“Cutting a Pancake with an Exotic Knife” might seem an unusual title for a piece of academic research. But that was the focus of the paper posted online recently by two mathematicians.
The investigation’s challenge: Cut a pancake into as many pieces as possible, in the latest attack on a longstanding puzzle known as the Lazy Caterer’s problem.
Simple enough. But there were a few caveats: The pancake was infinite, spreading endlessly in every direction. And the simplest case involved an infinite, straight knife.
The mathematicians, Neil J.A. Sloane, the founder of the “On-Line Encyclopedia of Integer Sequences,” and the aptly named David O.H. Cutler, an undergraduate at Tufts University, engaged in a lot of trial and error to negotiate the tricky task of optimally placing not just a straight knife but also a series of weirdly shaped knives on the pancake.
“You want to find the sweet spot,” Dr. Sloane said.
The pancake problem was famously discussed a few decades ago in a classic book, “Concrete Mathematics: A Foundation for Computer Science,” based on a Stanford course of the same name. The notion of “concrete mathematics” was meant as an antidote, of sorts, to new trends in “abstract mathematics” (a.k.a. “new math.”).
Dr. Sloane, who is a longtime visiting scholar at Rutgers, debuted the pancake problem research at an online experimental mathematics seminar, run by Doron Zeilberger, a mathematician at the university.
The general philosophy is, when you make a cut, you try to intersect all the previous lines, Dr. Sloane said during the seminar.
The knives deployed by Dr. Sloane and Mr. Cutler were bent — by the whims of their geometric curiosity — into unknifely shapes. And with these exotic knives, “any protruding arms or legs are made infinitely long,” Dr. Sloane said.
For instance, one of the many proposed cutting instruments was shaped like a lollipop with an infinite stem.
Another was the capital letter A, with some rigid design requirements. It was a “constrained A” knife, with the crossbar fixed horizontally such that when considered together with the A’s tip it forms the base of an isosceles triangle (two sides of equal length, two equal base angles).
Cutting a pancake with one constrained A produces three pieces. Two constrained A knives in just the right configuration produce 13 regions, like so:
Dr. Zeilberger observed that the notion of cutting an infinite pancake into a maximum number of pieces with a strangely shaped utensil can be explained to almost anyone,. “It’s so elementary,” he said in an interview. “What can be simpler?” But, he said, proving the conjectures that emerge is “highly challenging.”
The experimental aspect of this kind of research, Dr. Zeilberger said, involves no scientific laboratory per se. “The lab of mathematicians is a computer,” he said.
The computer is essential in figuring out how to maneuver the theoretical knives (the constrained A, and others) around the theoretical pancake in just the right way so as to find optimal configurations for the maximum number of pieces.
“You run out of room very quickly,” Mr. Cutler said in an interview.
With the help of a few friends, Mr. Cutler put together a “quick, sloppy piece of software” that searches for such configurations. One key ingredient in the program was a formula for polyhedra derived by Leonhard Euler in about 1750. (The formula states that the number of regions — in this case pancake pieces — minus the number of edges of the pieces, plus the number of vertices where edges intersect, equals one: R−E+V=1).
The program also used an optimization algorithm that generated a random initial arrangement of, say, three constrained A’s, then jiggled and wiggled the positions in small, random ways to find the maximum number of pieces.
A procession of integers
The numerical data that pours forth from this exercise is pure bliss for the likes of Dr. Sloane.
The computer results suggested that three, four and five constrained A knives produce 30, 53, and 83 pieces, respectively. One constrained A knife produces three pancake pieces. Two knives produce 13. And zero knives leaves the infinite pancake intact. So, a succession of 0, 1, 2, 3, 4, 5 knives generate 1, 3, 13, 30, 53 pancake pieces.
Such a numerical procession is called an integer sequence, an orderly list of whole numbers that follow a rule or pattern, like prime numbers: 2, 3, 5, 7, 11, 13 … ; or even numbers: 0, 2, 4, 6, 8, 10, 12 … The Fibonacci sequence — 0, 1, 1, 2, 3, 5, 8, 13, 21…; (in which each number is the sum of the preceding two numbers) — are related to structures in nature: the growth patterns of flower petals, the chambers of a nautilus cell, the branching of trees or, as discovered last year, vortices of light.
For Dr. Sloane the pancake partitioning process brought to mind a possible application, though only as fodder for a joke. “I can’t help think that some gerrymandering rule might appreciate these constructions,” he said.
‘Infinitely far away!’
Of all the shapes the researchers considered, the constrained A was “the most difficult to analyze,” Dr. Sloane said. “Even the 13-piece solution for two cuts required a computer to find.” The difficulty arose because the pancake pieces are often microscopically tiny. As the authors described in the paper, some regions were “barely visible to the naked eye” and approached “the precision limit of the computer.”
The computer helped them “grope in the dark,” Dr. Sloane said. Experimentation by hand was another good source of ideas, he found. Using the analog skill of visualization, he went through a lot of scratch paper making drawings. Here is sketch of the initial pancake problem in the simplest case:
It produces the integer sequence: 1, 2, 4, 7, 11, 16…
Although the problem’s domain is always the infinite, Dr. Sloane, when sketching, sometimes chooses a finite representation framed by a circle. As he scribbled on one sketch, he said: “The circles are the horizon, infinitely far away!”
After the constrained A, the mathematicians tried a series of exotic knives, each generating a number sequence.
One was what they called a “hatpin” knife — a semi-infinite knife, infinite in only one direction, and thus not quite as efficient. “But it was surprising to note that with, say, five hatpins you could make as many pieces as with four infinite knife cuts,” Dr. Sloane said. For zero through five cuts, the sequence of pieces was: 1, 1, 2, 4, 7, 11…
Then came a V-shaped knife, which in total produced almost four times as many regions: 2, 7, 16, 29…
Another knife looked like a three-armed V, which Dr. Sloane said is called a “Wu,” according to the Unified Canadian Aboriginal Syllabics section of Unicode, a character-encoding standard.
One Wu knife produced three pancake pieces; two Wu knives produced 14 pieces; and three Wu knives produced 34 regions.
The mathematicians named still another knife shape “nunchucks,” after a martial arts weapon.
Notably, three nunchucks, as shown in Dr. Sloane’s sketch, also generated 34 pancake pieces:
But these hand drawings have limits as they start to get messy and difficult to decipher. Computer drawings provide some clarity. Here is a digitally illustrated nunchucks:
And the same configuration of three nunchucks cutting an infinite pancake:
A triple-coincidence theorem
Jean-Paul Allouche, a mathematician and an emeritus senior researcher at the National Centre for Scientific Research of the Sorbonne, digested the paper with “gourmandise,” as he described in an email to Dr. Sloane. “The subject itself is surprising,” Dr. Allouche said in an interview. Mathematicians sometimes pose questions that others would not ask, he said.
For Dr. Allouche, the visual aspect of the research brought to mind a dictum often quoted by mathematicians, which he paraphrased as “geometry is the art of true proofs starting from wrong pictures.” It is typically attributed to the French mathematician Henri Poincaré, who wrote, “Geometry is the art of correct reasoning from incorrectly drawn figures.”
What lent certainty and rigor to the investigation, he said, was the fact that a multitude of resulting number sequences matched entries, originating from other contexts, that were already on record in the On-Line Encyclopedia of Integer Sequences. The paper cited the encyclopedia 38 times.
Dr. Allouche said it was “almost magic” — a phenomenon that in a sense provided justification for the research. It reminded him of “when a new theory in physics both contains the previous theory and explains new experiments not covered by that previous theory,” he said. For example, relativity theory explains things that Newton’s laws could not explain, but it contains Newton’s laws.
“Finding the unexpected connections, that is the fun of math,” Dr. Allouche said.
For the researchers, a notable surprise arrived when they tried an exotic knife in an elongated “A” shape — not the constrained version but rather an “A” that could have a crossbar skewed at any angle.
Three of these “long-legged” A’s generated 34 pancake pieces, the same number of regions generated by three Wu’s and three nunchuks. All elicited the same integer sequence (A140064 in the encyclopedia): 1, 3, 14, 34, 63, 101 … “This triple coincidence was in fact not just a coincidence, it became a particularly nice theorem,” Dr. Sloane said.
The investigation continued with other “long-legged” letters, namely E, H, L, M, T, W, and X, among other shapes.
After the research was posted, further progress — a proof of a conjecture inspired by the research — was made by the mathematician Jonas Karlsson, who is based in Boden, Sweden will be added to the paper as a third author.
Dr. Sloane called the alphabetical configurations “long-legged letters” because the pursuit brought to mind two lines from a poem by William Butler Yeats, which serve as an epigram at the beginning of the paper:
Like a long-legged fly upon the stream
His mind moves upon silence.
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