Abstract

The Magic Snake (Rubik’s Snake) is a toy that was invented decades ago. It draws much less attention than Rubik’s Cube, which was invented by the same professor, Erno Rubik. The number of configurations of a Magic Snake, determined by the number of discrete rotations about the elementary wedges in a typical snake, is far less than the possible configurations of a typical cube. However, a cube has only a single three-dimensional (3D) structure while the number of sterically allowed 3D conformations of the snake is unknown. Here, we demonstrate how to represent a Magic Snake as a one-dimensional (1D) sequence that can be converted into a 3D structure. We then provide two strategies for designing Magic Snakes to have specified 3D structures. The first enables the folding of a Magic Snake onto any 3D space curve. The second introduces the idea of “embedding” to expand an existing Magic Snake into a longer, more complex, self-similar Magic Snake. Collectively, these ideas allow us to rapidly list and then compute all possible 3D conformations of a Magic Snake. They also form the basis for multidimensional, multi-scale representations of chain-like structures and other slender bodies including certain types of robots, polymers, proteins, and DNA.

References

References
1.
Fiore
,
A.
,
1981
,
Shaping Rubik’s Snake
,
Penguin Books
,
Harmondsworth, Middlesex, England
.
2.
Zeng
,
D.
,
Li
,
M.
,
Wang
,
J.
,
Hou
,
Y.
,
Lu
,
W.
, and
Huang
,
Z.
,
2018
, “
Overview of Rubik’s Cube and Reflections on Its Application in Mechanism
,”
Chin. J. Mech. Eng. (English Edition)
,
31
(
4
), p.
77
. 10.1186/s10033-018-0269-7
3.
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
), pp.
499
506
. 10.1142/S0217984998000603
4.
Iguchi
,
K.
,
1999
, “
Exactly Solvable Model of Protein Folding: Rubik’s Magic Snake Model
,”
Int. J. Mod. Phys. B
,
13
(
4
), pp.
325
361
. 10.1142/S0217979299000205
5.
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
,
Guanajuato, México
,
June 2011
.
6.
Ding
,
X.
, and
Lu
,
S.
,
2013
, “
Fundamental Reconfiguration Theory of Chain-Type Modular Reconfigurable Mechanisms
,”
Mech. Mach. Theory
,
70
, pp.
487
507
. 10.1016/j.mechmachtheory.2013.08.011
7.
Zhang
,
X.
, and
Liu
,
J.
,
2016
,
Prototype Design of a Rubik Snake Robot
, Vol.
36
,
Springer
,
Cham
.
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
,”
Proc. Inst. Mech. Eng., Part C: J. Mech. Eng. Sci.
,
233
(
9
), pp.
3137
3154
. 10.1177/0954406218805112
9.
White
,
P. J.
,
Revzen
,
S.
,
Thorne
,
C. E.
, and
Yim
,
M.
,
2011
, “
A General Stiffness Model for Programmable Matter and Modular Robotic Structures
,”
Robotica
,
29
(
1 SPEC. ISSUE
), pp.
103
121
. 10.1017/S0263574710000743
10.
Liu
,
J.
,
Zhang
,
X.
, and
Hao
,
G.
,
2016
, “
Survey on Research and Development of Reconfigurable Modular Robots
,”
Adv. Mech. Eng.
,
8
(
8
), pp.
1
21
.
11.
Liu
,
J.
,
Wang
,
Y.
,
Ma
,
S.
, and
Li
,
Y.
,
2010
, “
Enumeration of the Non-Isomorphic Configurations for a Reconfigurable Modular Robot With Square-Cubic-Cell Modules
,”
Int. J. Adv. Rob. Syst.
,
7
(
4
), pp.
58
68
. 10.5772/10489
12.
Stoy
,
K.
, and
Brandt
,
D.
,
2013
, “
Efficient Enumeration of Modular Robot Configurations and Shapes
,”
IEEE International Conference on Intelligent Robots and Systems
,
Tokyo
, pp.
4296
4301
. http://dx.doi.org/10.1109/IROS.2013.6696972
13.
de Gennes
,
P.
,
1979
,
Scaling Concepts in Polymer Physics
,
Cornell University Press
,
Ithaca, NY
.
14.
EL Hassan
,
M. A.
, and
Calladine
,
C. R.
,
1995
, “
The Assessment of the Geometry of Dinucleotide Steps in Double-Helical DNA; A New Local Calculation Scheme
,”
J. Mol. Biol.
,
251
(
5
), pp.
648
664
. 10.1006/jmbi.1995.0462
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