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The recently proposed map [5] between the hydrodynamic equationsgoverning the two-dimensional triangular cold-bosonic breathers [1] andthe high-density zero-temperature triangular free-fermionic clouds, bothtrapped harmonically, perfectly explains the former phenomenon butleaves uninterpreted the nature of the initial(t=0)singularity. This singularity is a density discontinuity that leads, inthe bosonic case, to an infinite force at the cloud edge. The map itselfbecomes invalid at times t<0 t < 0 .A similar singularity appears at t = T/4 t = T / 4 ,where Tis the period of the harmonic trap, with the Fermi-Bose map becominginvalid at t > T/4 t > T / 4 . Here, we first map—using the scale invariance of the problem—thetrapped motion to an untrapped one. Then we show that in the newrepresentation, the solution [5] becomes, along a ray in the directionnormal to one of the three edges of the initial cloud, a freelypropagating one-dimensional shock wave of a class proposed by Damski in[7]. There, for a broad class of initial conditions, the one-dimensionalhydrodynamic equations can be mapped to the inviscid Burgers’ equation,which is equivalent to a nonlinear transport equation. Morespecifically, under the Damski map, thet=0singularity of the original problem becomes, verbatim, the initialcondition for the wave catastrophe solution found by Chandrasekhar in1943 [9]. At t=T/8 t = T / 8 ,our interpretation ceases to exist: at this instance, all threeeffectively one-dimensional shock waves emanating from each of the threesides of the initial triangle collide at the origin, and the 2D-1Dcorrespondence between the solution of [5] and the Damski-Chandrasekharshock wave becomes invalid. 

In Galperin billiards, two balls colliding with a hard wall form an analog calculator for the digits of the number π . This classical, one-dimensional three-body system (counting the hard wall) calculates the digits of π in a base determined by the ratio of the masses of the two particles. This base can be any integer, but it can also be an irrational number, or even the base can be π itself. This article reviews previous results for Galperin billiards and then pushes these results farther. We provide a complete explicit solution for the balls’ positions and velocities as a function of the collision number and time. We demonstrate that Galperin billiard can be mapped onto a two-particle Calogero-type model. We identify a second dynamical invariant for any mass ratio that provides integrability for the system, and for a sequence of specific mass ratios we identify a third dynamical invariant that establishes superintegrability. Integrability allows us to derive some new exact results for trajectories, and we apply these solutions to analyze the systematic errors that occur in calculating the digits of π with Galperin billiards, including curious cases with irrational number bases.