Book: The Equation That Couldn't Be Solved: How Mathematical Genius Discovered the Language of Symmetry
Quotes of Book: The Equation That Couldn't Be
An interesting question is whether symmetry with respect to translation, and indeed reflection and rotation too, is limited to the visual arts, or may be exhibited by other artistic forms, such as pieces of music. Evidently, if we refer to the sounds, rather than to the layout of the written musical score, we would have to define symmetry operations in terms other than purely geometrical, just as we did in the case of the palindromes. Once we do that, however, the answer to the question, Can we find translation-symmetric music? is a resounding yes. As Russian crystal physicist G. V. Wulff wrote in 1908: "The spirit of music is rhythm. It consists of the regular, periodic repetition of parts of the musical composition...the regular repetition of identical parts in the whole constitutes the essence of symmetry." Indeed, the recurring themes that are so common in musical composition are the temporal equivalents of Morris's designs and symmetry under translation. Even more generally, compositions are often based on a fundamental motif introduced at the beginning and then undergoing various metamorphoses. book-quoteThe properties that define a group are:1. Closure. The offspring of any two members combined by the operation must itself be a member. In the group of integers, the sum of any two integers is also an integer {e.g., 3 + 5 = 8}.2. Associativity. The operation must be associative-when combining {by the operation} three ordered members, you may combine any two of them first, and the result is the same, unaffected by the way they are bracketed. Addition, for instance, is associative: {5 + 7} + 13 = 25 and 5 + {7 + 13} = 25, where the parentheses, the "punctuation marks" of mathematics, indicate which pair you add first.3. Identity element. The group has to contain an identity element such that when combined with any member, it leaves the member unchanged. In the group of integers, the identity element is the number zero. For example, 0 + 3 = 3 + 0 = 3.4. Inverse. For every member in the group there must exist an inverse. When a member is combined with its inverse, it gives the identity element. For the integers, the inverse of any number is the number of the same absolute value, but with the opposite sign: e.g., the inverse of 4 is -4 and the inverse of -4 is 4; 4 + {-4} = 0 and {-4} + 4 = 0.The fact that this simple definition can lead to a theory that embraces and unifies all the symmetries of our world continues to amaze even mathematicians. book-quote
