Two students disproved a well-known mathematical hypothesis
Mathematicians thought they were on the cusp of proving the hypothesis of ancient structures known as Apollonian circles. But the summer student project was her end.
About 2200 years ago a Greek geometer Apollonius of Perga wondered how the circles could be located relative to each other if they all touch each other at one point.
Summer Haag and Clyde Kertzer had high hopes for their summer research project. Striking an unexpected blow to the weak point of an entire field of mathematics was not one of them.
In May, Haag was completing her first year of graduate school at the University of Colorado at Boulder, while Kertzer was a student. Both were looking forward to a break in classes. Haag planned to explore new hiking and climbing routes. Kertzer, a Boulder native, wanted to play football and prepare for graduate school. But, as aspiring mathematicians-researchers, they also applied for a half-time summer research program in the group of mathematician Katherine Stenge.
Stange is a number theorist who calls himself a math “frog” – a person who delves deeply into the intricacies of one problem before jumping to the next. According to her, she is interested in “simple at first glance questions that lead to an understanding of the richness of the structure.” In her projects, she often addresses the elusive open problems of number theory, using computers to obtain large amounts of data.
Haag and Kertzer began the program on Haag’s 23rd birthday with a week-long course on Apollonian circle packing, an ancient study of how circles can fit harmoniously into one larger circle.
Imagine that three coins are arranged so that each of them touches the others. You can always draw a circle around them that will touch all three coins from the outside. Then you can start asking questions: How does the size of this larger circle compare with the size of the three coins? What size circle will fit in the gap between the three coins? And if you start drawing circles that fill in smaller and smaller gaps between circles, creating a fractal pattern known as packing, how will the sizes of these circles relate to each other?
Instead of thinking about the diameter of these circles, mathematicians use a measure called curvature, the reciprocal of the radius. So, a circle of radius 2 has a curvature of 1/2, and a circle of radius 1/3 has a curvature of 3. The smaller the circle, the greater its curvature.
Note to the picture: Apollonian circles are fractal objects that fill the voids between ever-shrinking circles.
Let’s start with three circles, each of which is in contact with two others.
Let’s draw a circle around them, touching each of them at one point.
We continue to draw circles so that each of them touches the other three.
If the curvature of the first four circles is integer, then the curvature of all subsequent circles in the package is guaranteed to be integer
Renaissance mathematicians proved that if the curvature of the first four circles is integer, then the curvature of all subsequent circles in the package is guaranteed to be integer. This is wonderful in itself. But mathematicians went even further, wondering what kind of integers appear as circles get smaller and smaller, and their curvature gets larger and larger.
In 2010, Elena Fuchs, a number theorist now at the University of California, Davis, proved that the curvature follows a certain relationship that causes them to fall into certain numerical cells. Shortly thereafter, mathematicians became convinced that the curvature must not only fall in one cell or another, but also that every possible number must be present in each of the cells. This idea became known as the local-global hypothesis.
“In many works, it was referred to as if it were already a fact,” Kertzer said. “We discussed it as if it should be proven in the near future.”
James Rickards, a Boulder mathematician who works with Stange and students, has written a code that allows you to explore any desired arrangement of circle packings. So when Haag and Kertzer joined the group on May 15, they decided to build some cool graphs showing how the robust local-to-global transition rule works.
At the beginning of their project, Summer Haag (left) and Clyde Kertzer built graphs based on the data, and suddenly found that something was missing from them.
In early June, Stange flew to France for a conference. When she returned on June 12, the team huddled around graphs showing that several cells appeared to be missing certain numbers.
“We haven’t investigated this phenomenon,” Rickards said. “I didn’t try to check if that was true. I knew it was true – I just assumed it was. And all of a sudden we’re faced with data that says it isn’t.”
By the end of the week, the team was confident that the assumption was false. The numbers they expected to appear never showed up. They developed a proof and posted their work on the scientific preprint site on July 6. archive.org.
Fuchs recalls a conversation with Stange shortly after the proof was finalized. “How much do you believe in the local-global hypothesis?” “Stange asked. Fuchs replied that of course she did. “Then she showed me all the data and I said, ‘Oh my God, that’s amazing,'” Fuchs said. “So I really believed that the local-global hypothesis was true.”
“Once you see it, it all makes sense,” said Peter Sarnak, a mathematician at the Institute for Advanced Study and Princeton University, whose first observations helped develop the local-global interaction hypothesis.
“This is a fantastic discovery,” added Alex Kontorovich of Rutgers University. “We all kick ourselves for not finding it 20 years ago when people just started playing with it.”
The work also revealed a crack in the foundation of other hypotheses in number theory. Mathematicians are left to wonder which of the widely held notions might collapse next.
History of circles
The Apollonian packing of circles got its name from the name of its probable creator, Apollonius of Perga. About 2200 years ago a Greek geometer wrote the book “Tangents” on how to construct a circle that is tangent to any three others. The book has been lost over time. But about 500 years later, the Greek mathematician Pappus of Alexandria compiled a collection that survived the collapse of the Byzantine Empire.
Apollonius of Perga is considered one of the greatest mathematicians of antiquity. His work, presented here in an Arabic translation of the ninth century, continued to develop the geometric ideas of Euclid, who lived about a century earlier.
Using only Pappus’s description, Renaissance mathematicians tried to trace the progress of work on the original. By 1643, René Descartes discovered a simple relationship between the curvature of any four circles that are tangent to each other. Descartes stated that the sum of all curvature squares is equal to half the square of the sum of all curvature values. This means that, having three circles, you can calculate the radius of the fourth, tangent. For example, if you have three circles with curvatures of 11, 14, and 15, you can plug those numbers into Descartes’ equation and calculate the curvature of the circle that will fit between them: 86.
In 1936, Nobel Prize-winning radiochemist Frederick Soddy noticed something strange when he was building packages using Descartes’ ratio. When the circles got smaller and the curvature got bigger, he expected to get horrible numbers with square roots or infinite decimal places. Instead, all curvature values turned out to be integers. It was a fairly simple consequence of Descartes’ equation, but no one noticed it for hundreds of years. This inspired Soddy to publish a poem in the scientific journal Nature that began:
No need for trigonometry
When the mouth kisses the mouth.
But if you take four circles,
So that they touch each other
That will be the opposite.
[For pairs of lips to kiss maybe
Involves no trigonometry.
’Tis not so when four circles kiss
Each one the other three.]
Possible and inevitable
After it was established that there are packings consisting of integers, mathematicians tried to find patterns in these integers.
In 2010, Fuchs and Catherine Sanden decided to develop workpublished in 2003. They noticed that if you divide each curvature in a given package by 24, then a certain rule emerges. For example, in some packages, curvature values can only be with remainders 0, 1, 4, 9, 12, or 16. In others, only with remainders 3, 6, 7, 10, 15, 18, 19, or 22. Thus, you can distinguish six different groups.
In studying the various categories of packings, mathematicians have noticed that for sufficiently small circles—those with large curvature—every possible number must occur in each type of packing. This idea is called the local-global hypothesis. Proving it became “one of my little mathematical dreams,” Fuchs says. “Like, maybe someday, in many years, I’ll be able to solve it.”
In 2012, Kontorovich and Jean Bourguin (who died in 2018) proved that almost all the numbers predicted in this conjecture actually occur. But “virtually all” does not mean “all”. For example, perfect squares are rare enough that, from a mathematical point of view, “virtually all” integers are not full squares, although, for example, 25 and 49 are. According to Kontorovich, mathematicians believed that the rare counterexamples that remained possible after the publication of Kontorovich and Burgen’s paper did not actually exist, mainly because it seemed that the two or three most well-studied circle packings fit very well locally. the global hypothesis.
Step on the gas
When Haag and Kertzer started working in Boulder this summer, Rickards was sketching out the idea on a whiteboard in Stange’s office. “We had a whole list,” says Rickards. They had four or five starting points for experiments. “Something you can just play with and see what happens.”
One idea was to compute all possible circle packings containing two arbitrary curvatures A and B. Rickards wrote a program that prints out a kind of guest list telling which integers came to A’s party.
Based on this program, Haag created a Python script that plotted for many simulations at the same time. It was like a multiplication table: Haag chose which rows and columns to include based on their remainders when divided by 24. Pairs of numbers that appear together in Apollo packing get white pixels; those that don’t meet get black pixels.
Haag looked at dozens of graphs, one for each pair of residuals in each of the six groups.
Katherine Stange, a mathematician at the University of Colorado at Boulder, uses what students have learned to prove that the local-global hypothesis is wrong.
They looked exactly as expected: a white wall dotted with black dots for smaller integers. “We expected the blackheads to disappear,” Stange said. Rickards added: “I thought it might even be possible to prove that they are disappearing.” He suggested that by looking at graphs synthesizing many packages together, the team would be able to prove results that were not possible when considering any one package in isolation.
While Stange was away, Haag plotted all pairs of residuals – about 120. No surprises. Then she moved on.
Haag plotted the interaction of 1000 integers. (The graph is larger than it appears in the description, since it involves 1 million possible pairs). She then increased the scale to 10,000 times 10,000. On one chart, the regular rows and columns of black spots refused to dissolve. This was not at all like what the local-global hypothesis would have predicted.
On Monday, after the return of Stange, the team gathered for a meeting. Haag presented her charts and everyone focused on the odd dot chart. “It was just a continuous pattern,” says Haag. And then Kate said: “What if the local-global hypothesis is wrong?”.
“It’s like a pattern. It must continue. So the local-global hypothesis must be false,” Stange recalls. “James was more skeptical.”
“My first thought was that there must be a bug in my code,” said Rickards. “It was the only sensible solution that came to my mind.”
Half a day later, Rickards changed his mind. The pattern excluded all pairs in which the first number is of the form 8 × (3n ± 1)2, and the second is 24 times any square. This means that 24 and 8 never occur in the same package. The numbers that one might expect to meet there do not occur.
“I was just delighted. It’s not often that you come across something that really surprises you,” Stange said. “But that’s the magic of the data game.”
IN July article a rigorous proof is given that the regularity observed by them persists indefinitely, refuting the old hypothesis. The proof is based on a centuries-old principle called quadratic reciprocity, which is related to the squares of two primes. Stange’s team found that the principle of reciprocity applies to circle packings. This explains why circles with some defined curvature cannot be tangent to each other. This rule, called “obstruction”, applies to all packaging. “This is a completely new thing,” says Jeffrey Lagarias, a mathematician at the University of Michigan and co-author of a 2003 paper on circle packing. “They found it brilliantly,” Sarnak said. “If these numbers really appeared, they would violate the principle of reciprocity.”
Consequences
A number of other hypotheses in number theory may now be called into question. Like the local-global hypothesis, they are difficult to prove, but they have already been shown to be true in almost all cases, and it is generally accepted that they are true.
James Rickards, a Boulder mathematician working with Stange, developed a program that allowed the team to explore different circle packings.
For example, Fuchs studies Markov triples – sets of numbers that satisfy the equation x2 +y2 +z2 = 3xyz. She and others have shown that certain types of solutions are associated with prime numbers greater than 10.392. Everyone believes that this pattern should continue indefinitely. But in light of the new result, Fuchs allowed herself to feel doubtful. “Maybe I’m missing something,” she said. “Maybe everyone is missing something.”
“Now that we have a single example of a false statement, the question arises: is it possible that this is false for other examples as well?” Rickards said.
There is also the Zaremba hypothesis. It states that a fraction with any denominator can be expressed as continued fraction, which uses only the numbers 1 to 5. In 2014, Kontorovich and Burgain showed that Zaremba’s conjecture holds for almost all numbers. However, the surprise associated with the packing of the circle undermined the credibility of Zaremba’s hypothesis.
If the packaging problem is a harbinger of things to come, then computational data can be a tool to destroy it.
“It always fascinates me when new mathematics is born from simple data analysis,” Fuchs said. “Without it, it’s very hard to imagine that one would stumble upon it.”
Stange added that none of this would have happened without the summer project, which had little hope. “Randomness plays a huge role in discoveries and the attitude to research as a game,” she said.
“It was pure coincidence,” Haag said. “If I hadn’t taken a big enough step in the research, we wouldn’t have noticed.” This work foreshadows the future of number theory. “Understanding mathematics can be gained through intuition, through evidence,” Stange said. “And we used to trust this method because we spent a lot of time thinking. But you can’t argue with the data.”
