This month's topics:

Polygonal billiards

Quanta magazine published "New Shapes Solve Infinite Pool-Table Problem" by Kevin Hartnett, on August 8, 2017. The particular pool tables discussed here are polygonal and friction-free, so that a ball, once set in motion, will bounce around forever. The question discussed is, where does it go? For some, optimal, tables, either the ball repeats a closed path over and over or it eventually gets arbitrarily close to every point on the table. There is no third option.

A regular pentagon gives an optimal billiard table: every path either is periodic, like the one shown in blue, or as time goes to infinity it fills up the entire surface, as the orange one does. Images for this item adapted from Hartnett's article.

For a non-optimal table, there is a third possibility: a billiard-ball path may fill up an entire region of the table while leaving other parts of the table untouched.

This non-convex, square-cornered table is non-optimal. This path will eventually fill up the entire area except for the upper-left and lower-right corners.

As Hartnett explains, "If you're given a polygon, there's no easy way to tell if it has optimal dynamics. For this reason, many basic questions about billiard paths remain unanswered." Even triangles are not completely understood. Hartnett lists the eight optimal triangle shapes (unknown if there are more) and reports the recent discovery of two new optimal quadrilaterals, work (unpublished at this moment) due to Alan Eskin (Chicago), Curt McMullen (Harvard), Ronen Mukamel (Rice) and Alex Wright (Stanford). "The new article identifies two new families of quadrilaterals with optimal dynamics: quadrilaterals whose angle ratios are 1:1:1:9 and 1:1:2:8. Both resemble darts, with two fins in the back and a point up front, and together they're the first quadrilaterals with optimal dynamics to be found in a decade."

Representatives of the two new families of quadrilaterals with optimal pool-table dynamics. The angles are the labeled multiples of 30$^{\circ}$.
We read how the new quadrilaterals (McMullen calls them "rare jewels"), were nailed down at the end of a complicated collaborative process. It started as an offshoot of research Wright had been doing with the late Maryam Mirzakhani, whose emphasis on the importance of studying a moduli space (here, for example, the space of all translation surfaces made from quadrilaterals with rational angles) was fundamental. Hartnett waxes lyrical: "To find these jewels, these four mathematicians used an elegant set of methods that allow mathematicians to reimagine the claustrophobic, rebounding world of a billiard table as an elegant universe of smooth curves arcing unimpeded through space. There, the far-out future of the billiard path can be apprehended at a glance -- while at the same time, perfect billiard tables end up serving as clues about the nature of the exotic higher-dimensional space in which they appear."


Preserving helicity

The abstract of "Complete measurement of helicity and its dynamics in vortex tubes" (Science, August 4, 2017) begins with "Helicity, a topological measure of the intertwining of vortices in a fluid flow, is a conserved quantity in inviscid fluids but can be dissipated by viscosity in real flows." As they suggest, helicity is a physical concept with a very mathematical definition. If a fluid flow has velocity vector field $\bf{u}$, its vorticity field is the curl $\nabla\times\bf{u}$, and the helicity of the flow is the integral of the dot-product $\bf{u}\cdot\nabla\times\bf{u}$ over the total extent of the flow. It has been known since the 1960s that helicity is conserved in flows with no viscosity. The authors, M. W. Scheeler (Chicago), W. M. van Rees (Harvard), H. Keida, D. Kleckner and W. T. M. Irvine (all at Chicago), show by (ingenious) experiment that under certain circumstances helicity can remain constant even in a viscous fluid. They worked with "small collections of thin-core vortices." These are vortex rings formed when the flow has a periodic orbit along which the vorticity field is non-zero, so nearby orbits twist around that one. "Examples of commonly encountered thin-core vortices are found in aircraft wakes, in insect flight, ... ." A familiar and visible example would be a very skinny smoke ring.

Thin-core vortices turn out to be "an ideal model system for studying helicity dynamics, because they allow helicity to be broken into distinct geometric forms." More specifically, when one is modeled as a flux tube, "a bundle of individual filaments, analogous to the construction of a twisted rope," then the total helicity can be completely captured by the three topologically related forms of winding: twisting, linking, and writhing.


Twisting (top), linking (center) and writhing (bottom) are the three geometrically distinct ways that nearby field lines can be related in a flux tube. They are topologically equivalent: the purple link in the center image can be rotated in space and slid into position to give the top configuration, and as the bundle in the bottom image is straightened out, the writhe converts into twist. "[T]hese three forms of winding completely capture the total helicity." Images courtesy of William Irvine.

In one of the authors' experiments, the conservation of helicity is illustrated by conversion of twist into writhe: "we considered a simple system in which a helically wound vortex loop leapfrogs with a second, planar vortex ring. As the pair evolves, the helical loop is periodically stretched (compressed), causing the coils to become looser (tighter) and the writhe to decrease (increase), whereas the planar ring remains writhe-free throughout the evolution, allowing the investigation of general writhe-changing deformations. ... We compressed a helically wound vortex loop by simultaneously generating a smaller concentric planar ring within it. We measured a rapid increase in the writhe along with simultaneous production of negative twist to roughly conserve total helicity."


helices leapfrogging
In this experiment, a vortex ring with a circular core and one with a helical core are moving in the same direction and "leap-frogging." Tracking the paths of the individual blobs of dye planted in the cores allows a precise measurement of the total helicity. Larger image. Image courtesy of William Irvine.

In another experiment, an isolated helical loop was left to evolve alone. The initial helicity was measured to be zero, meaning that negative twisting was balancing the writhe. "We observed that ... the helicity trends away from zero toward the writhe of the helix, instead of remaining constant." Their interpretation is that "the twist component of the helicity is dissipated by viscous effects over time. ... Once viscosity has produced a zero-twist state, the helicity remains roughly constant and equal to the writhe as the helix continues to evolve ... ." As the findings are described in the editor's preface, "They show that twisting dissipates total helicity, whereas writhing and linking conserve it. This provides a fundamental insight into tornadogenesis, atmospheric flows, and the formation of turbulence."

Mirzakhani obituary in Nature

Nature ran a full-scale obituary notice for Maryam Mirzakhani in their issue dated September 7, 2017. It is very unusual for a pure mathematician to be so honored: the most recent ones were John Nash (June, 2015) and Alexandre Grothendieck (January, 2015). Kasra Rafi (Toronto) starts: "Maryam Mirzakhani was one of the greatest mathematicians of her generation. She made monumental contributions to the study of the dynamics and geometry of mathematical objects called Riemann surfaces. Just as impressive as her theorems was her ability to push a field in a new direction by always providing a fresh point of view. Her raw talent was rare, even among the most celebrated mathematicians, and she was known for having a taste for difficult problems." And he ends: "The mathematics community has lost one of its greatest minds much too early, and I have lost a friend."


Tony Phillips
Stony Brook University
tony at