Taffy, Trees, and Tangles

TL;DR

This work links three combinatorial objects—taffy pulling machines, the Calkin-Wilf tree, and Conway's rational tangles—by embedding them in a common four-way tree of fractions. It develops a taffy-number framework, including forward, reverse, and generalized moves, showing that taffy pulls are classified by $Q(t)$ and that every fraction (including $1/0$) appears in the four-way tree, with paths corresponding to continued fractions and computable via the Euclidean algorithm. A geometric bridge is established by rotating taffy diagrams to align with tangle diagrams, revealing that taffy pulls and rational tangles are physically distinct representations of the same underlying structure. The paper also uncovers algorithmic means to locate fractions in the tree, proves equivalence of generalized pulls to canonical representatives, and motivates broader connections to knot theory and representation other than the three primary objects studied.

Abstract

We study the relationship between three combinatorial objects -- a taffy pulling machine, the Calkin-Wilf tree of all fractions, and Conway's rational tangles. After introducing these objects, we develop a taffy analogue for Conway's characterization of rational tangles, and we give a direct geometric connection between rational tangles and taffy pulls.

Figures (21)

• Figure 1: A three-peg taffy pulling machine.
• Figure 2: Visualizing a right turn followed by a left turn.
• Figure 3: Depicting a sequence of taffy pulls made by the Nitz machine.
• Figure 4: Counting the numbers of left and right layers.
• Figure 5: A left turn "adds" the right layers to the left layers.

Theorems & Definitions (8)

• Lemma 1
• proof
• Lemma 2
• proof
• Lemma 3
• proof
• Theorem 4
• proof