The rubiks cube is a well known puzzle that has remarkable group theory properties. Two moves are considered same, if the final configuration after the moves are the same. Group theory and the rubiks cube harvard mathematics. In this lecture, we will introduce the concept of a group using the famous rubiks cube. In mathematics, a group is a set equipped with a binary operation that combines any two elements to form a third element in such a way that four conditions called group axioms are satisfied, namely closure, associativity, identity and invertibility.
More than 100 million cubes have been sold worldwide. In mathematics and abstract algebra, group theory studies the algebraic structures known as. Group theory in crystallography tutorial tu dresden. Group theory physics institute of bonn university, theory. This is because we can identify a legal position with a sequence of cube moves. One of the most familiar examples of a group is the set of integers together. About 150 years earlier, in the late 1820s and early 1830s, a french teenager named evariste galois developed a new branch of mathematics. The rubiks cube group the set of all possible moves on a rubiks cube form the rubiks cube group. The popular puzzle rubiks cube invented in 1974 by erno rubik has been used as an illustration of permutation groups. Group theory and the rubiks cube harvard department of. The solution to the cube can also be described by group theory 5. Functions will provide important examples of groups later on. The thesis first introduces the structure and notations of the cube and the mathematical background including permutation and group theory. Introduction to group theory and permutation puzzles march 17, 2009.
Introduction to group theory and permutation puzzles. Crystal system point group viewing direction space group type cubic. The next part provides the architecture and the essential features of the software. However, since di erent sequences of cube moves may result in the same legal position, we see that there will be many group relations. By 1982, rubiks cube was a household term, and became part of the oxford english dictionary. The objective of this project is to understand how the rubik s cube operates as a group and explicitly construct the rubik s cube group. The di erent transformations and con gurations of the cube form a subgroup of a permutation group generated by the di erent horizontal and vertical rotations of the puzzle 2. Group theory 4 applications in crystallography and solid state chemistry. References douglas hofstadter wrote an excellent introduction to the rubik s cube in the march 1981 issue of scienti c american.
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