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Making Mathematics with Needleworks

Frozen Tundra shared a few book titles with me a while back.  I decided to check out the book, Making Mathematics with Needlework by Sarah-Marie Belcastro and Carolyn Yackel. 

Making Mathematics with Needlework
This is what the Library Journal had to say about the book:
First, there's an overview intended for both crafters and mathematicians, so it should be understandable to mathematicians who don't know anything about crafting and also for crafters who don't know anything about mathematics. Then, there's a section of detailed mathematics which is intended for mathematicians. All of the authors have made an effort to include basic information so that mathematical enthusiasts who are not professional mathematicians can follow the bulk of the material. The third section of each chapter contains teaching ideas, and these range from elementary-school level to graduate level. Finally, every chapter has a project, with instructions written for and tested by crafters.

 After reading it I know I'm not quite the target, but if you are looking to improve your mathematics teaching with your knitting skills, this thought provoker is for you.  The authors admit the book is ". . .structured to be of interest to mathematicians, mathematics educators and crafters. ..", and I'd argue it is in that order. 

The introductions to each of the 10 major sections are mathematical in nature.  I did not know the Mobius band is really a circular scarf with a twist or that I had been knitting Torus when all I thought I was doing was making a donut.  (Thanks Lex for talking me through this stuff.) 

I spent the largest amount of time looking over Chapter 2 about picking up stitches and using the Doiphantine euqations.(FYI - a Doiphantine equation is an indeterminate polynomial equation that allows the variable to be integers only.)   The precise approach to determining how many stitches to pick up when placing a sleeve in an armhole was nearly overwhelming for me.   The problem was placing 82 stitches in 96 rows.  Now all I'd do is divide 82 by 96 and determine that I pick up 5 stitches over every 6 rows and fudge at the start and finish to make it fit.  But the authors pose: "The mathematically minded knitter, however, asks, 'How do we define, precisely, what it means for picked-up stitches to be evenly spaced?'"   I thought I just did that, didn't I?  Well, no I didn't because I used that fudge factor.  The Doiphantin equation would help to give one more exact and whole numbers.

This is how the mathematically keen solve this problem:

1.  Calculate R = (r-s) + 1  where R is how many stitches to pick up, r is the number of rows and s is the number of stitches to pick up.

2.  Using division with remainder, compute s/R to obtain the quotient, l, and the remainder, b.

3.  Define a = R - b.

This tells us that we should pick up l stitches and then skip a row end a times, and pick up l + 1 stitches and then skip a row end b times.  After a full page of mathematics-speak about determining the exact way to solve this problem it is concluded:  "In practice, a knitter would probably place it just off-center."  From my POV this means even with all this preciseness, the mathematically-inclined fudge, perhaps less than me, but fudge all the same.

Just in case you don't get the idea here's a page with equations talking about the Mobius:

Part of Page 20
Truly impressive but much more accurate than I'll ever get in my knitting.  If you want to be exacting and fully understand the mathematics about shaping your fiber art (not all projects refer to knitting) you will enjoy this book.  Thanks Frozen Tundra for the tip.  Reviewing Making Mathematics with Needlework was a reminder of how far away I've drifted from my 11th grade math class.

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