Geometry to the Next Level

The Pythagorean theorem for Extra Dimensions is a new look at geometry.

Copyright © 2008 by Benjamin Bradshaw

 

You may know the Pythagorean theorem, A² + B² = C² (also known as the square root of A² + B² = C). This theorem works very well, but this only applies to right triangles and two dimensions. But there is a way to use the Pythagorean theorem in more than two dimensions.

            To understand this better, I will describe a different way to think about the Pythagorean theorem. The Pythagorean theorem was not designed for right triangles. It works for right triangles, but that’s because it has a right angle next to the two vectors. The Pythagorean theorem was designed to combine vectors. Vectors are the component quantities at right angles to each other that add up to the original line. The reason for doing this is to combine lines that are not vectors of each other. To add them together, you break each one up into its component vectors, add the vectors together, and then recombine the resultant vectors to get the resultant line (see picture below).

Thinking about vectors now, how would you combine the vectors if another vector were added sticking out in the third dimension? They are all vectors to each other because they are at right angles to each other. If you think about it, the resultant line of the original vectors is also at a right angle to the third vector, so the first resultant line is a vector compared to the first vector. Therefore, you should be able to combine the two to make another resultant line, the resultant line from all three vectors. In terms of A (vector 1), B (vector2), C (vector 3), and D (resultant line of all three vectors) the formula for the resultant line in three dimensions is (√ (A² + B²)) ² +C² = D². The square root of something squared is itself, so the formula could be rewritten as A² + B² + C² = D².

Scientists do not know if there are more than three dimensions, but for the argument let’s pretend that there are. As you may have guessed, the Pythagorean theorem for four dimensions is A² + B² + C² + D² = E². The general rule of thumb for the Pythagorean theorem is the sum of all of the variables squared for all of the dimensions of the vectors you are using equals the resultant line squared.