Time to get Cubic S2E11

Wake! For the Sun, who scattered into flight

The Stars before him from the Field of Night,

Drives Night along with them from Heav'n and strikes

The Sultan's Turret with a Shaft of Light

-Omar Khayyam

From Al-Khwarizmi to Khayyam

After completing my work on completing the Square I stumbled upon the interesting polymath Omar Khayyam. He is mostly known for his poems but in the history of math has many contributions. The quote above is actually from his poem Rubaiyat. Khayyam was one of the first mathematicians to use analytical geometry centuries before Descartes, he found a partial solution to cubic equations, and  he independently found Pascal's triangle. Seeing that Khayyam completed the Cubic I wanted to use Al-Khwarizmi's method of completing the square to be extended into another dimension, the Cube. My teacher and I first set out finding a solution using only a white board and we got quite far but much time was wasted on drawing and redrawing cubes on the board. Here were our results:

Apparently its pretty difficult to represent a three dimensional object on a two dimensional plan. But we were able to come up with an equation for finding the factors of a cube: 

x = -b/3  + (d + 3(x*b^2/9) + b^3/9)^(1/3)

It still needs a lot of work but its a good start. another way of writing the equation is:

(x + b/3)^3 = d + 3(x*b^2/9) + b^3/9

Working with cubes is definitely a different ball game so my teacher and I decided to the best way to solve our problems is with blocks. I was tasked to build a cube. here's what the process looked like:

Measuring the Cut

Sawing the blocks

Gluing the blocks together

The Cube without its top on

The final cube

I'm still not finished everything needs a new coat of paint and I need to do a final write up on completing the cube. But each piece of the cube seen above physically represents a part og the equation solved above.