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Pythagorean Theorem found on clay tablet 1k years older than Pythagoras (2009)

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Re: Pythagorean Theorem found on clay tablet 1k years older than Pythagoras (2009)

#131

Post by snerc »

dr_dshiv wrote: post #10

Pythagoras was specifically known for accumulating the wisdom of diverse cultures—supposedly he met Thales, was initiated as Egyptian priest in Hermopolis, spent time in Babylon after being captured, and was initiated into every mystery cult he could. And as a boy on his home island of Samos, he would have been exposed to the building of the largest stone temple in Ancient Greece (to Hera) and the incredible engineer…

"The relationship between math and spirituality was very strong back then!" According to those whose communication ended up being indelible. I wonder how we'll be able to preserve digital info for millennia?

Re: Pythagorean Theorem found on clay tablet 1k years older than Pythagoras (2009)

#132

Post by drexlspivey »

loondri wrote: post #86

It's interesting that Pythagoras gets credit for a theorem he may not have discovered, especially when there's proof that Babylonians knew it 1000 years earlier. This challenges the idea that ancient Greek mathematicians were always ahead of others.

It’s interesting that people in a CS forum don’t seem to know what a theorem is. The tablet shows that the relationship was known, a theorem needs a proof

Re: Pythagorean Theorem found on clay tablet 1k years older than Pythagoras (2009)

#133

Post by posterboy »

fsckboy wrote: post #33
hyperthesis wrote: post #28

Earlier quoted context omitted.

Agree it's closer but still an instance.

depends on the method they used to come up with the formula. It's not obvious that the sum of a "more complicated than you can do in your head" series of fractions would be either the square root of two or the the diagonal of a square, let alone both. This is a fragment of a single clay tablet. If they had a systematic method of coming up with this convergent series, there might have been a stack of tablets showing s…

Proof by construction, the Curry-Howard correspondance.

Re: Pythagorean Theorem found on clay tablet 1k years older than Pythagoras (2009)

#134

Post by larschdk »

mcv wrote: post #114
travisjungroth wrote: post #46

Earlier quoted context omitted.

IMO, it’s better with tau. e^(i * tau) = 1 You lose the 0, but isn’t it a bit odd the 0 is there in the first place? Normally we’d reduce it to: e^(i * pi) = -1 Which obviously isn’t as nice in this case. And hey, if we’re allowed to break the conventions, you can have the 0 back easily. e^(i * tau) = 1 + 0

Is there an easy explanation of why these are true? I already struggle grasping e^i, and I completely don't understand what e to an irrational power even means, let alone why it would be -1. Why the circle circumference ratio has anything to do with this is completely beyond me.

3Blue1Brown: https://youtu.be/v0YEaeIClKY?si=BQ2W64DWIuhHoUuX

Re: Pythagorean Theorem found on clay tablet 1k years older than Pythagoras (2009)

#135

Post by Beijinger »

seanhunter wrote: post #111
Beijinger wrote: post #77

My math is a bit rusty, for a moment I was wondering about the proof of his theorem but then I realized that I messed up his theorem with the angles in a triangle that sum to 180°, which does not always hold up.

If the angles in a triangle don't add up to 180o then I think that means the triangle is not on a flat plane. An example would be if you took a point on the equator of the earth, moved East or West 1/4 of the earth's circumference and took another point and then projected those two points North until they met at the pole, you would have a triangle with 3 interior right angles.

Yes. Pythagoras theorem holds up in non-euclidian space but the sum of the angles is 180 degree does not.

Re: Pythagorean Theorem found on clay tablet 1k years older than Pythagoras (2009)

#136

Post by dr_dshiv »

darkerside wrote: post #128
dr_dshiv wrote: post #66

Earlier quoted context omitted.

I don’t know of any ancient sources claiming that Pythagoras killed Hippasus. Seems unlikely to me! In any case, Hippasus was a fascinating figure: “Aristoxenus (Fr. 90 Wehrli = DK I 109. 31 ff.) reports that Hippasus prepared four bronze disks of equal diameters, whose thicknesses were in the given ratios, and it is true that, if free hanging disks of equal diameter are struck, the sound produced by, e.g., a disk ha…

I'm pretty confused. How do you think an average 8 year old would feel?

Well, they continue to introduce Pythagorean theorem through many stages of elementary math, so 8 is pretty early. But.. here could be a section of a textbook:

Pythagoras & The Pythagoreans: Mathematics, Music, and Mystery

Pythagoras was an ancient Greek mathematician and philosopher who lived around 500 BC. He traveled widely and gathered knowledge from diverse cultures, like Egypt. He also founded his own school of men and women in Italy. There, he taught that numbers held the key to understanding the whole universe. Pythagoras is best known for the Pythagorean theorem.

Fun Fact! There are stories that when Pythagoras discovered his famous theorem, he celebrated in a big way. Some say he sacrificed 100 oxen, while others claim it was just an ox made of flour. The Pythagoreans were famously vegetarian, so what do you think?

Math & Music: Pythagoreans explored the relationship between math and music. They discovered that musical notes have mathematical relationships. Hippasus, a member of the Pythagorean community, used bronze disks to show that musical notes are connected to mathematical ratios. This is considered one of the first scientific experiments!

Activity: Using a stringed instrument, like a guitar, try plucking the strings when pressing at 1/2 the string or 2/3s the string. Try different fractions. Can you hear the mathematical relationships in the sounds?

Re: Pythagorean Theorem found on clay tablet 1k years older than Pythagoras (2009)

#137

Post by photochemsyn »

mcv wrote: post #114
travisjungroth wrote: post #46

Earlier quoted context omitted.

IMO, it’s better with tau. e^(i * tau) = 1 You lose the 0, but isn’t it a bit odd the 0 is there in the first place? Normally we’d reduce it to: e^(i * pi) = -1 Which obviously isn’t as nice in this case. And hey, if we’re allowed to break the conventions, you can have the 0 back easily. e^(i * tau) = 1 + 0

Is there an easy explanation of why these are true? I already struggle grasping e^i, and I completely don't understand what e to an irrational power even means, let alone why it would be -1. Why the circle circumference ratio has anything to do with this is completely beyond me.

1. e^x, sin(x) and cos(x) can each be expanded out into an infinite sequence of fractions (Maclaurin series), each fraction of the form x^n / factorial(n). This relies on differential calculus, Taylor series, theory of limits, convergence etc.

2. The e series fractions contain all the integers in the numerator (x^0, x^1, x^2, etc), while the sin series has only odd integers and the cosine series has only even integers. Also, e series terms are all additive while the trig functions series alternate adding and subtracting each successive fraction.

3. Introducing complex numbers (i = square root of negative one), we can generate the series for e^ix, which can be shown to be equal to sin(x) + i * cos(x). Note that introducing i into the e series means we generate a negative term for the even fractions in the e series (squaring i gives us -1), which is why i is so necessary here.

4. Solving e^ix for x = pi, using sin(x) + icos(x), we get -1.

Mathologer:

https://www.youtube.com/watch?v=-dhHrg-KbJ0

and

https://www.youtube.com/watch?v=DoAbA6rXrwA

As far as why an exponential function like e^x should have anything fundamental linking it to trigonometric functions like sin(x) and cos(x), it is rather strange.

Re: Pythagorean Theorem found on clay tablet 1k years older than Pythagoras (2009)

#138

Post by hnuser123456 »

somenameforme wrote: post #39
dr_dshiv wrote: post #10

Pythagoras was specifically known for accumulating the wisdom of diverse cultures—supposedly he met Thales, was initiated as Egyptian priest in Hermopolis, spent time in Babylon after being captured, and was initiated into every mystery cult he could. And as a boy on his home island of Samos, he would have been exposed to the building of the largest stone temple in Ancient Greece (to Hera) and the incredible engineer…

Oh how can you skip the most fun tale of them all. [1] Its authenticity is dubious, but there's probably at least parts of truth in it. In a nutshell, Pythagoras started a cult based around numbers, and the pseudo-divine purity of rational numbers - of which everything can be represented. Hippasus, a member of said cult, however managed to compellingly demonstrate that the square root of 2 could not be a rational num…

4/1-4/3+4/5-4/7+4/9...

Re: Pythagorean Theorem found on clay tablet 1k years older than Pythagoras (2009)

#140

Post by usrusr »

detourdog wrote: post #107
usrusr wrote: post #68

Earlier quoted context omitted.

Why would you need numbers for right angles? You need a straight line and a string. But I agree with the second paragraph: there's a huge difference between a procedure that is handed down as part of "this is how we estimate a building project" and a theorem that is declared universal truth and base for all kinds of other theorems. Even if they are exactly the same thing.

How useful are universal truths compared to getting shit done? I only see theorems as useful for complex societies. The son of a gem dealer would have the time to work out universal truths. The reputation of everyone doing it without numbers would reveal the pattern of the universal truth. Finally this looks like it was all done using cuneiform. Which brings up questions of notation and the language to describe a squ…

Getting shit done is the antithesis of progress because it goes hand in hand with "if it was good enough for my father, it will be good enough for me, who are you to disrespect (the methods of) my father!"
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