A Major Flaw in the Pythagorean Theorem

The Pythagorean Theorem has been called the crown jewel of mathematics. It has received countless proofs and it serves as a building block for many others. People are so eager to find proofs which support it’s truth because it agrees with the ways that they are already using it. However, all of the wishful thinking doesn’t change the fact that A² + B² != C² from a less-than-infinite perspective.

Nobody can say that the theorem is absolutely true or false because the answer is always both, depending on context. The world of duality is a strange place, it’s based upon the notion that it’s impossible to be right outside of the moment. For the sake of completeness, duality has to be mentioned here so that readers understand that this information is in no way hostile to the Pythagorean Theorem, rather it’s presented to demonstrate that logic has another side which is largely unexplored. It was bewildering for people to accept that the Earth is round (two opposite ways to reach a destination) and similarly, it will be difficult to accept that the answer is always both. Proofs are only valid from the context of a particular direction.

• X/1: It is TRUE when cycles are taken towards infinity because the error becomes so negligible that it’s virtually nonexistent. That is, the proportions of A² and B² approach 1 at the limit.
• 1/X: It’s FALSE when considering that the error is always there, no matter how insignificant that it becomes. Neither A² and B² are perfect squares because infinity can not be reached in practice.

The only way to get the bits to equate on graph paper is to sacrifice a cathet from each of the orthogonal squares. That is, A² and B² cannot be counted as perfect squares if the objective is to precisely equate their summation to C².

The first 6 cycles of the “Discrete Pythagorean Theorem” with an imaginary C², beginning with A² = 1, B² = 1, and C² = 0.

Below is an alternative way to depict C², one in which infinitesimal points are represented as the bits’ center-points rather than the inverse — at the intersections, where the memories are not.

This version comes with an extra bit, (C²+1). This can be connected with the overlap/handshake of the “unit one” where the origin is established. This has been directly related to the reason why primes show up with properties of (n-1) on various types of time/space processes (such as Fermat’s Little Theorem). In short, it’s the consequence of a doubly-counted “fence post error” which is shared by adjacent cycles from an imaginary, time-based dimension.

Cycles #3, #4, #5 of the “Discrete Pythagorean Theorem” with a rational C².

Pythagorean Triples

Everyone loves the Pythagorean Triples because they balance out without any errors. However, they are only half of the story because when A² and B² do not equate it means that there’s not a perfect way to construct C² out of square-bits/building blocks (i.e. depict on graph paper). It can be said that C² becomes irrational in such cases and this invokes an off-center oscillation. This is analogous to the way that photons wiggle their way to a destination, or the why square roots oscillate over an infinitesimal boundary between even/odd cycles. The Pythagorean Triples are only half-true (it’s always both), due to the underside of a wave which they are comprised from.

Here’s a matrix demonstrating the growth of Pythagorean Triples when A² and B² are equal. It shouldn’t be too hard for the reader to see how each row is derived from the previous one via summation. Notice that the “error” balances out to zero, meaning that the oscillation is perfectly centered.

When A² and B² are equal, the square root’s oscillation is perfectly balanced.

Here is another matrix, but this time the cycles begin with Pythagorean Triples. Consequentially, the oscillation is unbalanced because the errors do not cancel out. When one side of the wave falls upon zero it can be expected that the other side will contain a square number. Interestingly, the Pythagorean Triples produce errors which are derived from a high share of prime numbers.

When the matrix starts from a Pythagorean Triple, the error oscillates between ZERO and SQUARE (often prime²).

Finally, here’s a matrix which starts relative calculations from randomly chosen values. There are three important things to take away.

1. The oscillation will always be off-center when A² and B² do not equate.
2. The summation of errors will always result in a square number (or zero in the case of a rational C²).
3. The size of the square, which results from the summation of errors, is directly related to the difference between A and B.

A matrix of Pythagorean combinations which begin from randomly chosen values.

What I have demonstrated here is the tip of a very large iceberg and its tentacles touch an endless amount of other subjects. Stay tuned from more articles on Medium.com to learn about the strange world of duality.