![]() Numbers they expected to appear never did. And then suddenly, we’re faced with data that says it’s not.”īy the end of the week, the team was confident the conjecture was false. I knew it was true - I just assumed it was true. “We weren’t investigating this phenomenon,” Rickards said. When she returned on June 12, the team huddled around charts that demonstrated how a few buckets seemed to be missing certain numbers. Stange flew to France for a conference in early June. So when Haag and Kertzer joined the group on May 15, they thought they’d create cool plots of the reliable local-to-global rule kicking in. James Rickards, a mathematician at Boulder who works with Stange and the students, had written code to examine any desired arrangement of circle packings. “We discussed it as though it was going to be proven at some point in the near future.” “Lots of works referenced it as though it were already fact,” Kertzer said. The idea came to be known as the local-global conjecture. Shortly afterward, mathematicians became convinced that not only must the curvatures fall into one bucket or another, but also that every possible number in each bucket must be used. In 2010, Elena Fuchs, a number theorist now at the University of California, Davis, proved that curvatures follow a particular relationship that forces them into certain numerical buckets. But mathematicians have taken the problem a step further by asking questions about which integers show up as the circles get smaller and smaller and the curvatures get larger and larger. Renaissance mathematicians proved that if the first four circles have a curvature that’s an integer, the curvatures of all the subsequent circles in the packing are guaranteed to be whole numbers. ![]() The smaller the circle, the bigger the curvature. So a circle with radius 2 has curvature 1/2, and a circle with radius 1/3 has curvature 3. Rather than think about the diameter of these circles, mathematicians use a measure called curvature - the inverse of the radius. Then you can start to ask questions: How does the size of that bigger circle relate to those of the three coins? What size circle will fit into the gap between the three coins? And if you start to draw circles that fill in progressively smaller and smaller gaps between circles - creating a fractal pattern known as a packing - how do the sizes of those circles relate to one another? You can always draw a circle around them that touches all three from the outside. Imagine arranging three coins so that each one touches the others. Haag and Kertzer began the program on Haag’s 23rd birthday with a weeklong primer on Apollonian circle packings - the ancient study of how circles can harmoniously squeeze into one larger circle. ![]() Her projects often poke at number theory’s elusive open problems by using computers to generate large data sets. She is interested in “simple-seeming questions that lead to a richness of structure,” she said. Stange is a number theorist who describes herself as a mathematical “ frog” - someone who delves deep into one problem’s intricacies before hopping to another. But as aspiring research mathematicians, they had also applied for a half-time summer research program in the group of the mathematician Katherine Stange. Kertzer, a Boulder native, wanted to play soccer and prepare his grad school application. Haag planned to explore new hikes and climbing routes. Both looked forward to a break from classes. In May, Haag was finishing her first year of graduate school at the University of Colorado, Boulder, where Kertzer was an undergraduate. Blindsiding an entire subfield of mathematics was not one of them. Summer Haag and Clyde Kertzer had high hopes for their summer research project.
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