The list of things named after Leonhard Euler on Wikipedia runs to about 1500 words, and, I would hazard, omits several such things.
So how to settle on one? I’ve come down on one of the greatest “low barrier, high ceiling” problems there is: it’s a conjecture so simple, you can grasp it as soon as you’ve got a sense of Pythagoras’s theorem, but so complicated that people have been searching for perfect bricks since… well, since before Euler really got going((Stigler’s Law of Eponymy states that nothing in science is named after its originator, and was first stated by Merton. The mathematical version is that everything is named after the second person to discover it, or else everything would be named after Euler. This example goes the other way.)).
What is an Euler brick?
An Euler brick is a cuboid that has integer side lengths, such that the diagonal of each face is also an integer.
For example, Paul Halcke discovered the smallest Euler brick in 1719: it has edges of length 44, 117 and 240. A moment’s work with Pythagoras((or several moments’ work if you, fearing the ninja, decline a calculator)), the rectangular face with sides 44 and 117 turns out to have a diagonal of length 125; the 44 and 240 face has a diagonal of length 244; and the 117 and 240 face a diagonal of length 267.
Euler studied these and came up with several ways to parameterise subsets of bricks, but not all such bricks. So the second most interesting question (for me) about such bricks is, is there a parametric formula to generate all of them?
The first most interesting question relates to perfect Euler bricks. A perfect Euler brick, in addition to its face diagonals, also has an integer main diagonal, from one corner to its opposite. The question: does a perfect Euler brick exist, or can it be proved not to? (If it does, its sides are extremely long.)
Why is it important?
I would class this problem as annoying more than important. It’s something very easy to get your teeth into, but very hard to make significant progress with. Perhaps its importance lies in turning interested young geometers into number theorists? Or perhaps there’s just nothing I like more than an unsolved problem.
Who was Leonhard Euler?
This is a bit like a football blog asking “who is Pelé?”, or a music blog asking “who was Bach?” Leonhard Euler (1707-1783) was one of the… no, scratch that. He was the most important mathematician of all time. There’s barely a corner of maths he didn’t touch, and several corners that he built himself.
If I get started on his achievements, I’ll never finish – much like his written works, which are still being published today, nearly 250 years after he died.
A selection of other posts
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