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Rational Tangles

A Mathematical Dance for Audiences from Middle School to Abstract Algebra

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Bryan Stutzman, Ph.D.

  • Instructor of Mathematics: NCSSM-Morganton
  • Courses Currently Teaching
    • Calculus 2
    • Data Science
    • Operations Research
  • Background
    • Educational
      • NCSSM -> Duke (EE/Econ) -> GT (IE/OR)
    • Professional
      • CAPS Logistics -> Elementary -> Private School -> NCSSM-Morganton

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Rational Tangles

  • I first saw at a Julia Robinson Festival at UNC Charlotte
    • Middle School Girls Audience
    • Michael Pillsbury: Randolph Middle
  • Formulated by John Conway (1969)
  • Accessible at many levels
  • Uses
    • General Audience Event
    • Deeper study (curricular or extracurricular)

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Basic Dance

  • 4 Dancers
  • 2 Ropes
    • Color of Ropes and Identity of Dancers is Insignificant
  • 2 (3) Moves
    • Display
    • Twist
    • Rotate

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How to classify the dance

  • What state are we in?
  • What do the dance moves do?
  • Are we right?
  • Are we sure?

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What we found out

  • Original formation = 0
  • Twist T
    • +1
  • Rotate R
    • Opposite Reciprocal
  • Questions
    • Are all rational numbers possible?
    • Is it possible to not be able to get back to 0?
    • Are there inverses?
      • R
      • T

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Math circles

  • Tom Davis
    • Homepage link
    • YouTube Playlist of similar presentation
    • 12 Page paper of observations on Rational Tangles

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Abstract Algebra and Rational Tangles

  • Matthew Salomone, Bridgewater State University

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Rational Tangles and Unraveling DNA

PBS Infinite Series

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Connection to Greatest Common Divisor

  • See Tom Davis Paper
  • Euclid’s Algorithm
    • GCD of (a,b) (a<b) is same as GCD of remainder c of b/a and a
  • Similarly Work from fraction -a/b to 0 with moves of +1 and -1/(a/b)

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What does a Tangle Look Like

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Questions

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Thank You