## Introduction

This is the page for my 2018 highschool math mentorship program.

## Week 1 : Project Summary

The project will produce:

• An article for BISFA:
• Apply Storer theory to James Murphy’s figures
• Catalog figures using Storer theory
• Some nice variations of common figures
• Explaining Storer theory
• Teaching some figures

The theories we will need to learn:

• Knot theory
• Graph theory

There are several skills or talents that students will acquire through the program:

• Learn some string figures / tricks
• Teach someone else string figures
• Learn a string story
• Analyze figures
• Reconstruct figures

Participants will be expected to:

• Cancel meetings early
• Come to meetings on time
• Write something every week (questions, findings, ideas, etc.)
• Be present at meetings on a regular basis
• Present their findings regularly

Timeline: Presentation on Wednesday May 2nd 16:30-18:30 (88 days away).

Homework:

• Write questions arising from readings
• Practice $\underline{O}.A: \overrightarrow{1}(\underline{2f}) : N1 : \Box 5 |$(Video of this figure is in this post.)

## Week 2 : String Positions

This week we reviewed the readings Systemology 1-29 and learned about string positions and linear sequences.

Homework:

• Read Chpt. 1 Brokhos p30-50
• For each of the four “very different” positions: write out the calculus, draw their schema, and write their linear sequences.
• Practice:
• Carrying Wood (Navajo)

$\underline{O}.A: \overrightarrow{12}(\underline{5n}) \# : \Box 5 : N1 : N2 : >\!\!1 \textrm{ and catch the base of } 1\Delta \textrm{ (extend palms away)}$ There is a video in this post.

• Siberian House (Big Diomede Island)

$\underline{O}.A:\ >\!\! \overrightarrow{1\infty} \rightarrow 2345: \overrightarrow{1}(2345n) : \overrightarrow{1}(\underline{2345f}) : \Box 2345\infty : \textrm{(extend)} : \Box 2 : \textrm{(extend)}$. There is a video in this post.

## Week 3 : String Positions Continued

We explored string positions and linear sequences by looking over the homework on the four “dissimilar string positions” together. We also talked about the issue of orientation in complex crossings.

Solutions:

The four string positions assigned in last weeks homework can be found here.
These are not definitive, and final, because you can wiggle the string a little bit.
There might even be mistakes!

Homework:

• Teach us a new string figure. This includes telling us: its name, where it is from, how to make it (in your own words), and pointing out anything neat about it. You might want to write down the Calculus to construct it. Assignments:
• Construction analysis (schema and linear sequence for each move) of Outrigged Canoe: $\underline{O}.A: \overrightarrow{1}(\underline{2f}) : N1 : \Box 5 |$
• Practice:
• Bokola (Fiji): $\underline{O}.A: \overrightarrow{2 \infty} \rightarrow 3 : \overrightarrow{1 \infty} \rightarrow 2 : \overrightarrow{M} ( \underline{2n} )\ \# : \stackrel{\Longleftarrow}{R23} (\underline{ML}) \# : \stackrel{\Longrightarrow}{L23} (\underline{MR} )\# : \Box M : \|\ \overleftarrow{5 \infty} \rightarrow 23 : \textrm{N} (\ell 2) : \textrm{N} (\ell 3)$
• (Find a good comfortable extension)

## Week 4 : Anthropologist’s Figures

We explored a couple string figures from the anthropological literature.
There was no film, this time. Megan Shaw was a special guest at mentorship.

Hillarey chose to present Kopu from Andersen’s Maori string figures (p64).

Caroline chose to present A Mahara Raiatea from Handy’s String figures from the Marquesas and Society Islands (p58).

Solutions:

The crossing analysis of Outrigged Canoe assigned in last week’s homework can be found here.

Homework:

• Read Chpt. 2 Osage Diamonds p50-90.
• Write the linear sequence for your figure.
• Write the linear sequence with complex crossings for your Koura.
• Get a cork bulletin board and push-pins for pinning figures.
Pics or it didn’t happen. For example, here is my cork board.

Practice:

• The Heart Sequence Construction for Koura (Storer p40):

$\underline{O}.A : <<\! 5\infty :: \overrightarrow{1\infty}(2 \infty) : \underrightarrow{1 \infty}(5 \infty) : \overleftarrow{1\infty} \downarrow (2 \infty) : \underleftarrow{1 \infty} \rightarrow 1 :: \Box 2 | \textrm{(gently)}$

We met and discussed possible plans for the future of the project.
We decided that Hillarey and Caroline would translate French string literature to English and Joseph would translate Murphy’s figures from English to Storer Calculus. Parker will give a talk about knot theory, and relate it to the formal study of string figures.

Homework:

## Week 6 & 7 : Project Work Period

Hillarey presented her work on “The Flames of the Oil Lamp” (taqicit cugararte’) from Victor’s Angmagssalik Figures. Joseph started to present his work on Murphy’s article: Using String Figure to Teach Math Skills I: The Diamonds System. Parker’s talk on knot theory was delayed until next week.

Homework:

• Hillrey will write a good draft of her translations of Game 17 (taqicit cugararte’) from Victor’s Angmagssalik Figures and continue work on Games 1 (orercenna) and Game 6 (paδi).
• Joseph will write continue working on the Calculus for Murphy’s Diamonds System and begin on: Murphy, J.R. (1998) “Using String Figures to Teach Math Skills — Part 2: The Ten Men System.” Bulletin of the International String Figure Association 5:159-209.
• Parker will give a presentation on knot theory. It will introduce the technicalities of defining knots and their formal representations. The goal will be to state Reidemeister’s Theorem. It would be lovely to get to the dynamite paper: Lackenby, Marc (2015), “A polynomial upper bound on Reidemeister moves“, Annals of Mathematics, Second Series, 182 (2): 491–564

## Week 8: Knot Theory

Joseph presented his work on translating James Murphy’s Diamond System in to Storer Calculus. He taught us how to: store two diamonds, perform the Rastafarian addition, and extend a figure via the Murphy Power Lift. Hillarey presented her work on “The Paddles” (paδi) from Victor’s Angmagssalik Figures. Parker gave a talk introducing formal knot theory: we showed that the unknot and trefoil are distinct using Ralph Fox’s 3-colourability invariant. As an encore, Parker proved that there are no knotted 1-dimensional curves in 4-dimensional space. Blurry photos of the blackboard are available here.
Parker gave Hillarey and Joseph some small knots for reference. Joseph also got some two coloured strings for his work with Murphy’s systems.

Homework:

• Read: Chapter 2 Osage Diamonds.
• Provide a construction for Goeritz’s Unknot.
• Identify your knot in Livingston’s Knot Table (available here).
• Hillarey will write a good draft of her translations of Game 6 (paδi) from Victor’s Angmagssalik Figures and continue work on Games 1 (orercenna).
• Joseph will write continue working on the Calculus for: Murphy, J.R. (1998) “Using String Figures to Teach Math Skills — Part 2: The Ten Men System.” Bulletin of the International String Figure Association 5:159-209.
• Parker will prepare another talk on knot theory with an emphasis on unknot results. The goal will be to state the “Storer Calculus Completeness Conjecture” suggested by Jaimie Thind. He will edit Hillarey’s draft of Victor’s Game 17 and Joseph’s Calculus for the Diamonds System.

Research:

• Try to define “string figure” in the language of knot theory. Your definition should include schema and/or linear squences.
• Prove that Storer Calculus can general all possible string figures.
• Think about writing a short essay called “Mathematics and String Figures”.

## Week 9 & 10: Project Work Period

We met to discuss progress on our final project. We have finalized our goals for the final poster. Joseph will translate the first two articles of Murphy’s systems in to Storer Calculus. Hillarey will translate Victor’s Angmagssallik figures: Game 1, 4, 6, and 17 in to English. These are: a fox, a thoracic cage, paddles, and the flames of the oil lamp.

We discussed the possibility of continuing our work past the final poster presentation in order to prepare our materials for publication in BISFA. We are all in favour of this possibility.

Parker presented his work on the Goeritz Unknot. He was able to reconstruct the (2,1)-Goeritz Unknot, top left of the pin board, but unable to reconstruct the original unknot in the bottom left.

He presented the “reconstruction algorithm” from: Sapiens, H. (1999) “Fun with Newkirk 2: Deconstruction and Analysis of a Novel String Figure.” Bulletin of the International String Figure Association 6:253-260. Parker will continue to work on the original Goeritz Unknot.

We went to the computer lab and played around with LaTeX typesetting.

We want the text on our final poster to be LaTeX’d.

Our mock-up of the final poster is available in this post.

## Week 11: Final Poster Work Session

We got together and worked on our final poster presentation. A demonstration of the final poster presentation is available in this post. Megan Shaw helped out with the preparation.