A Different Model For Khyyam's Cubic Solution

While working on Khyyam's Cubic solution in class, my table distributed each piece of the cube to a different person so we would all have something to work on. I was given the triangular pyramid, the piece that was 1/6 of the whole cube. I really struggled while trying to visualize the whole cube put together with three individual parts. I looked multiple times at the model at the front of the class, but I still couldn't make very much sense of it. I was extremely frustrated by the time that class was done, and I didn't get anywhere! It seemed like I had a different vision in my head as to how the pieces should go together. I decided to work on it for my daily work. I continued to try to understand the model that we did in class, but every time I tried to sketch the net of the piece I confused myself all over again. So, I decided to try something I knew I could visualize. I didn't know if it was going to work, but I thought I would try it. This is what I came up with:

This made more sense to me than the model we had in class, and it still had a triangular prism and a triangular pyramid, but it had a rectangular pyramid instead of a square pyramid. From here, as a continuation of my daily work, I decided to draw the nets of each piece and actually construct the figure and see if it would work.

I started with the net of the triangular prism (the one in the top left corner in the image above). I first started by drawing smaller nets that were to-scale of the actual figure I was going to create, I used the scale were 6cm = 1 4/10cm, and (72)^(1/2)cm = 2cm. I visually pulled the each triangular base flat first, and the I knew the square their bases were attached to was the 6cm x 6cm square face of the cube. I then visually unraveled the sides of the prism and laid it out flat. So my smaller scale version the net of the triangular prism looked like this:

 Similarly, I did the same thing to create the net of the rectangular pyramid. I knew that the base of the rectangular pyramid was the was the same dimensions as the rectangle in the net of the triangular prism because that was were the base of the rectangular pyramid would rest when I put the cube together. So, I visually just pulled down each of the sides to create the net. I knew that the smaller triangular side in the back was going to have the two 6cm sides on the outside because hypotenuse of the triangle was the measurement of the side of the base. It was a little tricky making this one, because I had to make sure that the lines were not only 6cm but also that they formed a 90 degree angle. the other two triangles on the smaller end of rectangular base I just laid flat, but then the last triangle was a little more tricky.I knew when I put the other 3 triangles together that the last triangle would be made with two of the hypotenuses and the last side of the rectangular base. So, I knew it was an equilateral triangle. This is the net of the rectangular pyramid that I came up with:

Lastly, I had to make the smallest and trickiest piece. I knew that the base of the triangular pyramid would lay on the equilateral triangle from the rectangular pyramid, so the base was an equilateral triangle with side lengths of (72)^(1/2)cm. It made it simple from there because I knew that each side of the base was going to be the hypotenuse of the triangles creating the sides. However, it was very tricky because again I had to make sure that were the 6cm sides met was a 90 degree angle. To make sure, I erected a perpendicular line from the midpoint of each side of the equilateral triangle. This created a line for the two 6cm side lengths to meet to ensure that they created a 90 degree angle. This is what I came up with for the net of the triangular pyramid:

Once I got the nets sketched that were to-scale, I made the bigger nets and folded each piece of the cube. Here are the 3 separate piece after I folded and taped them:

They all turned out fairly well and fit together! Here is the succession of adding each piece:

And the final product!

It kind of looks like the pieces didn't all fit perfectly in the picture, but that is because of my taping abilities. They really did fit snugly when I held them. 

Overall, from this, I felt like this reinforced the idea that there isn't only one way to do things. I think it was challenging trying to come up with a different way to make the pieces that still works and fits together. I am also very satisfied that I did do this because I became a lot less frustrated knowing that the way I was trying to construct it and was thinking about it in class really did work!