Pythagorean cuboids

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Is it possible to have a cuboid where
1) the length of each side is of integer length,
2) the diagonal of each face is also of integer length,
3) the internal, corner to opposite corner diagonal is also of integer length?
If not, why not?
My maths education stopped far too early to get to this level of complexity, but curiosity persists.

1) No.
2) Yes (I assume you mean "Angle, and not "Length").
3) No.
4) Because, Trigonometry!

I suspect that he does mean "length", and not "angle". Inches and not degrees.

But the answer is probably the same: you probably cannot have all of those conditions met within the same rectangular solid because of trig. But I am not a mathematician, and Pythagorus is about the only thing I remember from geometry class.

You can have a two dimensional rectangle work out that way. One side is three units, another four units, and the hypotenuse would be five units.

And then you could construct a three dimensional rectangular solid consisting of four sides with those dimension rectangles, with a three unit square on the top and another three unit square on the bottom. But the hypotenuse of those two squares would not be in integer units, and neither would the internal hypotenuse that connects the bottom left corner of the front facing rectangle with the top right corner of the back rectangle (if that's what your talking about). Ditto with the identical one connecting the right and left sides.

There maybe a way to do a rectangular solid so all of that works out into integer units, but I suspect that it would be impossible.

Last edited by naturalplastic on 05 Oct 2019, 6:20 pm, edited 1 time in total.

I can confirm, I do mean length. Here are a couple of examples that fail at least one requirement:

A cuboid measuring 5×12×16 will have one pair of faces with diagonal for triangles measuring 5×12×13, another pair with triangles of 12×16×20, but the remaining pair would not have all integers at 5×16×SQRT(281), and the internal diagonal would likewise not be an integer at SQRT(425), assuming I can type correctly without the predictive text feature messing me about, again.

A cuboid measuring 1×2×2 will have no faces with integer diagonal measurements but an internal diagonal of 3.

What do you mean by “Cuboid”?

There are only 5 regular, convex Pythagorean solids: the Tetrahedron, the Hexahedron, the Octahedron, the Dodecahedron, and the Icosahedron

The Tetrahedron has 4 sides, 4 corners, and 6 edges. Angles are 60 degrees.
The Hexahedron has 6 sides, 8 corners, and 12 edges. Angles are 90 degrees.
The Octohedron has 8 sides, 6 corners, and 12 edges. Angles are 60 degrees.
The Dodecahedron has 12 sides, 20 corners, and 30 edges. Angles are 144 degrees.
The Icosahedron has 20 sides, 12 corners, and 30 edges. Angles are 60 degrees.

The Tetrahedron has no corners in opposition.
The Hexahedron is also known as the “Cube”.
The Octohedron is essentially two pyramids connected base-to-base.

Prolly his own made-up term for what I think that you're supposed to call a "rectangular solid". Something that's not perfect enough to be a three dimensional version of a square (ie a cube), so it settles for being a three dimensional version of a rectangle.

Like he calls "the hypotenuse" "the diagonal".

So, effectively, for a rectangular prism of dimensions AxBxC, you're wanting to know if there are numbers A, B, and C such that:

A, B, and C are integer;
the square roots of A²+B², A²+C², and B²+C² are integer; and
the square root of A²+B²+C² is also integer?

In that case, you're looking for a perfect cuboid or perfect Euler brick. No such examples have been found in mathematics to date. However, it has also not yet been completely proven that no such rectangular prism exists. If one does, the sides would be extremely large integers (exceeding half a trillion units).

More information on Euler bricks at Wikipedia: https://en.wikipedia.org/wiki/Euler_brick

[quote="Dial1194"]In that case, you're looking for a perfect cuboid or perfect Euler brick. /quote]

Thanks, Dial1194. I'll follow that up.

Interesting ... I wonder if Kubrick's Monolith would classify as a Euler Brick. It's dimensions are 1 x 4 x 9.

I'm too tired to do the math right now.

Its face diagonals would be the square root of 17, the square root of 82, and the square root of 97.

The smallest Euler brick known has dimensions of 44 x 117 x 240.