Saturday, February 20, 2010

Richmond Math Salon - Symmetry

Lots of people showed up, right at 2. The adults played with the math games even more than the kids this time. The kids love the trampoline, but they still came back in to play with polydrons, mirror books, Blink!, Set, and tangrams. And to read You Can Count on Monsters. I'll tell more later. I want to get this post up so Madeline, and others, can comment here. (Hey Madeline, you're awesome too!) [Now edited - photos and activity details added.]

I had just gotten my big box of polydrons yesterday, and was excited to share them. We had oodles of fun with them. The first thing I built when I opened the box on Friday was a tetrahedron. Each face is an equilateral triangle. There are 4 faces total - a pyramid on a triangular base. The picture you see here is not a tetrahedron, because the base is a square.

Then I built an octahedron, using 8 of the same equilateral triangles. I don't have photos yet, so here's a generic octahedron. My son built a cube. So we had 3 of the 5 Platonic solids. On a Platonic solid, every face is exactly the same, and each face is a regular polygon. (Regular polygons have all sides the same length, all angles equal. What we call a side on a 2 dimensional polygon is an edge on the 3 dimensional solid.) There's one more criteria: At each vertex (think corner), the same number of faces have to meet. To me, it feels wild that there are only 5 possible ways to do this.

As we sat down to build, I said I wanted to build an icosahedron (20 faces) and a dodecahedron (12 faces). D built an icosahedron pretty quickly. One of the parents was trying to build one, and thought he'd used the wrong triangles. (He hadn't.) I was excited - I got to explain one reason we're limited to just 5 Platonic solids. I showed him (and a few others) how you can see that the angles in an equilateral triangle are 60 degrees, and 6 of them make a whole circle. So putting 6 together, they'll lie flat. (It looked like the picture here, on the top, but not on the sides.) We need to put less of them together at the vertex if we want it to poke up. As we talked about changing it, he mentioned having lived in a yurt. So we kept his yurt, and started over for the icosahedron, which has 5 triangular faces at each vertex. I had a lot of trouble getting the last face snapped onto mine, and D helped me.

So 3 triangular faces at each vertex makes the tetrahedron, 4 at each vertex makes the octahedron, 5 at each vertex makes the icosahedron (20 faces total). You can't use more because 6 faces at each vertex would lie flat, and you can't use less because 2 faces at a vertex won't have space in between. The cube has square faces, 3 to a vertex. We can't make something with 4 squares meeting a a vertex; same problem, it would lie flat. Next shape is a pentagon, with 5 sides. If you put three of those together at each vertex you'd end up with 12 sides. D wanted to do that, but we didn't have enough pentagons. I'll have to order those separately.

I got these polydrons very cheaply through Educator's Outlet. (When you go to the page I've linked to, you'll see the prices slashed, but then when you check out, they're cut even more. I paid about $30 for what would normally cost a few hundred, I think.)

Mirror Books
I got this idea from Maria Droujkova (who posts at Natural Math). Tape two little mirrors together. Draw something and set the mirrors in a V just behind. You get a kaleidoscope. Draw a straight line, and set the mirrors on it. As you adjust the angle between them, you get all sorts of polygons. We also played with writing upside down, writing in cursive and then drawing the mirror reversal of the writing, making a simple path and trying to follow it with the pen while looking in the mirror (way hard!), and looking at our faces in the mirrors (use a 90 degree angle to see your face as others see it, which doesn't happen in a regular mirror). These were all suggestions from participating parents, yeay!

Blink! and Set are the games I put out every month, so people can sit right down to something as others are arriving. (I see that Out of the Box sold the rights to Blink! to Mattel in 2007.) The tangrams also come out pretty often. The book, You Can Count on Monsters, by Richard Schwartz, is about prime and composite numbers, but can actually be enjoyed by very young kids, because it has a page for each number, 1 to 100, and each one has a different monster on it. It's a delightful concept.

I had a few other activities planned, but never got around to them. I wanted to do the rep-tiles activity explained here. And I wanted to do an activity where you find a line of symmetry in an odd-shaped figure, that I found In James Tanton's book, Math Without Words.

[Note to participants who post comments: This is a public space, so just use initials if you mention other kids.]


  1. Thx so much sue i had so much fun and i can't wait to come back
    i love your chickens and the trampoline

  2. $10 and $6 a set!! (for over 110 pieces in each set) Thank you!!

  3. I was excited, too. Good toys usually cost a lot. I found out through someone on Living Math Forum (a Yahoo group for mostly homeschooling parents to discuss math).

    After I got my 4 sets in the mail on Friday, I immediately ordered 2 more. I might call them today to order another. We didn't have enough pentagons to build a dodecahedron. It's the only platonic solid we couldn't build.

  4. Great post Sue. I love your simple explanation of triangles and 360 degrees in a circle and how that limits building choices with triangles.


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