Summer Fun 2014 - Marshamallow and Toothpick Geometry - Finding the Cubes in a Hypercube

I did a little bit of research, and it turns out the octagon full of squares, and triangles, and stars which the girls had so much fun coloring in with sidewalk chalk paint...

...is really the outline of the shadow of a tesseract - not the glowing blue cube from The Avengers, or the 5th dimensional wormhole Meg Murray travels through in L'Engle's  A Wrinkle In Time, but a 3-dimensional representation of a 4-dimensional hypercube, "that is to a cube, as a cube is to a square".  I'm totally not making this stuff up, either.

What we found really interesting though, is that what all that gobbledegook means, is that besides being filled with squares, and triangles, and stars, our painted picture is also filled with cubes.  Can you see them?

We couldn't see any cubes at first.  So, we built a couple 3D models out of toothpicks and marshmallows, that we could hold, and turn, and flip around...

...by starting with two cubes...

...and then connecting the respective corners - "front top left" to "front top left", "front bottom left" to "front bottom left"...

...and so on, until they were all connected and...

...we could smoosh, and pull...

...and flatten the shape, looking for the cubes.  How many can you find?

We found eight.

And, if I understand correctly, eight is the total.  Can you see them now?

It's great to be a homeschooler.

Resources:

A Wrinkle in Time by Madeleine L'Engle.

Flatland, a Romance of Many Dimensions by Edwin Abbott Abbott.

Carl Sagan's Comsos - Tesseract.

Wikipedia - Hypercube and Tesseract.

Animated Hypercube by Nicholas Mee.