A True 3D Pythagorean Theorem
a² + b² = c²
You have heard it a bunch of times when it comes to triangles but what does it even mean?
The formula is the sum of two squares to form a bigger square and it’s used for triangles is because the lengths of the three squares can be used to form a shape (a triangle) whilst following the formula.
What would a 3D version of this look like then?
a³ + b³ = c³ isn’t it because it can never result in a positive whole number according to Fermat’s Last Theorem.
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a² + b² + c² = d² is used to find the diagonals in a cube or cuboid but it’s not exactly what I’m aiming for.
When I think of true 3D version of the Pythagorean Theorem, I think of this, a³ + b³ + c³ = d³
What this formula means is, the sum of three cubes to form a bigger square
A very loved number is the square root of 2, which is the diagonal of a unit square and the length of the square gotten from combining 2 unit squares
If you do the same thing with the formula I proposed then you get the cube root of 3 (1.44224957)
From my many attempts I haven’t found a way to visualise this formula in the same way the Pythagorean Theorem can be visualised using a triangle
The only real use I’ve gotten from this formula is a better understand of volume and how they can be represented
