Week 8: Knitting and Knotting
I enjoyed the article "Adventures in Mathematical Knitting" by Sarah-Marie Belcastro. She began her adventure during graduate level mathematics courses, knitting away simply to keep herself awake while she listened to her professor. Eventually she began translating ideas she was learning about into knitted forms, starting with the Klein bottle.
I have to admit that I didn't really get the significance of the Klein bottle at first. Picture an esophagus and stomach where the top of the esophagus has been pushed through the side of the stomach and fused to produce an opening on the bottom. I had to watch a couple of Numberphile videos to know why it is such a fascinating object to mathematicians. Even though it looks like it has an outside and an inside, it does not. It also has no edges. You could (in theory) travel to all locations on its surface without crossing any edges. That is kinda cool. The mobius loop is related to it. It looks like the mobius loop has two edges and two sides to its surface, like any sort of loop. But to travel from one position on the loop to the other side and then back again, you will only ever cross one edge. Anyway, here is a link to the most useful of the Numberphile videos I watched: Numberphile: Klein Bottle.
Ok--so here is Sarah-Marie in her mathematical topology course working away on a craft perfectly suited to exploring surfaces. She began by knitting a Klein bottle, a design she continues to tinker away with (twenty years and counting so far!), adding insights from exploring mobius loops and improving the bottle's aesthetic appeal every iteration. The goal is always this: where possible, make the mathematical properties that are special about the objects an intrinsic result of the design, rather than fudged at the end to make them appear special. She also played with mobius loops (of course: who wouldn't want a nice mobius loop scarf?) and objects that divide the universe into an inside and and outside (such as the doughnut-shaped torus and the sphere).
I am not a knitter or a crafter of any sort (unfortunately), but I can certainly see the appeal of trying to figure out whether it is possible to represent abstract geometric forms in ways that are true to the mathematics. What other craft allows you to create 3-dimensional shapes by producing 2-dimensional object skins? Knitting is uniquely suited for this particular exploration.
Onto the knotting portion of this blog entry!
To start, it's lacing, not knotting. I couldn't help but bend my word choice to produce alliteration in my title. I have a nice pair of Fluevog boots that could use some freshening-up with some zippy blue laces.
Before lacing, I have to acknowledge the math. Apparently there are 1,814,400 'tight' lacing combinations (where each eyelet is connected to two eyelets on the other side of the shoe) for a shoe with 7 eyelets. When you increase the eyelets, you increase the number of possible tight lacing combinations. So, if you had a shoe with two eyelets, you would have 0 tight lacing combinations. If you had a shoe with 4 eyelets, you would have only 1 combination (I'm pretty sure). If you had a shoe with 6 eyelets, you would have...hmmm. I need to grab a note pad. Hang on...(a couple of minutes later)...if I've done it correctly, then you would have 27 combinations. If you have 8 eyelets, then you would have...(a few minutes later)...I think I was wrong with the 6 eyelet combinations. I forgot about the reflections. So maybe 54? Of course, there will be ones with perfect reflective symmetry between the eyelets, so I'll have to subtract those. Uh oh. Maybe this is more complicated than I thought!
Ok. This is a really good little pattern problem. I could definitely give this one to middle schoolers and beyond. It has that nice low floor of simply drawing and counting. If you are asking yourself, "What am I noticing here?" there are all kinds of tantalizing hints that a more general pattern is within reach.
Given the millions of possible combinations for a shoe with 7 eyelets, I was pretty sure I could come up with something beyond the standard lacing pattern that would work. Here are my attempts to creatively lace my precious Fluevogs:
Note the reflective symmetry. Again, I can imagine middle schoolers having fun creatively lacing their shoes, showing translations, reflections...perhaps even rotations? That would be interesting. By the way, I always felt constrained in my designs by which eyelets were open. I didn't want to jump up too far with both ends of the lace because that meant I would have to go back down to fill empty eyelets. Maybe I shouldn't have worried about that?
Again...another fun and fascinating dive into the hybrid world of mathematics and art!
Hello Jen,
ReplyDeleteI love trying different lacing combinations too because of their interesting looks and different levels of comfy on my foot (I like my ‘walking’ shoes loose but ‘running’ shoes tighter). I thought I would share one of my favorite videos here with you if you ever come across or need some creative ways to tie your shoes: https://www.youtube.com/watch?v=GhzGrc88Iy0&t=385s
I love reading your thoughts while doing the math in your post, it made them personal and added such a friendly attribute to your sharing :) I always knew there are more possible lacing combinations with a greater number of eyelets, but never quite sure how to calculate that. Do you mind sharing how you calculate the number of lacing combinations with the number of eyelets?
Well...my combination work started off very simply. I drew tight lacing options for one eyelet (noticing that there were three possibilities). For each of those possibilities, there were three more possibilities once you consider options another eyelet (so far, that is 3 x 3). Then for each of those possibilities, there are three more once you consider a third eyelet (now we have 3 x 3 x 3 or 3 to the third power). Then I doubled that (because there are three other eyelets on the opposite side with the same number of possibilities that differ only in that they are a reflection of the first three eyelets). Then I wondered about patterns that are already bilaterally symmetrical. I thought I'd need to subtract 1 for every bilaterally symmetrical option. I haven't figured out how many there are, but I bet a group of middle schoolers could!
DeleteHey Jen, I saw the discussion by you and Zaman and thought I'll come back to comment here. If I understand both of your reasonings correctly, Zaman's approach left out the eyelets already used on the other side, and yours included them? With Zaman's approach for your 6 eyelets example, the calculation would start with 3 x 2 (taking out the option going back to the same eyelet on the other side) instead of 3 x 3. Hopefully that helps!
DeleteHi Jen,
ReplyDeleteI have not seen a physical Klein bottle. Like you, I did not appreciate its significance from the information I had about the weird looking thing. On the other hand, I always found Möbius strip fascinating and have drawn lines on it, and cut it into half, etc. You provided the link to a Numberphile video, so I decided to watch it. As I started watching, my first reaction was Oh No... Not that guy. I had seen his video about the Seven Bridges of Konigsberg. I couldn’t stand him. Unrelated to the topic: They say speaker enthusiasm can be contagious. I found that there has to be a limit to showing enthusiasm, and excessive enthusiasm could be distracting. Anyway, I watched the video to the end. Then I watched some more videos. I learned a little about the Klein bottle, and also about the presenter, Clifford Stoll. He is an interesting guy. Actually, now that I feel like I know him better, I don’t mind his style. I really enjoyed his video Cutting a Klein Bottle in Half- Numberphile. (https://www.youtube.com/watch?v=I3ZlhxaT_Ko). Cutting the glass Klein bottle, and also unzipping a Klein bottle hat, he shows that a Klein bottle is two Möbius loops joined together. I intend to make a Klein bottle from two Möbius loops and hopefully will have better appreciation for this remarkable object.
Sarah-Mari Belcastro knitting a Klein bottle is amazing. I used to watch my grandma knit mittens (Like Bernie Sanders mittens). Grandma did not get a formal education and had no concepts of geometry, as we know, but came up with beautiful designs. I wish I had learned something about knitting from her. It may be too late for me to learn, but hopefully the curriculum in BC, and globally, includes elective courses on knitting, weaving baskets, etc.
By the way, what a fantastic color combination you have for your boots, laces, and socks. My shoes are either black or brown with laces of the same color. Your non-conventional but symmetrical lacing style is great.
Thanks, Zaman. I love those Numberphile videos. Funny about his style: I don't mind it, but yes, he is a little over-enthusiastic :) It is never too late to learn to knit. My knitting skills are very basic, but one day I'd love my grandchildren to appreciate my mittens, too. I'd actually like to make myself a pair of socks to rival my very colourful ones.
DeleteThanks Jen and the group for an interesting discussion! Jen, I really enjoy your writing and appreciate how you take us through your mathematical exploration of knitting and knotting (lacing). Cool lacing design with 7 eyelets! And I agree that creative shoe-lacing could be a fun activity for middle schoolers!
ReplyDelete