## #MathArtChallenge 100: Balloon Polyhedra

In the first, allow me to thank each and every one of you who has participated in the #MathArtChallenge in the last few months. This is my “last” post. Meaning, I don’t promise to make more Math Art Challenges, but there’s always the chance that something will come up…

All of the #MathArtChallenge-s will continue to be up on this blog, and I really hope that you’ll make use of them in your classes or in your fun time or however brings you joy.

Today, you get Balloon Polyhedra. There’s actually several papers written about this, so go check them out.

I intended to make all the Platonic solids and all the Archimedean solids, but frankly, after what I did do today, my hands are sore from tying and the balloons didn’t fit in my house any more.

I also intended to take some videos of these, but there are few unbreakable rules in the universe and one of them is: when the neighbor kid sees your balloons, you have to give them to her. (She wore them all in outfits and was just gleeful. It was pretty great to see.)

Depending on how you use this activity, you may engage with different standards. Here are a few suggestions for how you might integrate the 8 mathematical practices. Feel free to add your own suggestions in the comments!

1.) Make sense of problems and persevere in solving them. How many balloons are needed for each structure? How will you connect them such that you don’t have the balloons double-backing on themselves?

6.) Attend to precision. In creating the models, you’ll have to pay close attention to how each balloon is segmented and which segments connect in which ways to other balloons.

7.) Look for and make use of structure. These often require some serious planning, but their creation leans on your understanding of and the understanding you build of the structure of the edges and vertices of the various solids.

## #MathArtChallenge 96: Not so much rational tangles as… a 2 link challenge.

Soooooooo, I planned on having this be rational tangles, but that didn’t work out. If you want to skip to the challenge, scroll down to THE CHALLENGE (it’s obvious, I promise).

Still, I wanted to do rational tangles because I desperately wanted to carve out some time for myself to play more with knot theory and because I think that rational tangles are spectacular (even if I don’t fully understand them yet) But, as they say, I bit off more than I could chew tonight. (Although seriously, if you want to play with me and rational tangles, let me know, I’d love to give poor Professor Noodle Arms a break from my incessant questioning.)

I also wanted to have this be on rational tangles because I’d get the chance to introduce you to Candice Price and Mariel Vázquez, who are both working on biological topology with DNA and knot theory (my heart!)

I was introduced to Dr. Price because she features in an episode of My Favorite Theorem wherein she explains rational tangles and the Conway’s rational tangle theorem. She also happens to be a co-founder of MathematicallyGiftedAndBlack.com which is a wonderful resource for helping to diversify your knowledge of mathematicians. Dr. Price also mentions Dr. Vázquez in the My Favorite Theorem episode, which led me to this gem, and honestly, be still my heart, if I’m ever able to meet either of these women, I’ll act like any normal person meeting Beyoncé.

Unfortunately, I don’t think I can make rational tangles work as a math art challenge tonight (maybe a surprise bonus challenge once I get my feet under me) BUT I can get you closer. At least knot theory is involved.

# THE CHALLENGE

So here’s the challenge: Using two links – these are CLOSED loops (Use an extension cord for each – you can plug them into themselves, and they’ll work beautifully) – see how many totally unique arrangements you can make. I’ll start you off. Below are a bunch of 2 link knots…(someone’s going to yell at me for that language, just do so nicely in the chat and I’ll adjust)… each unique from the others. Can you come up with others?

Things to ponder:

• Can you prove there are an infinite number of 2-link knots?
• Aren’t Dr Price and Dr Vázquez amazing?
• If you removed one of the links, (cut and pulled away) how many of the arrangements would leave an unknotted link behind? (This, admittedly, is directly related to my Bridges submission this year. Totally worth your time to peruse the whole gallery.)

Depending on how you use this activity, you may engage with different standards. Here are a few suggestions for how you might integrate the 8 mathematical practices. Feel free to add your own suggestions in the comments!

1.) Make sense of problems and persevere in solving them. How can we know that we’ve gotten “all” of them?

7.) Look for and make use of structure. What types of links can we find and classify? Can any of those types be classified in more than one way?

## #MathArtChallenge 93: Archimedean Solids from Vertex descriptions

The Challenge: Using a vertex description, build yourself one, two… up to all 13 of the Archimedean solids.

Materials needed: Card stock and tape (painter’s tape is great, or masking. Other stuff will work, but I’ve had more success with the paper-y tapes.) OR Magnatiles, but those can get pretty pricey.
Math Concepts: structure, polyhedra, angles, 3D structure

Platonic solids are 3D shapes with congruent regular faces. There are 5.

Archimedean solids are 3D shapes with regular (all congruent side lengths and angle measures) faces and identical vertices. There are 13.

So the idea is to get yourself a bunch of equilateral triangles, quadrilaterals, pentagons hexagons (and octagons or decagons if you’re feeling ambitious), and start building!

Below is a timelapse of me building the {3,6,6}, {3,4,4,4} and the {3,5,3,5}.

I think it’s fascinating how challenging it is to predict the finished sizes and number of faces. There are definitely ways to do it, though, so for students who want a challenge, see if they can figure out how many of each face they’ll need without looking it up or building it first.

Here are the vertex descriptions:

Platonic Solids

• {3,3,3}
• {4,4,4}
• {5,5,5}
• {3,3,3,3}
• {3,3,3,3,3}

Archimedean Solids

• {3,6,6}
• {3,4,3,4}
• {3,8,8}
• {4,6,6}
• {3,4,4,4}
• {4,6,8}
• {3,3,3,3,4}
• {3,5,3,5}
• {3,10,10}
• {5,6,6}
• {3,4,5,4}
• {4,6,10}
• {3,3,3,3,5}

To make it a bit easier on you, I have a sheet of 2 inch side length shape PDFs for you:

2in equilateral triangles

2in squares

2in pentagons

2in hexagons

One caveat to this vertex notation: the pseudo-rhombicuboctahedron:

This is actually an activity I have done with students before. Megan Schmidt and I got to run a summer camp for a week last summer and it was just glorious. It was so much fun watching students try to puzzle their way through making these shapes and then drawing connections between them. Some questions to consider:

• How many faces or edges will each shape have?
• How are the shapes related to each other? What connections do you see?
• What other materials might you use?
• Can you identify the shapes that are chiral? (They have a right or a left turning?)
• Why are there only 13 Archimedean shapes? Why only 5 Platonic shapes? Can you find more? Why or why not?

Depending on how you use this activity, you may engage with different standards. Here are a few suggestions for how you might integrate the 8 mathematical practices. Feel free to add your own suggestions in the comments!

2.) Reason abstractly and quantitatively. Just based on the vertex description, can you figure out how many of each polygon is needed to complete the polyhedra?

7.) Look for and make use of structure. What combinations of polygons are possible? Which are not?

## #MathArtChallenge Day 18: Celtic Knots

The #MathArtChallenge is just a fun, simple way to engage our brains during this time of unease. All tasks are low tech: paper, pencil, maybe string. Nothing fancy. I would LOVE to see what you come up with. Post on social media with the hashtag #MathArtChallenge!

THE CHALLENGE: Create a Celtic knot, and do some wondering about why you got the number of links that you did. Can you predict how many links you’ll get? (I wrote another blog post on this a while ago, but only go there if you need more examples, because I reveal a lot of the good stuff in it: Knots, Links, & Learning)

MATERIALS NEEDED: Paper, pencil. If you have grid paper, that might help, and here is some special grid paper you can use courtesy of Justin Aion.

Here is a visual set of instructions, and below that is a video tutorial.

## #MathArtChallenge Day 17: PERMUTOHEDRONS

The #MathArtChallenge is just a fun, simple way to engage our brains during this time of unease. All tasks are low tech: paper, pencil, maybe string. Nothing fancy. I would LOVE to see what you come up with. Post on social media with the hashtag #MathArtChallenge!

THE CHALLENGE: Create your very own order 4 Permutohedron (see video below).

MATERIALS NEEDED: Paper, pencil, patience, and if you want to go 3D, you sure can.

To see an answer, scroll down. (To not spoil the fun, you have to scroll down pretty far.)

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Here you go. This is a flattened version of the answer, but it’s really satisfying to find. If you were to build this in 3D you would get a truncated octahedron. Every permutation should be connected to 3 other permutations.

## #MathArtChallenge Day 13: Overlapping Circles

The #MathArtChallenge is just a fun, simple way to engage our brains during this time of unease. All tasks are low tech: paper, pencil, maybe string. Nothing fancy. I would LOVE to see what you come up with. Post on social media with the hashtag #MathArtChallenge!

THE CHALLENGE: Can you figure out all of the ways that 3 circles overlap?

MATERIALS NEEDED: Curiosity and patience. Whatever medium you like! Please watch ONLY the first 1 minute and 20 seconds of this video. If you watch more, the answer will be given away! Don’t ruin it for yourself! It’ll be so satisfying if you do it yourself!