## The Greatest Parallelepiped Ever

In general, it’s unusual for a rectangle to have sides and diagonals whose lengths are all** integers **(i.e., whole numbers). Consider the following three rectangles, all of width 3:

Looking at the different lengths, we see one place where the diagonal length is an **integer**, but in the other cases, the diagonal length happens to be a **non-terminating**, **non-repeating **decimal (i.e., irrational). Indeed, the diagonal length will be an integer exactly when the length and width are part of a **Pythagorean triple**, but compared to the alternative, this is uncommon. (While there are infinitely many occurrences of this, we can still meaningfully consider it uncommon).

Now, imagine the situation in three dimensions. A rectangular prism (think of a cardboard box) has 12 sides, 12 **face diagonals**, and four **space diagonals**. It would be extremely unusual for all of those 28 lengths to be integers. Even if we didn’t limit ourselves to rectangular prisms, but we allowed for the box to be *slanted* in all directions (that is, a *parallelepiped*), it would still be a numerical miracle for all those lengths to be integers.

Well, meet the perfect parallelepiped!

This was discovered by a couple of mathematicians at Lafayette College in Pennsylvania, using brute-force computer trials. It looks like they found some others, too. So thank you, Clifford Reiter and Jorge Sawyer, for giving me an extra credit problem for my next exam!

*Click here to see more in Geometry.*

I thought about just making a bunch of equations, and seeing if it would be possible to find this Parallelepiped without a computer. Its proving more difficult then expected.

That’s a lot of vector algebra.

Is it perfect only because it has integral lengths?

Yes, that’s the definition of a

perfectparallelepiped. Apparently the guys linked to above were the first ones to show that these things existed.This demonstration at WolframMathWorld is pretty cool:

http://demonstrations.wolfram.com/PerfectParallelepipeds/