Pencil Cup:

Our dodecahedrons got less poppy over time. That's okay. We made more! But what can be done with a flabby dodecahedron? Here's an option. Cut off one of the pentagons from one of the pieces.

Then tape everything together (instead of using a rubber band to hold them together). Voila!

Dragon Egg:

Another option is to tape together the top piece and the bottom piece separately, making two cup shapes.

Then tape together one edge of the top piece to one edge of the bottom piece, making a hinge. Your child won't have any trouble finding things to put in their new dodeca-box. Or, find a small toy and let it hatch out!

Tiling the Plane:

When we trace our pentagon onto the cardboard, notice that there are little gaps between the outer pentagons.

When shapes fit together snugly, with no space between, and can cover an entire piece of paper (no matter how big), this is called "tiling the plane." (A "plane" is the math term for a flat 2-dimensional surface, like a piece of paper.)

Let your child try tracing their pentagon so that the shapes are right up against each other (no overlap) and there are no gaps between them. Don't make them try too long, because they'll quickly discover this is impossible.

So what shapes can tile the plane? Cut out different shapes to try. Also see our BA Playground game Cookie Cutter for more on this topic.

Euler's Formula:

A 3-D shape like your Pop-Up Dodecahedron is called a polyhedron. Leonhard Euler, considered one of the greatest mathematicians of all time, discovered a simple formula about polyhedra that even young math beasts can easily verify.

How many pentagon-shaped flat pieces does your dodecahedron have? (It has twelve.) The math term for these flat pieces is "faces." One way to think of the number of faces is to imagine that your dodecahedron were a die. How high would you be able to number it?

Now count the number of "vertices" your dodecahedron has. These are the pointy corners that poke out. There are a lot, so it may help to color them with a marker as you go. There are 20 vertices.

Finally, count how many "edges" it has. These are the straight-line pieces where the pentagons meet. Again, there are quite a few, so keep track by marking them with a marker as you count.

Your dodecahedron has 30 edges.

Now, what do you get when you add up the number of faces and vertices, then subtract the number of edges? Our calculation is: 12 + 20 - 30 = 2.

To see why this is remarkable, find some other polyhedra to examine. You may not realize this, but you likely have some other polyhedra around your home. A die is a cube, which is a kind of polyhedron. Count the number of faces, vertices, and edges. A cube has 6 faces, 8 vertices, and 12 edges. Again, add the number of faces and vertices, then subtract the number of edges: 6 + 8 - 12 = 2. We got 2 again?

You may find more polyhedra. Here are a couple more we found.

The red block has 5 faces, 6 vertices, and 9 edges. Euler's Formula gives us 5 + 6 - 9 = 2. The orange block has 8 faces, 12 vertices, and 18 edges. Euler's Formula gives us 8 + 12 - 18 = 2. You'll get 2 every time!

Be careful when searching for polyhedra. Euler's Formula only applies to what are called "convex" polyhedra, meaning shapes that don't have any parts that dent inward. Any block that has a curvy shape is also not a polyhedron. Here is an example of a couple of shapes that wouldn't work.