Pythagorean Theorem, in three dimensions

 

It occurred to me: is the Pythagorean Theorem or a similar statement valid in space (but modestly remaining in 3-dimensional space in the first step)?

Of course, a plane exists in any number of dimensional spaces, and the Pythagorean Theorem can be interpreted and valid in that plane. But if we are already in space, let's break the planar triangle. Which isn't that hard. What else would a spatial "triangle" be if not a tetrahedron? And what else would the spatial "right triangle" be. On the other hand, it is necessary to clarify what we consider a right-angled tetrahedron, because there can be several interpretations. We accept the definition that a right-angled tetrahedron is a tetrahedron whose three angles belonging to one of its (right-angled) vertices are all right angles. Such a tetrahedron is part of a cube.)

Another tricky question seemed to be what to apply the result of the claim. The original Pythagorean Theorem applies to the sides of a triangle, which are one-dimensional figures. In addition, if the subject has been transferred from the two-dimensional space to the three-dimensional one, it seems reasonable to transfer the elements of the result, i.e. instead of one-dimensional (straight sections) they should be two-dimensional (plane triangles). It is interesting that we use the word side for both, but the sides of the latter sides are the edges of the tetrahedron.

Another pressing issue was the interpretation of squaring. The square of the sides of a triangle is a simple, "natural" square. But what is the square of a plane figure? And there would be no reason to expect a cubic lift instead of a square lift.

The specific study quickly gave the surprisingly simple and "familiar" result:

 

Theorem of the right-angled tetrahedron: The sum of the squares of the areas of the three right-angled sides is equal to the square of the area of ​​the fourth side.

 

The proof is relatively simple, but can be done with work requiring attention and patience. I rather convinced myself of the truth of the theorem by programming a general model (the power of habit).

What a denial, confirmation of the result was a great pleasure for me. There are two possibilities: either this result is already known, or no one has done it yet. My joy does not depend on it.

It is a bit absurd, but this is how we are today in science, technology, and creation: it takes more time and energy to prove that you are the first than to achieve the result itself. Unfortunately, life is too short to waste it proving superiority. In this, we should rather count on those who find their dubious joy in being able to refute the primacy of others.

 

 

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