, entails
pages and pages of tiny details.
The other direction, reduction, is cuter:
here billions of steps were left to computer.

DISCHARGE-ing, REDUCE-ing, what's that get?
An awkward and large unavoidable set.
Kempe's methds were extended, as though on a leash
considerably lengthened by Birkhoff and Heesch.
Plus a tribe of coworkers who nobly increased
the list of reductions without help in the least
from computers.  Some people say "It's a cheat.
Your proof is in part an unprintable feat."

One year, TWO years, THREE-and-FOUR,
Better throw in a dozen more.
The years roll by, the proof is firm,
Jut a few little errors to make them squirm!
But all soon are fixed, JUST LIKE NEW,
the critics now have better THINGS to do.
A nice short proof is NOT in sight,
Better to avoid all colors and PAINT things white.


  FOUR                              Any   flat    map
  FOUR                              we draw we may
  FOUR                              color  with  four
  FOUR                              c o l o r s   only.

        
For a variety of math-song lyrics, including work by Tom Apostol and Tom Lehrer, visit this page maintained by Math Professor Joel W Robbin of the University of Wisconsin.