#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.

Materials Needed: twisting balloons, pump, patience
Connected Math Concepts: graph theory, geometric structure

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 97: Tomoko Fusé’s Bird Tetrahedron Origami

The Challenge: Make an origami sculpture using Tomoko Fusé’s text Multidimensional Transoformations Unit Orgiami. (I used instructions on pgs 134, 138-139.)

Materials Needed: Paper (origami paper is handy, but any paper will work) and scissors/paper cutter.
Math Concepts: 3 dimensional building, angles, space filling, rotations, proportions

I love this particular type of origami because you get to determine the shape and scope of the final piece. You’ve basically made yourself some origami legos. If you have a lot of time, you’ll be able to fill space (well, there’ll be holes) with this unit & connectors, and could therefore, theoretically build a really dramatic sculpture, much like one could also do with legos. Just this is with paper!
For mine, I opted for the relatively simple tripe helix with a double rainbow. It took about 8 hours of folding and construction. We’re all a ways a way from this, but someday, post-pandemic, this would be a REALLY lovely project to do as a whole class or in groups.

To build this sculpture, you need square paper, scissors and 3 different folds:

  • Bird tetrahedron (technically a di-triangular pyramid)
    • 3 pieces of nxn square paper per di-triangular pyramid.
  • Horizontal connectors.
    • If your bird units are made with nxn paper, these are n/4 x n/2 paper. (1/8th the area of the nxn paper) 2 connectors needed to connect each 2 bird units. 6 needed to connect 3 units.
  • Vertical connectors.
    • If your bird units are made with nxn paper, these are n/2 squares. (or 1/4th of the original nxn paper) 3 connectors will connect 2 bird units.

Folding the bird units:

Putting the bird units together:

Folding the horizontal connectors:

Putting bird units together horizontally:

Folding the vertical connectors:

Putting bird units together vertically:

Questions to consider:

  • What forms are possible? What are excluded?
  • How much space is actually taken up?
  • Is it possible to get a square grid going here? A cubic one?
  • How many transformations are actually happening?
  • How many triangles appear in each of the tetrahedral units?
    • Each di-pyramid is actually 3 square faces folded along their diagonal to make 2 45-45-90 triangles. What do you notice about that construction? What might it tell you about the angles between the faces?

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!

6.) Attend to precision. (This gets pretty annoying if you’re not at least working toward being precise when folding.)

7.) Look for and make use of structure. How can these units be put together? What is possible and what is not?

8.) Look for and express regularity in repeated reasoning What do you notice about how the pieces do and do not fit together? What generalities can we determine?

#MathArtChallenge 94: Twelvefold Islamic Geometric Rosette from Samira Mian

12fold rosette full

The Challenge: Make yourself a 12-fold Islamic Geometric Rosette.

Materials needed: Compass, straight edge, paper, pencil, colors
Math Concept: geometry, intersections, tessellating, polygons, proportions

All of the instructions I followed here can be found at Samira Mian’s website. If you aren’t already familiar with how much I respect and admire this woman, you probably haven’t been following me particularly closely. I couldn’t do the last 10 without nodding to Samira. Her work is an absolute gift to the mathematics community. You should probably definitely go take her first and second online classes.

12fold rosette detail

Here’s the time lapse of me making it:

Things to ponder:

  • Check out how different the rosette looks when it’s in a 4 fold or a 6 fold tiling (you can see both at Samira’s page). What shapes change? How are the tilings similar or different?
  • What other ways could you highlight shapes? Is there a way to have a laced pattern with alternating shapes colored or empty? (Positive and negative space)

(p.s. I know this is a day late. I was at the EduColor summit yesterday and it was just necessary for me to spend my time there. I’ll still try to post another later today.) 

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 does this rosette change when it is tessellated in 4fold vs 6fold symmetry?

6.) Attend to precision. What do you notice about this construction compared with other Islamic constructions?

#MathArtChallenge 93: Archimedean Solids from Vertex descriptions

IMG_1969

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 76: Decagon & Pride Flag

Here is my first attempt at a more thoughtful #mathartchallenge. Pushback, thoughts, additions you’d like to see all welcome. You can comment below, or here, or talk to/DM me on twitter.

June is Pride month. I try to always acknowledge and honor that in my classroom. Last year, I put up this display in my classroom window:

This year, it made sense to create a rainbow #mathartchallenge to post for my kids to see. Having seen pride flags with Black and Brown stripes, noting it as a nod to a more inclusive pride, I picked a decagon design that I wasn’t totally sure how to create from Arts and Crafts of the Islamic Lands , and voila.

Except, of course, that I messed up. I didn’t do my research. ANY research. The original rainbow flag is 6 colors: red, orange, yellow, green, blue and violet. There are new designs that include Black and Brown stripes, and as of 2018, designed by Daniel Quasar, there is this flag:

Daniel Quasar flag

This flag includes not just the Black and Brown stripes, but also nods to the transgender pride flag. You can read about it here. 

I also want to speak the names of Tony McDade, a Black trans man killed by police, and Nina Pop, a Black trans woman killed in her apartment, both within the last few weeks. 

When I posted my ill-informed rainbow, I hadn’t necessarily planned it to be a #mathartchallenge. However, seeing the collaboration of Tina Cardone and Xi Yu, I think it makes for a good exercise. If you follow the tweet, you can see their solution – the tutorial Xi refers to is also here.

You may also notice they actually tagged their creation with “Queer #BlackLivesMatter”, which I failed to do.

Here is a better version.

MAC 75 Decagon pride flag

Better, I think because it honors (as best I could match) the actual colors of the pride flag, including those represented by the transgender flag. Not great yet, because as any decagon has 10 parts, I just ran out of space for all the colors (an 11 sided shape is rather challenging to construct). I would like to note that in the trans flag, the white color represents gender neutral or non-defined gender. Given the white paper, it seemed the easiest shade to not include. By no means do I mean to exclude gender neutral or non-defined gender people in this representation.

If you choose to participate in this #mathartchallenge, here are some questions I’d like you to reflect on:

  • The larger shape here is a decagon (10 sided shape). Where else do you see the theme of 10 appear here?
  • How many? What did you count and how did you count it?
  • What angles are represented here?
  • What shapes do you see that are not decagons?
  • How do the shapes interact with each other? What can you say about the relationships between them?
  • Can you find an 11 sided design that would better represent Daniel Quasar’s 11 color-flag?
  • How do symbols, like flags, interact with identity at large? With your personal identity? What thoughts about representation are present or missing here?

#MathArtChallenge Day 29: Paper Roll Polyhedra

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: Make your very own paper roll polyhedra. All inspired by this tweet from Mr. Allan shared by Clarissa Grandi

MATERIALS NEEDED: Paper rolls. Can be toilet paper or paper towel rolls or you can make your own!

Here are some that I made:

2020-04-12 14.40.31And now is a great time to explore some polyhedra if you haven’t yet done so! Here are some websites shared by Mark Kaercher that are great.

https://polyhedra.tessera.li/

https://www.templatemaker.nl/en/

#MathArtChallenge Day 28: Paper Tube Designs

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!

Today, courtesy of the extraordinary Mark Kaercher:

 

#MathArtChallenge Day 25: Tessellations

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: Great your very own tessellation from a small piece of cardboard.

  1. Start with a square (Cereal or pop box cardboard works great) Mine was 2”x2”
  2. Cut a piece from one side and tape it to an adjacent side.
  3. Repeat for remaining 2 edges.
  4. Trace!

MATERIALS NEEDED: Pencil, paper, and cardboard from a cereal box or a pop box or something similarly thin but stiff enough to trace.

#MathArtChallenge Day 24: Polygon Midpoints

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:

  1. Draw a polygon.
  2. Mark the midpoint of each side.
  3. Connect the midpoints of each side to make a new polygon.
  4. Repeat.

Don’t sleep on the quadrilaterals here. They do something rather surprising and beautiful!

MATERIALS NEEDED: Whatever you prefer!

2020-04-08 08.13.50

#MathArtChallenge Day 21: Agra Fort Islamic Design

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 this design from the Agra Fort.

Day21 MAC still

MATERIALS NEEDED: If you have a compass, use this tutorial from Samira Mian. If you do not have a compass, you can use my instructions for making this from a regular grid below that.

Day21 MAC grid instructions