![]() Next come the three genuine ways to flip a mattress. MathematiciansĬall it the “identity element,” so I’ll denote it by the symbol I. It plays the same role for group theory that zero does for addition of numbers, or that 1 does for multiplication. Still, it’s very important to include in the group. It certainly satisfies all the rules, but it’s not much help in prolonging the life The first is the “do-nothing” transformation, a lazy but popular choice that leaves the mattress untouched. ![]() With these rules in place, let’s see what transformations qualify for membership in this exclusive little group. Has to fit snugly on the rectangular bed frame (that’s what stays the same). In the case of a mattress, the transformations alter its orientation in space (that’s what changes) while maintaining its rigidity (that’s the constraint). Taken together they form a “group,” a collection of transformations whose relationships define the shape’s most basic architecture. More precisely, they look for all the transformations that leave a shape unchanged, given certain constraints. It while keeping something else about it the same. ![]() But group theorists focus more on what you can do to a shape - specifically, all the ways you can change Normally we think of symmetry as a property of a shape. Wide range of phenomena, group theory is necessarily abstract. It addresses something the two cultures share - an abiding fascination with symmetry. As these examples suggest, group theory bridges the arts and sciences. ![]()
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