The Perfect Parallelepiped

Published by patrick honner on

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!

a 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!

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patrick honner

Math teacher in Brooklyn, New York


Ahmed Gouda · October 24, 2010 at 1:13 am

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.

MrHonner · October 24, 2010 at 7:58 am

That’s a lot of vector algebra.

DHC · October 25, 2010 at 6:55 am

Is it perfect only because it has integral lengths?

MrHonner · October 25, 2010 at 5:48 pm

Yes, that’s the definition of a perfect parallelepiped. Apparently the guys linked to above were the first ones to show that these things existed.

This demonstration at WolframMathWorld is pretty cool:

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