Zometool and 3d Geometry

We gave my youngest son a Zometool set for Christmas and we’ve all been having a great time with it.  He’s built some basic shapes following their guide as well as an impressive army of “Zome bots”  that have occupied the play room waiting to attack my feet without warning.  We’ve also had some fun playing with some 3D geometry.

One interesting problem to revisit was one from the NY Times Wordplay column in September:

http://wordplay.blogs.nytimes.com/2013/09/23/pyramid-2/

(and also from Cut the Knot here:  http://www.cut-the-knot.org/Curriculum/Geometry/ThreePyramids.shtml )

The problem is easy enough to state, and I’ll borrow the wording from Cut the Knot – “A solid square-base pyramid, with all edges of unit length, and a solid tetrehedron also with all edges of unit length, are glued together by matching two triangular faces. How many faces does the resulting solid have?”

At first glance it seems that joining two solids along one congruent side will result in a new solid having a number of sides equal to two less than the sum of the side of the two original solids (since you’ve covered up two sides).  In this specific case that reasoning leads you to believe that the number of sides in the new solid will be 7.  However, there is a surprise . . . .

The second fun project was constructing a Excavated Dodecahedron, which I happened to see while flipping through an old geometry book last night.  That book – “The Penguin Dictionary of Curious and Interesting Geometry” by David Wells is really fun to play around with.  I’d recommend it to anyone interested in geometry.

The book has a nice picture of an Excavated Dodecahedron (which it calls a “deltahedron”), but the animated gif on the Wikipedia page is easier to understand:

http://en.wikipedia.org/wiki/Excavated_dodecahedron

My youngest son and I spent the morning yesterday building a small version of the shape from the blue pieces in his Zometool kit:

Definitely a fun morning, but the shape was so interesting that we thought it would be fun to try to make a larger version.  The larger version revealed a hidden (to us anyway!) icosahedron inside of the shape.   I’m not sure if that icosahedron will show up well in the pictures or not, but we tried to highlight it using small red pegs going from the center to each of the 12 verticies.  The edges of the icosahedron are the shorter blue pegs.  Here’s a top view and a side view:

Finally, here’s our short talk about the project and a few other fun 3d shapes:

All in all, the Zometool set has been a hit.