Negative refraction and tiling billiards

Davis, Diana; DiPietro, Kelsey; Rustad, Jenny; Laurent, Alexander St

doi:10.1515/advgeom-2017-0053

Cited by 9 publications

(33 citation statements)

References 16 publications

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“…They may be periodic or drift-periodic (invariant under a non-trivial translational symmetry of the tiling). Such behaviors have been seen before in a number of tilings [4]. However See Figure 2 for an example of a portion of a trajectory which appears to fill part of the plane, but misses a periodic family of open triangles in the center of upward-pointing triangles.…”

Section: Introductionsupporting

confidence: 62%

“…The connection between metamaterials with a negative index of refraction and the problem on planar tilings was made by Mascarenhas and Fluegel [22]. Davis, DiPietro, Rustad and St Laurent named the system tiling billiards and explored several special cases of the system, including triangle tilings and the trihexagonal tiling [4]. They found examples of periodic trajectories in the trihexagonal tiling, constructed families of drift-periodic trajectories, and conjectured that dense trajectories and non-periodic escaping trajectories exist ( [4], .…”

Section: Tiling Billiardsmentioning

confidence: 99%

“…Concurrently with our work on the trihexagonal tiling, the first author with Baird-Smith, Fromm and Iyer in [3] studied tiling billiards on triangle tilings, showing that trajectories on these tilings can be described by interval and polygon exchange transformations, and resolving additional conjectures from [4]. These systems are quite different: if a trajectory visits a single tile twice then the trajectory is periodic.…”

Section: Tiling Billiardsmentioning

confidence: 99%

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Periodicity and ergodicity in the trihexagonal tiling

Davis

Hooper

2018

Comment. Math. Helv.

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We consider the dynamics of light rays in the trihexagonal tiling where triangles and hexagons are transparent and have equal but opposite indices of refraction. We find that almost every ray of light is dense in a region of a particular form: the regions have infinite area and consist of the plane with a periodic family of triangles removed. We also completely describe initial conditions for periodic and drift-periodic light rays. arXiv:1609.00772v2 [math.MG] 18 Jul 2018Theorem 1.2 (Ergodic directions). If θ ∈ E then the flows T θ,+ and T θ,− are ergodic when the domains X θ,+ and X θ,− are equipped with their natural Lebesgue measures.By remarks above this implies that the set of non-ergodic directions has Hausdorff dimension less than 1. Example 1.3 (θ = π 4 ). The angle θ = π 4 lies in E. To see this observe that θ = θ + π 3 = 7π 12 ∈ [ π 3 , 2π 3 ]. By definition u θ is the unit tangent vector based at i pointed into ∆ at angle 5π 6 from the vertical. The billiard trajectory is periodic and is depicted in Figure 2. This billiard trajectory repeatedly travels above the line where y = 1 √ 3 and so by the Theorem above the flows T θ,+ and T θ,− are ergodic. A trajectory of T θ,+ is shown on the left side of Figure 2. We do not know if this trajectory equidistributes.

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Section: Introductionsupporting

confidence: 62%

Section: Tiling Billiardsmentioning

confidence: 99%

Section: Tiling Billiardsmentioning

confidence: 99%

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Periodicity and ergodicity in the trihexagonal tiling

Davis

Hooper

2018

Comment. Math. Helv.

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“…Here are some remarks on the qualitative behavior that were clear from the previous work on triangle tiling billiards [DDRSL16,BDFI18]. First, drift-periodic orbits occur only when ∆ has rationally dependent angles (i.e.…”

Section: Qualitative Behavior Of Orbits In Triangle Tiling Billiardsmentioning

confidence: 88%

“…He supposes that the black and white tiles have different refraction coefficient indices, k 1 and k 2 , and relates the dynamics to interval exchange transformations in the case when k 1 k 2 > √ 2. As far as we know, the first published (in 2016) works on tiling billiards are [DDRSL16] and [G16]. Although, one could say that the story of tiling billiards starts almost thirty years earlier, in 1989 with the work of Nogueira [N89] on interval exchange transformations with flips.…”

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confidence: 99%

Triangle Tiling Billiards and the Exceptional Family of their Escaping Trajectories: Circumcenters and Rauzy Gasket

Hubert

Paris-Romaskevich

2019

Experimental Mathematics

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Consider a periodic tiling of a plane by equal triangles obtained from the equilateral tiling by a linear transformation. We study a following tiling billiard : a ball follows straight segments and bounces of the boundaries of the tiles into neighbouring tiles in such a way that the coefficient of refraction is equal to −1. We show that almost all the trajectories of such a billiard are either closed or escape linearly, and for closed trajectories we prove that their periods belong to the set 4N * + 2. We also give a precise description of the exceptional family of trajectories (of zero measure) : these trajectories escape non-linearly to infinity and approach fractal-like sets. We show that this exceptional family is parametrized by the famous Rauzy gasket. This proves several conjectures stated previously on triangle tiling billiards. In this work, we also give a more precise understanding of fully flipped minimal exchange transformations on 3 and 4 intervals by proving that they belong to a special hypersurface. Our proofs are based on the study of Rauzy graphs for interval exchange transformations with flips.

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No-slip billiards with particles of variable mass distribution

Ahmed

Cox

Wang

2022

Chaos: An Interdisciplinary Journal of Nonlinear Science

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Astute variations in the geometry of mathematical billiard tables have been and continue to be a source of understanding their wide range of dynamical behaviors, from regular to chaotic. Viewing standard specular billiards in the broader setting of no-slip (or rough) collisions, we show that an equally rich spectrum of dynamics can be called forth by varying the mass distribution of the colliding particle. We look at three two-parameter families of billiards varying both the geometry of the table and the particle, including as special cases examples of standard billiards demonstrating dynamics from integrable to chaotic, and show that markedly divergent dynamics may arise by changing only the mass distribution. Furthermore, for certain parameters, billiards emerge, which display unusual dynamics, including examples of full measure periodic billiards, conjectured to be nonexistent for the standard billiards in Euclidean domains.

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Negative refraction and tiling billiards

Cited by 9 publications

References 16 publications

Periodicity and ergodicity in the trihexagonal tiling

Periodicity and ergodicity in the trihexagonal tiling

Triangle Tiling Billiards and the Exceptional Family of their Escaping Trajectories: Circumcenters and Rauzy Gasket

No-slip billiards with particles of variable mass distribution

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