Hey everyone,

I've been exploring geometric transformations and thought I'd share a new approach I've been working on for proving the Pythagorean theorem. I'm excited to hear your thoughts and get some feedback!

The Pythagorean Theorem Recap:

In any right-angled triangle, the sum of the squares of the two shorter sides equals the square of the hypotenuse:

My Geometric Proof Attempt:

1. Construct a Right-Angled Triangle:

Start with a right-angled triangle , where the angle is 90 degrees.

Let the sides be:

Hypotenuse

2. Build an Adjacent Rectangle:

Construct a rectangle adjacent to the side with dimensions .

3. Apply a Shearing Transformation:

Shear rectangle alongside to form a parallelogram without changing its area.

The top side of the rectangle shifts horizontally by length , creating a parallelogram with sides and .

4. Dissect the Parallelogram:

Draw a perpendicular from the shifted top corner of down to side , splitting into:

A rectangle with dimensions .

A right-angled triangle congruent to .

5. Analyze the Areas:

Area of rectangle : .

Area of parallelogram : (since shearing preserves area).

Area of rectangle : .

Area of triangle : .

6. Attempt to Establish the Relationship:

Since consists of and :

Simplifying:

Here's where I hit a snag. The simplification leads to , which doesn't hold true unless , which isn't generally the case.

Seeking Your Expertise:

I'm reaching out to see if anyone can help identify where my reasoning might have gone astray or how this approach might be adjusted to correctly prove the theorem.

Questions:

Has anyone seen a similar method or can point out where I might have erred?

Is there a way to modify this construction to make it valid?

Could this approach lead to a valid proof with some adjustments?

Why This Could Be Exciting:

Fresh Perspective: Exploring new geometric proofs can deepen our understanding and appreciation of fundamental theorems.

Collaborative Discovery: Your insights could help refine this approach or inspire new ones.

Looking Forward to Your Thoughts!

I'm excited to discuss this with you all and see where we can take this idea together. Thanks for reading, and I appreciate any feedback you might have!

Edit: After some reflection, I realized that my area calculations might be off, and perhaps the construction needs tweaking. Any suggestions are welcome!

Hi everyone,

I made a long documentary on Pythagoras on YouTube. If anyone is interested, here is the link: PYTHAGORAS: Behind The Math And Myth https://youtu.be/Ac2gYn9hylw

Also, I will highly appreciate any feedback and criticism on it.

Thank you!

Can anyone here help me out and tell me if, number one, it’s an actual quote from Pythagoras. Number two, what is the dose material?

Hi,

I'm reading Russell's The History of Western Philosophy and have finished the chapter on Pythagoras.

Just like most people, I scratched the surface of Pythagoras' work in math classes at school, but it's the first time I come across a work that tries to show his philosophy as something universal (ie, applicable to a lot of fields other than maths).

Given the above, I've got a question - am I right to think that Pythagoras' philosophy is universal as it promotes acknowledging what's evident and then deducing the less evident from it (as supposedly done by Pythagoras' successors)?

Ian and Nigel Gilcrest are musing on that...Theres a new book "When the Dog Speaks". Maybe there was more east/west cultural interchange than we ever thought...

Wow

Hello. Im wonder how buidling or shelter and garden of this community looked like because I cant find informations on this topic on internet. They had communical accommodation or lived in cells? It was a big building? How pythagorean garden looked like? Was this garden surrounded by trees? Location was near mountains or sea?

Hail Pythagoreans.

Does the following reflect an accurate description?

​

1. Monad (1): Represents unity, the origin of all things, and the divine. It's a symbol of the essence of existence and the source of all numbers.
2. Dyad (2): Represents the principle of "otherness" or difference. It's the first number after the Monad and introduces the concept of relations since a relation requires at least two entities. In this sense, it was associated with diversity, division, and the physical world.
3. Triad (3): Symbolizes harmony, since it encompasses the beginning (Monad), middle (Dyad), and end, creating a whole. It's also associated with concepts such as wisdom and understanding, and it was seen as a reconciling or mediating figure between unity and diversity. Some have connected this with the idea of birth, life, and death.
4. Tetrad (4): Represents solidity and was associated with the four elements: earth, water, air, and fire. It was also related to the four seasons, four directions, etc. Pythagoreans considered it a perfect number since it is equal to the sum of its divisors (1+2), excluding itself.
5. Pentad (5): Represents health and life. It's associated with marriage (the sum of the first male number, three, and the first female number, two) and also corresponds to the five senses.
6. Hexad (6): Represents creation and is often associated with space and time. The six faces of a cube represent spatial dimensions, and the six periods of a day (dawn, morning, noon, afternoon, dusk, night) represent time.

Is it safe to say the principle that binds these structures/beliefs is symmetry?

I cannot remember the values of U and V in the Pythagorean triple 36,77,85. Can anyone help me out?

I was just watching something centered on him, and such was mentioned, that some could view him as such today.

I just hoping for someone who has good knowledge of his history, and some society construct ideals history, to give their take on such question. Also if communist isn't what you best classify him, which category would be best/closest? I kind of feel weird asking this, since I myself never asked such of me, I feel best to be none, but I also don't push or condone any system, since I can't even, when I'm not knowledgeable at all in the subject. I can't even think of the terms to come up with. However I always see those terms being brought in a bad light, and at least from this introduction to Pythagoras, where people are neutral, tickled my curiosity.

The other reason for not being able to easily pick up the terms I'm after, is that english is my 3rd language, and I was self taught, and never schooled, where you usually learn those terms. I have not practice much my mother language for decades as well, so even on such language the terms scapes me. So sorry for not being any more clear.

The Enneagram is an occult symbol used in certain circles.

Its first appearance was by the hand of Gurdjieff, an armenian mystic from the late 19th to mid 20th century, who was quite ambiguous about its origins, stating that he learned of it during his travels in central Asia. He did not make any assertion over its origins. Later, it was utilized by Oscar Ichazo, a Bolivian spiritual teacher.

Now, there's a whole charade involving copyrights and intellectual theft after some people started using Ichazo's theories. The lawsuit is quite famous. In the end, the other authors managed to convince the judge that the origin of the enneagram as taught by Ichazo was ancient, making it go back to the desert fathers, the sufi, the Kabbalah etc. The accusation is that he clearly took it from Gurdjieff without any accreditation. Ichazo denied all of this and he had the following to say:

"The enneagram figure, which the Gurdjieffians affirm that I took from their Master, is in fact one of the forms known as 'seals,' which were produced by the Pythagorean school (500 BC), and the Platonic mathematicians (300 BC) who studied the internal relation of numbers with geometrical forms, giving to each number, not only their characteristics, but their internal interrelations. "

Is there any truth to this?