# American Institute of Mathematical Sciences

December  2021, 41(12): 5537-5549. doi: 10.3934/dcds.2021087

## Hearing the shape of right triangle billiard tables

 School of Mathematical Sciences, Peking University, Beijing 100871, China

* Corresponding author: Jiazhong Yang

Received  December 2020 Revised  April 2021 Published  December 2021 Early access  May 2021

Fund Project: The paper is supported by NSFC-12071006

In this paper, we give a positive answer to the problem that whether one can identify the shape of a right triangle billiard table by a single bounce sequence. Moreover, a convenient method to calculate the shape of polygons is given in this paper, too.

Citation: Yang Shen, Jiazhong Yang. Hearing the shape of right triangle billiard tables. Discrete and Continuous Dynamical Systems, 2021, 41 (12) : 5537-5549. doi: 10.3934/dcds.2021087
##### References:
 [1] J. Bobok and S. Troubetzkoy, Does a billiard orbit determine its (polygonal) table?, Fundamenta Mathematicae, 212 (2011), 129-144.  doi: 10.4064/fm212-2-2. [2] J. Bobok and S. Troubetzkoy, Code and order in polygonal billiards, Topology and its Applications, 159 (2012), 236-247.  doi: 10.1016/j.topol.2011.09.007. [3] A. Calderon, S. Coles, D. Davis, J. Lanier and A. Oliveira, How to hear the shape of a billiard table, preprint, arXiv: 1806.09644, 2018. [4] M. Duchin, V. Erlandsson, C. J. Leininger and C. Sadanand, You can hear the shape of a billiard table: Symbolic dynamics and rigidity for flat surfaces, preprint, arXiv: 1804.05690, 2019. [5] G. Galperin, Non-periodic and not everywhere dense billiard trajectories in convex polygons and polyhedrons, Comm. Math. Phys., 91 (1983), 187-211.  doi: 10.1007/BF01211158. [6] G. Galperin, T. Krüger and S. Troubetzkoy, Local instability of orbits in polygonal and polyhedral billiards, Comm. Math. Phys., 169 (1995), 463-473.  doi: 10.1007/BF02099308. [7] P. Hooper, Periodic billiard paths in right triangles are unstable, Geom. Dedicata, 125 (2007), 39-46.  doi: 10.1007/s10711-007-9129-9. [8] M. Kac, Can one hear the shape of a drum?, Amer. Math. Monthly, 73 (1966), 1-23.  doi: 10.1080/00029890.1966.11970915. [9] A. Katok and B. Hasselblatt, Flows on surface and related dynmical systems, in Introduction to the Modern Theory of Dynamical Systems, Cambridge University Press, (1995), 470–479. [10] H. Masur and S. Tabachnikov, Rational billiards and flat structures,, in Handbook of Dynamical Systems, Elsevier, 1 (2002), 1015–1089. doi: 10.1016/S1874-575X(02)80015-7. [11] Y. Shen, Hearing the Shape of Some Polygon Billiard Tables, Ph.d. Thesis, Peking University, 2021. [12] S. Tabachnikov, Billiards, Panor. Synth., 1 (1995), vi+142pp. [13] G. W. Tokarsky, Galperin's triangle example, Comm. Math. Phys., 335 (2015), 1211-1213.  doi: 10.1007/s00220-015-2336-6. [14] S. Troubetzkoy, Recurrence and periodic billiard orbits in polygons, Regular and Chaotic Dynamics, 9 (2004), 1-12.  doi: 10.1070/RD2004v009n01ABEH000259. [15] S. Troubetzkoy, Periodic billiard orbits in right triangles, Annales de l'institut Fourier, 55 (2005), 29-46.  doi: 10.5802/aif.2088.

show all references

##### References:
 [1] J. Bobok and S. Troubetzkoy, Does a billiard orbit determine its (polygonal) table?, Fundamenta Mathematicae, 212 (2011), 129-144.  doi: 10.4064/fm212-2-2. [2] J. Bobok and S. Troubetzkoy, Code and order in polygonal billiards, Topology and its Applications, 159 (2012), 236-247.  doi: 10.1016/j.topol.2011.09.007. [3] A. Calderon, S. Coles, D. Davis, J. Lanier and A. Oliveira, How to hear the shape of a billiard table, preprint, arXiv: 1806.09644, 2018. [4] M. Duchin, V. Erlandsson, C. J. Leininger and C. Sadanand, You can hear the shape of a billiard table: Symbolic dynamics and rigidity for flat surfaces, preprint, arXiv: 1804.05690, 2019. [5] G. Galperin, Non-periodic and not everywhere dense billiard trajectories in convex polygons and polyhedrons, Comm. Math. Phys., 91 (1983), 187-211.  doi: 10.1007/BF01211158. [6] G. Galperin, T. Krüger and S. Troubetzkoy, Local instability of orbits in polygonal and polyhedral billiards, Comm. Math. Phys., 169 (1995), 463-473.  doi: 10.1007/BF02099308. [7] P. Hooper, Periodic billiard paths in right triangles are unstable, Geom. Dedicata, 125 (2007), 39-46.  doi: 10.1007/s10711-007-9129-9. [8] M. Kac, Can one hear the shape of a drum?, Amer. Math. Monthly, 73 (1966), 1-23.  doi: 10.1080/00029890.1966.11970915. [9] A. Katok and B. Hasselblatt, Flows on surface and related dynmical systems, in Introduction to the Modern Theory of Dynamical Systems, Cambridge University Press, (1995), 470–479. [10] H. Masur and S. Tabachnikov, Rational billiards and flat structures,, in Handbook of Dynamical Systems, Elsevier, 1 (2002), 1015–1089. doi: 10.1016/S1874-575X(02)80015-7. [11] Y. Shen, Hearing the Shape of Some Polygon Billiard Tables, Ph.d. Thesis, Peking University, 2021. [12] S. Tabachnikov, Billiards, Panor. Synth., 1 (1995), vi+142pp. [13] G. W. Tokarsky, Galperin's triangle example, Comm. Math. Phys., 335 (2015), 1211-1213.  doi: 10.1007/s00220-015-2336-6. [14] S. Troubetzkoy, Recurrence and periodic billiard orbits in polygons, Regular and Chaotic Dynamics, 9 (2004), 1-12.  doi: 10.1070/RD2004v009n01ABEH000259. [15] S. Troubetzkoy, Periodic billiard orbits in right triangles, Annales de l'institut Fourier, 55 (2005), 29-46.  doi: 10.5802/aif.2088.
After a reflection, $P_{n}$ becomes $P_{n+1}$ in Case ($i$) and $P_{n-1}$ in Case ($ii$)
From $B(\gamma^+) = ({r}, {b}, {r}, {g}, {b}, {r}, {b}, {r}, {b}, {g}, {r}, \cdots)$ to get the return sequence $R(\gamma^+) = (1, 2, 1, 2, \cdots)$ and then to the get the level sequence $L(\gamma^+) = (0, 1, 0, -1, 0, \cdots)$
The visual diagram of $\theta_{n}$, $\theta_{n+1}$ and $\theta_{n+2}$
Cases ($i$) and ($ii$) correspond to $LM_n = ({r}, {g}, {r})$ and $\theta_{n}\in(\alpha, \pi-\alpha)$; Cases ($iii$) and ($iv$) correspond to $LM_n = ({r}, {b}, {r})$ and $\theta_{n}\in[0, \alpha)\bigcup\, (\pi-\alpha, \pi] = (-\alpha, \alpha)\, (\bmod\, \pi)$
In Case ($i$), $\theta_{n_0} = \pi-\alpha$. In Case ($ii$), $\theta_{n_0} = \alpha$
The left-hand part is a right triangle billiard table $Rt$ and a periodic orbit: $p_{0}$ $\longrightarrow$ $p_{1}$ $\longrightarrow$ $p_{2}$ $\longrightarrow$ $\cdots$ $\longrightarrow p_{9}$ $\longrightarrow$ $p_{10}$ $\longrightarrow$ $\cdots$ in $Rt$. The right-hand part is the unfolding floors of the corresponding rhombus of $Rt$ along the periodic orbit
The left-hand part is a right triangle billiard table $Rt$ and a periodic orbit: $p_{0}$ $\longrightarrow$ $p_{1}$ $\longrightarrow$ $p_{2}$ $\longrightarrow$ $\cdots$ $\longrightarrow$ $p_{6}$ $\longrightarrow$ $\cdots$ in $Rt$. The right-hand part is the unfolding floors of the corresponding rhombus of $Rt$ along the periodic orbit
 [1] Xiaohui Liu, Jinhua Wang, Dianhua Wu. Two new classes of binary sequence pairs with three-level cross-correlation. Advances in Mathematics of Communications, 2015, 9 (1) : 117-128. doi: 10.3934/amc.2015.9.117 [2] Zhe Zhang, Jiuping Xu. Bi-level multiple mode resource-constrained project scheduling problems under hybrid uncertainty. Journal of Industrial and Management Optimization, 2016, 12 (2) : 565-593. doi: 10.3934/jimo.2016.12.565 [3] Richard Hofer, Arne Winterhof. On the arithmetic autocorrelation of the Legendre sequence. Advances in Mathematics of Communications, 2017, 11 (1) : 237-244. doi: 10.3934/amc.2017015 [4] Yixiao Qiao, Xiaoyao Zhou. Zero sequence entropy and entropy dimension. Discrete and Continuous Dynamical Systems, 2017, 37 (1) : 435-448. doi: 10.3934/dcds.2017018 [5] Walter Briec, Bernardin Solonandrasana. Some remarks on a successive projection sequence. Journal of Industrial and Management Optimization, 2006, 2 (4) : 451-466. doi: 10.3934/jimo.2006.2.451 [6] Mario Roldan. Hyperbolic sets and entropy at the homological level. Discrete and Continuous Dynamical Systems, 2016, 36 (6) : 3417-3433. doi: 10.3934/dcds.2016.36.3417 [7] Alexander Bobylev, Åsa Windfäll. Boltzmann equation and hydrodynamics at the Burnett level. Kinetic and Related Models, 2012, 5 (2) : 237-260. doi: 10.3934/krm.2012.5.237 [8] Kai-Uwe Schmidt, Jonathan Jedwab, Matthew G. Parker. Two binary sequence families with large merit factor. Advances in Mathematics of Communications, 2009, 3 (2) : 135-156. doi: 10.3934/amc.2009.3.135 [9] Matthew Macauley, Henning S. Mortveit. Update sequence stability in graph dynamical systems. Discrete and Continuous Dynamical Systems - S, 2011, 4 (6) : 1533-1541. doi: 10.3934/dcdss.2011.4.1533 [10] Wenjun Xia, Jinzhi Lei. Formulation of the protein synthesis rate with sequence information. Mathematical Biosciences & Engineering, 2018, 15 (2) : 507-522. doi: 10.3934/mbe.2018023 [11] Thierry Horsin, Mohamed Ali Jendoubi. On the convergence to equilibria of a sequence defined by an implicit scheme. Discrete and Continuous Dynamical Systems - S, 2021, 14 (8) : 3017-3025. doi: 10.3934/dcdss.2020465 [12] Zhen Li, Cuiling Fan, Wei Su, Yanfeng Qi. Aperiodic/periodic complementary sequence pairs over quaternions. Advances in Mathematics of Communications, 2021  doi: 10.3934/amc.2021063 [13] Ji-Woong Jang, Young-Sik Kim, Sang-Hyo Kim. New design of quaternary LCZ and ZCZ sequence set from binary LCZ and ZCZ sequence set. Advances in Mathematics of Communications, 2009, 3 (2) : 115-124. doi: 10.3934/amc.2009.3.115 [14] Wenbing Chen, Jinquan Luo, Yuansheng Tang, Quanquan Liu. Some new results on cross correlation of $p$-ary $m$-sequence and its decimated sequence. Advances in Mathematics of Communications, 2015, 9 (3) : 375-390. doi: 10.3934/amc.2015.9.375 [15] Fanghua Lin, Dan Liu. On the Betti numbers of level sets of solutions to elliptic equations. Discrete and Continuous Dynamical Systems, 2016, 36 (8) : 4517-4529. doi: 10.3934/dcds.2016.36.4517 [16] Francisco Sánchez-Sánchez, Miguel Vargas-Valencia. Games with nested constraints given by a level structure. Journal of Dynamics and Games, 2018, 5 (2) : 95-107. doi: 10.3934/jdg.2018007 [17] Jun Wu, Shouyang Wang, Wuyi Yue. Supply contract model with service level constraint. Journal of Industrial and Management Optimization, 2005, 1 (3) : 275-287. doi: 10.3934/jimo.2005.1.275 [18] David L. Finn. Convexity of level curves for solutions to semilinear elliptic equations. Communications on Pure and Applied Analysis, 2008, 7 (6) : 1335-1343. doi: 10.3934/cpaa.2008.7.1335 [19] Marina Dolfin, Mirosław Lachowicz. Modeling DNA thermal denaturation at the mesoscopic level. Discrete and Continuous Dynamical Systems - B, 2014, 19 (8) : 2469-2482. doi: 10.3934/dcdsb.2014.19.2469 [20] Alexander Zeh, Antonia Wachter. Fast multi-sequence shift-register synthesis with the Euclidean algorithm. Advances in Mathematics of Communications, 2011, 5 (4) : 667-680. doi: 10.3934/amc.2011.5.667

2021 Impact Factor: 1.588

## Metrics

• HTML views (291)
• Cited by (0)

• on AIMS