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Topology Explains Why Automobile Sunshades Fold Oddly (2009) (arxiv.org)
57 points by ColinWright 20 days ago | hide | past | favorite | 30 comments



This is the same way you coil a bandsaw blade, which has the dubious advantage of having two readily distinguishable edges. When coiled correctly, all of the teeth face the same way in the coil.

You can actually repeat the coiling process on a bandsaw blade one of two ways (theoretically, there are probably more, practically, if you have blade long enough to need to, please invite me over so I can ogle your saw):

1. Repeat it with the whole 3 coil loop, which gives you a 9 coil loop. Uncoiling without injuring oneself is a bit tricky, as bandsaw blades are quite springy, and it's hard to uncoil the blade in two discrete steps one it's in a nine coil loop.

2. Gather the three coil section to one side (or equivalently, extend one of the three coils) giving the gathered section about 1/3 the total size of the exteneded section, then repeat the process with the exteneded loop in one hand and the gathered coils in the other yielding a five coil loop. It's still tricky to get the whole thing uncoiled in 2 discrete steps, but you've stored a bit less energy in the coils of the blade and it's somewhat less hazardous.

I wouldn't suggest trying either of these with a 93 1/2" blade (the standard size for 14" saws), but method 2 works well with a 162" blade.

I took a knot theory class in college and there are actually two characteristics, twist and writhe, that are quantifiable and can be exchanged by coiling and uncoiling (massive handwaving, bad terminology, etc. That class was a long while ago). If you've ever taken a garden hose in a flat coil (with all the coils in the same direction) and pulled it out without unwinding it, you've enjoyed an illustration of this.


I think this is hat parent is talking about as my imagination was insufficient in imagining it.

https://www.youtube.com/watch?v=KRarn9st2bg


Twist and ‘writhe’ also explain the counterintuitive ‘roadie wrap’ method for coiling long wires - where alternate loops are reversed, introducing twists, but resulting in a wire that can be uncoiled smoothly by pulling one end - or holding one end and throwing the coil.


A spool works too. If you have a spool you don't get the twists. The twists are incurred by coiling wire by adding to the sides of the coil without a rotation of the coil itself. If you use a rotating spool you won't have such an issue.


It also puts less strain on the wire, making it last longer. This is really important in say a TV studio, where the cable that connects a camera can cost thousands of dollars.


yes, the naive way of bundling a wire into a coil strains the wire just like if you were to take both ends and twist them in opposite directions.


I am fascinated by a line of study that leads to a knot theory class. What school/major was that part of, at what college?


Honestly, I majored in compsci. Between the required first and second semester calculus and two math electives I took, I was one class from a math minor. Knot theory looked interesting, had no prerequisites, and seemed like I could pass it.

It has had zero applicability to anything I’ve done since, except for coiling bandsaw blades and climbing rope.

I hope that wasn’t a terribly disappointing answer to your question.


It wasn't disappointing. So knot theory was part of the school of mathematics? That's fascinating. I love academia - those little foibles and weird little intricacies - they've all been poured over and thought about and re-evaluated. It's just so fun, from a higher-order thinking perspective.


Yeah, the class was through the math department. I share your amazement at just how fractally complicated the world is that you end up with such specialized fields!


There's pop-up tents with a even more interesting topology. It's always fun to watch (semi-drunk) people try to fold up theirs: https://www.youtube.com/watch?v=0gp5C1dYtuA


What is interesting is to see the evolution of the system used to fold pop-up tents.

The most successful tent in France is, by far the Quechua 2 second, by Decathlon. Go to any festival in a country where Decathlon is present and they will make up the majority of the camping ground (make sure to remember which tent is yours!).

The tricky thing, of course is to fold them back. Over the years, instructions have changed, the design also changed slightly to make it easier. In the latest version, you just have to pull a cord and you are more than halfway there.

All these improvements are not just for helping the customer. Decathlon has a very liberal return policy, and they had to deal with a huge number of broken tents because people didn't fold them correctly.


Decathlon is one of my favorite stores. I used to go there quite often when I lived in Europe/North Africa, and now that they came to Canada its quite handy.


After a bit of practice it’s relatively easy to fold those tents back up, but explaining someone else to do it is a different story.

I know how to fold them on feel, but I’m clueless about what I’m actually doing.


My wife bought one of those, or something like it. Even while reading the instructions the whole time, I had about the same amount of trouble that that guy did. I guess it's surprisingly hard to write out text instructions for them.

They probably should have included a lot of graphics a la IKEA instead. We did eventually get it, but it was a team effort and I'm pretty sure we've never opened it again.

Edit: Oh, and I'm really good at the car-shade version of them, so I didn't think it was going to be hard at all going into it.


IKEA instructions are great indeed. Sadly that type isn't appropriate in many cases.


We have a beach sun shade with the same sort of folding mechanism. Anyone can unfold it - you basically just unzip it and it explodes into the final product, but I'm always the assigned "Sun shade folder".

Maybe I don't understand it well enough myself, because I've only successfully taught one person. The key insight is definitely that you're going for three loops, not two or four.


Considering only the loops, I think it's actually the same topology, just with two loops instead of one. Assuming I understood the end of the video correctly :-)

I think it's also required that the two loops be the same circumference, unless something really tricky is happening with the fabric to allow them to be coiled concentrically but with different centers.


Totally off-topic question, but is this website name “arXiv” supposed to be pronounced like “archive”?


Yes. From the Greek letter Chi.

https://en.wikipedia.org/wiki/Chi_(letter)


Neat. Thank you :)


I don't have a definitive answer, but I've always assumed so.


Very cool and digestible paper! Followup question: Is it possible to fold the shade into n loops for every odd value of n?


Yes, see @mauvehaus's top-level comment [1] for a proof sketch. The idea is to either treat the folded shade as an unfolded shade and repeat the process, or to make one loop sufficiently large that it can be folded again independent of the other loops. Either way, imagine taping the original loops together, since they do not participate in future folds.

[1] https://news.ycombinator.com/item?id=24407572


Ah yes, with the second method you can always add 2 loops. The first method only gives you 3^n though.


Why do people publish in this awkward obsolete Letter-size paper format that is hard-to-impossible to read on modern computer screens?

Who is going to print out "College Mathematics Journal"?


Printing papers out is a great way to let your optic nerve do the work so you can save your short term memory for comprehension rather than simply recalling what the symbols are.

I absolutely love printing out shortish papers single sided and sticking every single page in order on a poster board or wall. Our eyes are made for flicking from one thing to the next, it works really well. And also highlighting and other note taking is much more direct and intuitive in this format. I also discovered I prefer a columnar format.

if I had a monitor that could display 18 pages at once at 12-14pt and was as easy to annotate as using a pen or marker then maybe I wouldn't do this, but I don't.


Probably mathematicians just like to use LaTeX (or TeX? idk the difference). You could download the source in the other formats page[0] and convert it to whatever format you want using pandoc[1].

[0]: https://arxiv.org/format/1205.4797

[1]: https://pandoc.org/


This is a PDF. You can zoom in it. I'm not wearing my glasses right now, so I just zoomed until the letters were several centimetres wide.

Unless you meant that there's not enough words per lines?


People who read it and are aware that screens aren't great for reading comprehension.




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