Research Paper Sample: The Rubik's Cube in Relation to Group Theory

Rubiks cube is a structure made up of twenty-seven cubies, contained in six cubicles. Only twenty-six cubies are visible. Corner cubes have three different colors and add up to eight in number. All the six faces of the cube are customarily colored. Furthermore, twelve cubes on the edges have two faces visible, again, of different colors. Only the six remaining cubes at the center of each cubicle have a single side and shares the color of that cubicle. The temptation is often to use the colors to try to solve the Rubiks cube puzzle. In addition, the need to be systematic comes inherently, what skips people who take on this puzzling cube, however, is that one needs to be able to have a way of describing the individual cubies (Chen, 2017). Attempts fail as the cube jumbles all the more.

Mathematics has a way of finding its way in to everything from art to music to shapes. Group theory has a perfect example of defining itself through the patterns of the Rubik's cube. Group theory branches from a class of algebra that does not rely on a specific operation or set (Nash, 2016). Functions and consequently groups are central to an understanding of Group theory. A function is a rule that assigns a particular element in a set, say H to only one other component of a set, J, while a group is a set G with an operation. Guided by the Rubik's cube, a group describes the set of actions that fulfill specific properties. Successfully presentation of the Rubik's cube move in a group can help develop methods for solving the Rubiks cube.

The Mathematics of the Cube

The puzzle of the cube is a mathematical problem. The popular toy concept is symmetry and patterns, and there exist features that provide evidence to affirm that (Jeevanjee, n.d.) The first is that moves performed come in a fixed fashion. Any move puts corner cubes at corners and edge cubies as edge cubies. Trying to make half an action is impossible. Moreover, it is possible to reverse a move once made. Another observation is that every step made has deterministic properties. Finally, all these actions made in ninety-degree turns take any combination. The four characteristics of the cubes actions are enough proof of mathematical aspects (Nash, 2016). The study uses the features described above as the rules (or axioms) to denote the Rubik's cube mathematically (Macauley, 2017). These are:

Actions are fixed

Actions are reversible

Actions made are deterministic

Activities can take any combination.

Making moves on Rubiks cube into Groups

A series of visualizations help in the quest to present the David Singmaster developed the notation used to name the faces of the Rubiks cube, and so there is no need to develop another. They are up (u), left (l), front (f), right (r), back (b), and down (d) (Chen, 2017). The description allows for locating each face of the cubies. The primary goal is to make it clear what is being explored using a common language. In this form, there is a basis upon which to derive logical conclusions.

Figure 1 Rubik's Cube notations of its six faces

The most basic action rotates a face once. It follows then that there are six basic moves all of which leave center cubies within the cubicle, and leaves corner cubies and edge cubies as corner cubies and edge cubies, respectively. This observation means any corner cubies can reside in any cubicle. Seven other cubies can also do this and the other six too. Then, there are 8.7.6.5.4.3.2.1=8! Possible permutation of the corner cubies, but since each corner cubie has three faces, 8 of them can fit in the cubicle three different ways, to make the total permutations 38 .8! That fact also holds true for the 12 edge cubies, 12! configurations whose two faces give 212 possible orientations giving 12!.212 permutations. Therefore, the Rubik's cube has infinitely possible 8!.38.12!.212 configurations from just six moves (Ryanheise.com, 2007).

If, one takes a set of moves of the cube into a group, G, elements of G are all the likely moves of the Rubik's cube (Provenza, n.d.) Consider this group to operate as follows; there are two moves T1 and T2; T1*T2 describes a step where you do T1 followed by T2. The evidence defines Gs closure properties, the reason being T1, T2 and T1*T2 are all moves. Further, having an empty move, e, then T*e= T leading to the conclusion G has a right identity. Also, taking T as a move, reversing the steps of that move gives T, meaning T*T==e, the meaning is G possesses the right inverse. Lastly, showing * is associative proves that G is a group of rotational symmetry

All possible configurations are valid, and there is need to work out a set of moves that can move someone from any logical configuration to back to the start hence solving the Rubiks cube puzzle. Starting with a solved Rubiks cube and distorting it through a few moves coupled with their inverses, one should be able to undo the mystery by restricting oneself only to a subgroup of the Rubiks cube group. Trying to understand a large number of configurations that result from all the possible permutations is not easy. The understanding of a group and then trying to understand smaller bits of it, in general, is group theory (Davis, 2017).

References

Macauley, M. (2017). [online] Math.clemson.edu. Available at: http://www.math.clemson.edu/~macaule/math4120-online.html [Accessed 10 Nov. 2017].

Chen, J. (2017). [online] Math.harvard.edu. Available at: http://www.math.harvard.edu/~jjchen/docs/Group%20Theory%20and%20the%20Rubik%27s%20Cube.pdf [Accessed 10 Nov. 2017].

Davis, T. (2017). Group Theory via Rubik's Cube. [online] Geometer.org. Available at: http://www.geometer.org/rubik/group.pdf [Accessed 10 Nov. 2017].

Jeevanjee, N. (n.d.). An introduction to tensors and group theory for physicists.

Nash, D. (2016). A friendly introduction to group theory. [North Charleston, S.C.]: [CreateSpace].Ryanheise.com (2007). Group theory | Rubik's Cube. [online] Ryanheise.com. Available at: https://www.ryanheise.com/cube/group_theory.html [Accessed 11 Nov. 2017].

Provenza, H. (n.d.). Cite a Website - Cite This For Me. [online] Math.uchicago.edu. Available at: http://www.math.uchicago.edu/~may/VIGRE/VIGRE2009/REUPapers/Provenza.pdf [Accessed 11 Nov. 2017].