Posts tagged moebius strip

Last-Minute Christmas Decorations and Tangled Lights

0

Days are flying by and everyone is busy preparing for the holidays. While you might not know what gift(s) you will be getting this year or whether the kids will catch the sniffles right before the party, one thing you can count on is tangled lights. Seriously, no matter how carefully we pack them, when we open the boxes next year, here they are, all knotted up. Turns out, it’s all about math and physics as opposed to our carelessness. Check out this NPR article for details. Here’s the best part – next time you have knotted up strings of lights, give them to your little one to undo and proudly tell your friends he was busy solving a knot theory problem.

In addition to tangled lights, another thing you can count on is a broken ornament or two or three. If your tree looks a bit bare, don’t rush to the store. Instead, consider some really cool last-minute DIY ornaments. Bonus is they are all about math. The easiest ones are paper chain garlands. This year, add a twist, (ok, half a twist) and turn them into Moebius chain garlands. Or bend pipe cleaners into shapes and grow Borax crystals on them. What shapes you choose is totally up to you. Inspired by Vi Hart’s Borromean Onion Rings video, we made this Borromean Rings ornament.

Or how about turning cardboard boxes you were about to throw away into star ornaments just like Malke and her daughter did on the Map is Not the Territory blog.

Finally, we’d like to share a holiday printable we found thanks to the link from one of the readers, Elena T. She printed it for her daughter to color days ago, but we just got around to it. This Christmas tree might seem like just a giant cute coloring page, but don’t let it fool you. It’s got lots of things going on, including some great math. Can you find examples of gradients, tessellations, pattern, rotational symmetry, radial symmetry, iconic quantities and a lovely Sierpinski triangle?

Share your holiday math with us on the blog and on our Facebook page.

Math Stories – Moebius Ants

1

This is a story inspired by Vi Hart’s “Wind and Mr Ug” video. I so wanted to tell it to my son myself, but my drawing skills fall way short. So instead we talked about ants.

Once upon a time two ants lived on a strip of paper. A strip of paper has… how many sides? how many edges? [I drew two ants on a strip of paper] Each ant lived on his own side of the strip. They never saw each other, but since ants have an excellent sense of smell, they smelled each other. And they really, really wanted to have a playdate or maybe a tea party one of these days. They tried visiting one another, but each time they reached an edge of their little flat worlds and would get scared.

Then one day something happened. There was much shaking and twisting and the ants got scared and closed their eyes and covered their heads and tucked their antennas. When the twisting and shaking stopped, they opened their eyes and saw something strange. Their flat world was no longer flat. Instead, it became cylindrical [At this point I glued the edges of the strip together to create a cylinder]

Hooray! said the ants. Maybe now we can visit each other! One of the ants, who was a bit braver and more adventurous than the other, immediately set out on a round-the-world trip in hopes of meeting his friend. He crawled and crawled along, leaving tiny prints behind him [I'm drawing ant's path with a marker]. Will he ever see his friend?

Soon the ant came to a set of tiny footprints. At first he got excited. Was that the marks left by his never-before-seen friend? Is he getting closer? But soon the ant realized that those were his own prints and he’s been crawling round and round his little world.

But what about the other ant? He too sat out on a journey, crawling along. Will he have better luck? [I'm drawing second ant's path with a different colored marker]. Nope, he too finds no one, just his own footprints. Their world had… how many sides? how many edges?

Poor tired ants needed a rest. But just as they were going to take a nap, their cylindrical world shook and twisted again. Again they got scared and closed their eyes and covered their heads and tucked in their antennas. [Here I cut the cylinder to turn it back into a strip; then I twist the strip and glue to form a Moebius strip]. When the twisting and the shaking stopped, they opened their eyes and looked at their strange new world. Maybe now that it changed they will be able to meet each other.

The first ant, the braver one, set out on his round-the-world trip once again. He walked up the hill and down the hill and across the valley [I'm tracing the ant's path with a marker] until… he saw the other ant! Hooray, the two cried and hugged each other. And then they walked back to the first ant’s home [a child is tracing the ants' path with a different color marker]. Their world was no longer flat. Was it a cylinder? Nope, it became something called a Moebius strip. How many edges does it have? How many sides?

And that was the story. But then we experimented some more. We made another cylinder and another Moebius strip, each with its own pair of ants. This time my son traced ants’ paths all by himself. Then I brought out the scissors and both worlds underwent another cataclysm, this time it was a continental drift (thanks, Ice Age 3, for the idea) as I cut the cylinder and the strip in half. Want to know what happened to the ants? Try it for yourself. It’s really very fun!

Knot Theory for Young Kids

0

A couple of weeks ago I discovered a wonderful math blog called Math Munch. Now, with a name like this, you know this is going to be good! And it is even for someone who is not a professional mathematician and who was pretty scared of math as a child (I’m pointing my finger at myself now, tsk-tsk-tsk). Basically, Math Munch is a weekly digest of the beautiful, surprising, strange, engaging, and fun math out there on the Internet.

As I was browsing the site, I came across a post about knots. I love knots because they are a) beautiful and b) because I have such a hard time following the instructions and learning how to tie them (something so simple, even a hagfish can do it). Well, I was in for a surprise – turns out, mathematically speaking, the knots we (hagfish and I) usually tie are not really knots at all since the knots we tie are not closed loops.

Intrigued, I followed up with Anna Weltman, who teaches math, folds paper, ties knots and co-authors Math Munch. I wanted to know what kind of activities can I do together with my 5-year old to further explore mathematical knots. Anna’s suggestions were so awesome, that I felt they needed to be shared with you. Note: I added names to each game just because bullet lists are boring. Feel free to rename the activities.

So here is Anna’s response to my question:

A big mathematical question that knot theory is really good for exploring is, “Can I turn this into that without breaking something about it that I think is important?” Another question that knot theory is good for is, “How can I make an object that will do this particular thing I want it to?” I’m not an expert on 5-year-olds, but here are some ideas I had for approaching those two questions through knots:

The Game of Moebius Strip

A fun way to make knots and links is to make moebius strips with different numbers of twists and cut them in half. You can make two different knots – a trefoil knot and a knot with 5 crossings – but cutting a 3-twist strip and a 5-twist strip in half, respectively. You can play with cutting differently twisted strips in half and describing the similarities and differences between the results.

The Game of Twisted Cords

Get some pretty slippery string – maybe lanyard/gimp, or headphones (though with only 1 phone would be best). Tangle it up – either haphazardly or methodically, your choice. Then, tape the ends together. Can you untangle it? Untangle it as much as you can and describe what you have in the end. You can do this a bunch of times and keep track of the untangled results. You can then try to make a tangle that you can untangle – ask, what should we do or not do with the ends of the string so that we are sure to be able to untangle it in the end?

The Game of Steer Roping

Get some string and a bunch of differently shaped objects – make sure some have at least 1 hole in them. Challenge your child to tie the string around the objects so that you can’t just slip it off. See if the child can describe what kinds of objects are tie-up-able and what kinds aren’t, and what kinds of tying are best for mastering the challenge. This is less knot theory than it is study of surfaces, but it involves knots!

The Game of Un-Twister

This game is really fun, but you need several people to play it. Stand in a circle. Everybody take hands, but not with the person standing next to you and not both hands with the same person. Then twist and climb and duck under each other’s arms – but don’t let go of each other’s hands! – until you’re standing in a circle again, completely untangled! If you have enough people, it sometimes happens that you end up in two circles. If you want to really analyze the game, you can make a sketch of how everybody took hands and map out the untangling process.

The Game of “I Am Knot Myself Today”

Another fun thing to do along those lines is to try to make a knot out of your arms, torso, and a stick that you can’t untangle without dropping the stick.

Oh, and Anna pointed me to this great Vi Hart video which seems like a perfect conversation starter.

Awesome applesauce! We’ve already tried the first game Anna suggested and it was a huge hit. I can’t wait to try the rest. Thank you so much, Anna!

If you haven’t done this yet, do check out Math Munch blog. Every single post is chock-full of links to beautiful and engaging math sites many of which could be explored  with young children.

Go to Top