A Rubik’s Snake with General Rotation Angles

Volume 5, Issue 6, December 2021     |     PP. 123-135      |     PDF (899 K)    |     Pub. Date: November 23, 2021
DOI: 10.54647/isss12179    92 Downloads     5975 Views  

Author(s)

Songming Hou, Program of Math&Stats and Center of Applied Physics, Louisiana Tech University, Ruston, LA 71272
Scott Atkins, Louisiana School for Math, Science, and the Arts, Natchitoches, LA 71457
Yu Chen, Program of Math&Stats, Louisiana Tech University, Ruston, LA 71272

Abstract
A Rubik’s Snake is a toy that has been around for 40 years. It is a system of serial chain. It traditionally only allows rotations of its pieces in 90 degree increments. In this paper we are not going to restrict ourselves to these limited number of incremental rotations. We prove a theorem about the sum of rotations of a Rubik’s Snake that closes on itself. We present an alternative representation for the Rubik’s Snake, different from the standard representation that the toy has. We will use matrices to study the rotations of the pieces of the Rubik’s Snake. In particular we will study two cases where all of the pieces of a Rubik’s Snake have the same rotation angle, proving a theorem comparing the first and last pieces of a Rubik’s Snake.

Keywords
Rubik’s Snake, robot design

Cite this paper
Songming Hou, Scott Atkins, Yu Chen, A Rubik’s Snake with General Rotation Angles , SCIREA Journal of Information Science and Systems Science. Volume 5, Issue 6, December 2021 | PP. 123-135. 10.54647/isss12179

References

[ 1 ] Fiore, A., 1981. Shaping Rubik’s Snake. Penguin Books, Harmondsworth, Middlesex, England.
[ 2 ] Fenyvesi, C., 1981. “Rubik’s snake of ‘infinite possibilities”’. The Washington Post.
[ 3 ] Jensen, G., 1981. “Now meet Rubik’s snake –‘bigger than Rubik’s cube!”’. United Press International.
[ 4 ] Iguchi, K., 1998. “A toy model for understanding the conceptual framework of protein folding: Rubik’s magic snake model”. Mod. Phys. Lett. B, 12(13), p. 499–506.
[ 5 ] Iguchi, K., 1999. “Exactly solvable model of protein folding: Rubik’s magic snake model”. Int. J. Mod. Phys. B, 13(4), pp. 325–361.
[ 6 ] Ding, X., Lu, S., and Yang, Y., 2011. “Configuration transformation theory from a chain-type reconfigurable modular mechanism-Rubik’s snake”. The 13th World Congress in Mechanism and Machine Science.
[ 7 ] Zhang, X., and Liu, J., 2016. “Prototype design of a Rubik snake robot”, Vol. 36 of Mechanisms and Machine Science.
[ 8 ] Liu, J., Zhang, X., Zhang, K., Dai, J. S., Li, S., and Sun, Q., 2019. “Configuration analysis of a reconfigurable Rubik’s snake robot”. Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science, 233(9), pp. 3137–3154.
[ 9 ] Detmvit, J., and Hartenberg, R., 1955. “A kinematic notation for lower-pair mechanisms based on matrices”. ASME Journal of Applied Mechanics, pp. 215–221.
[ 10 ] Yim, M., Roufas, K., Duff, D., Zhang, Y., Eldershaw, C., and Homans, S., 2003. “Modular reconfigurable robots in space applications”. Autonomous Robots, 14(2-3), pp. 225–237.
[ 11 ] Zhang, X., Liu, J., Feng, J., Liu, Y., and Ju, Z., 2020. “Effective capture of nongraspable objects for space robots using geometric cage pairs”. IEEE/ASME Transactions on Mechatronics, 25(1), pp. 95–107.
[ 12 ] Hull, T. C., and Belcastro, S.-m., 2002. “Modelling the folding of paper into three dimensions using affine transformations”. Linear Algebra and its applications, 348(1-3), pp. 273–282.
[ 13 ] Tachi, T., 2009. “Simulation of rigid origami”. Origami, 4(08), pp. 175–187.
[ 14 ] Li, Z., Hou, S., and Bishop, T., 2020. “Computational design and analysis of a magic snake”. J. Mech. Rob., 12(5), p. 054501.
[ 15 ] Hou, S., Chen, Y., and Li, Z., 2021. “Some mathematical problems related to the Rubik’s snake”. J. Mech. Rob., 13(1), p. 014502.
[ 16 ] Hou, S., Su, J., and Chen, Y., 2021. “Palindromic, periodic and Möbius Rubik’s snakes”. International Robotics and Automation Journal, 7(3), pp. 84–88.