Octahedrons, Tetrahedrons, and the Pyramid Puzzle

Last night I did a fun project with my older son on a problem from the 2006 AMC 10a. We were both a little tired and although we found a plausible solution to the problem, we didn’t really solve it. I realized later in the evening what a missed opportunity this was, so I thought about repeating the project tonight. As I thought more about it, I realized that there was also a connection to two other really amazing math ideas:

(1) The Pyramid Puzzle – which has a super fun history:

The Pyramid problem in the NY Times

(2) A cool post about 3D geometry from John Baez:

A neat John Baez post – look especially at the tetrahedral-octahedral honeycomb in 3 dimensions picture:

Having made these two extra connections in my mind overnight, I decided it was definitely necessary to repeat last night’s project. I thought it was going to be such a fun project that I included my younger son even though he’s not studied any geometry before.

Here’s a link to the original problem:

Problem 24 from the 2006 AMC 10a

The problem is fairly easy to state – if you form an octahedron by connecting up the centers of each side of a cube, what is the ratio of the volume of the octahedron to the volume of the cube? Although it is easy to state, it isn’t easy to solve – especially if you don’t know the formula for the volume of a pyramid! We’ll solve it with our Zometool set, though, and use no volume formulas at all. Along the way we’ll see some really cool 3D geometry!

Just like last night, we started with a simple shape – a 2x2x2 blue strut Zometool cube:

 

Next we constructed the octahedron inside of our cube and discussed what we noticed about the shape:

 

In the last movie we saw that our geometry problem had a lot of symmetry which allowed us to simplify the problem a little. The next thing that we did was to look at one of the smaller cubes that contained a smaller pyramid which was 1/8 th of the octahedron:

 

It turns out to be a little hard to see what fraction of our cube is taken up by the little pyramid, so we now decided to build a bit more. We put a few more green struts in our cube and found that our cube now had 4 little pyramids and a single tetrahedron. Interesting – is this progress? Hard to tell, actually, because we don’t know how the pyramids relate to the tetrahedron.

 

Next comes some really fun geometry. We construct the little pyramids and tetrahedrons in all of our small cubes and take a careful look at what our large cube looks like now. My younger son noticed that if we had lots of these large cubes we’d make a bunch of octahedrons if we connected them together. That’s the tetrahedral-octahedral honeycomb in the Baez article – I cannot believe that he saw that, but you never know what kids are going to notice!

After that bit of fun we talked about another little geometric miracle – there’s a large version of our small pyramid hiding in our shape. It is 8 times the size of our small pyramid (since all the edges are 2x as long as the edges of our small pyramid) and is composed of 6 small pyramids and one tetrahedron. That means that each tetrahedron has the volume of two small pyramids. Amazing!

 

Next we take a small step back and write down the information that we have so far. We already know how many small pyramids that it takes to build the octahedron, and we just figured out how many small pyramids it takes to make a tetrahedron. Turns out that’s enough to solve the original problem. No formulas, no algebra, just a little Zometool geometry. Incredible!

 

With the old AMC10 problem out of the way, we moved on to looking at the Pyramid puzzle. First we took a quick look at a square pyramid and the tetrahedron. The reason for this bit is that in solving the last problem we actually figured out the ratio of the volumes of these two objects because the square pyramid is the top of the octahedron.

 

I introduced the Pyramid puzzle at the end of the last movie, but wanted to give it a stand alone movie so that the boys thoughts wouldn’t be mixed up with the thoughts on the other volume problem in the last video:

 

Last but not least, we build the object described in the Pyramid puzzle and see for ourselves how many faces it has.

 

So, a super fun project. Pure geometry – no formulas at all – and then also a fun connection to a famous puzzle. Definitely one of the most exciting projects that we’ve ever done.

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