Chapter 2 Rational Tangles
While braids and knots have been well studied by mathematicians for at least the last century, two of the people most responsible for its popularization with math educators are mathematicians John Conway and James Tanton.
Conway, in search of "building blocks" for the construction of knots, developed his theory of rational tangles in the late 1960s. As we'll see in this chapter, tangles are something like a souped-up type of braids. We restrict ourselves to only two strands and only one orientation of twisting, which would threaten to take away all the interest from the structure except that we also permit rotation so we can interweave horizontal and vertical twists. From these two generating moves, a fixed twist and a fixed rotation, all rational tangles are by definition generated.
While the theory, classification, and applicability of rational tangles to the study of knots were all well developed, its recent history in math education is characterized by the writings of mathematicians Tom Davis and James Tanton, who developed a set of accessible, interactive activities to introduce students and teachers to tangles. These activities begin with a mathe-magical maypole ceremony known as the Rational Tangle Dance, whose surprising result creates the "why did that work?" that carries us all the way through the study of tangles and our first introduction to an algebraic invariant, the continued fraction of a rational tangle.