ABSTRACT
Preparation for <xref ref-type="book-part" rid="chapter5">Chapter 5</xref>
The purpose of this (rather fun) exercise is to assure the reader that a number is just a symbol, and context determines its usefulness. Conway’s rational tangles are a way to describe the various ways in which two strands of rope can be entwined. Place two ends of rope on a table, and label the corners of the table A, B, C, and D in a clockwise fashion. Now, place the ropes on the table, one going from point A to B and one from D to C:
A twist is performed by swapping the ends at B and C, passing the end that started at B over the one that started at C. A rotate consists of rotating the ends 90 degrees clockwise, A → B → C → D. Here are two twists followed by a rotate: https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781482243048/27066273-287e-4f87-9432-f815edbc9855/content/pg104_1.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/>
Conway associates with each two-rope tangle a rational number. The starting point is 0. Each twist adds 1, and each rotate takes the negative reciprocal. So, two twists and a rotate yield https://www.w3.org/1998/Math/MathML"> 0 → T 1 → T 2 → R − 1 2 . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781482243048/27066273-287e-4f87-9432-f815edbc9855/content/eq323.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/>
He proved that tangles are in one-to-one correspondence with rational numbers. Consequently, you can untangle the above tangle by doing another twist, a rotate, and then two more twists: https://www.w3.org/1998/Math/MathML"> − 1 2 → T 1 2 → R − 2 → T − 1 → T 0. https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781482243048/27066273-287e-4f87-9432-f815edbc9855/content/eq324.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/>
(By the way, rotating 0 yields https://www.w3.org/1998/Math/MathML"> − 1 0 = 1 0 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781482243048/27066273-287e-4f87-9432-f815edbc9855/content/eq325.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> .)
Prep Problem 5.1. Determine the sequence of twists and rotates to construct the tangle https://www.w3.org/1998/Math/MathML"> 3 7 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781482243048/27066273-287e-4f87-9432-f815edbc9855/content/eq326.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> . Next, how do you untangle it with more twists and rotates? Check your answer using two strings or shoelaces.
To the instructor: As alluded to in the previous chapter’s suggestions, cutcake and hackenbush from [BCG01, Ch. 2] provide good examples for numbers. Noam Elkies constructs chess endgame positions in [Elk96] that include https://www.w3.org/1998/Math/MathML"> ↑ , 1 2 , 1 4 , ± 1 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781482243048/27066273-287e-4f87-9432-f815edbc9855/content/eq327.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> , and +1. If the class has chess players, we recommend covering his material as a case study toward the end of this chapter.
From here on, students should be encouraged to use CGSuite to help attack problems (some of which are quite challenging to do entirely by hand) and to translate their observations into clear proofs.
Y’see it’s sort of a game with me. Its whole object is to prove that two plus two equals four. That seems to make sense, but you’d be surprised at the number of people who try to stretch it to five.
Dalton Trumbo in The Remarkable Andrew