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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)

#121

Post by seanhunter »

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.

So I haven't done complex analysis yet which I think you need to get the whole thing but I can get you some of the way there with basic trig.

If you take a unit circle and construct a radius to some point (x,y), if you drop a perpendicular line down to the x-axis, it's easy to see that the length of that perpendicular line is y and the distance you've gone across the x axis is x. So you have a right angled triangle where the hypotenuse is 1 (it's a unit circle) and the other two sides are x and y. Now consider the angle at the origin and call that theta.[1] You can do basic trig to show that the coordinates of your (x,y) point are (cos theta, sin theta), because sin is opposite (y) over hypoteneuse (1) and cos is adjacent (x) over hypotenuse. Ok cool. So x = cos theta and y = sin theta. If you measure in radians, then the angle of a full revolution is 2 * pi radians and the angle of a half revolution (180 degrees in other words) is pi. Now consider the point when you have gone around the unit circle 180o, Its coordinates are x=-1 and y=0. Remember this point - we'll come back to it in a minute.

Now imagine instead of your unit circle being just any old circle it's in the complex plane. This means that the x axis is the real part of some complex number and the y axis is the imaginary part. We now know that the coordinates of points on this circle are (cos theta, sin theta), but if you have a complex number z= a+bi, these correspond to a and b. So z = cos theta + i sin theta. Here's the bit where my current mathematical ability runs out of gas and you're just going to have to trust Euler, who showed that cos theta + i sin theta = e^(i theta).

Now remember our point from before where theta = 180 degrees? What was the angle in radians? It was pi. So e^(i pi) = -1 (because the real part of the number is the x coordinate, -1 and the imaginary part, the y coordinate is zero).

[1] Here's a diagram I made which will get you up to here https://www.geogebra.org/calculator/btz38m3c. My note about the trig of unit circle I made while studing is here https://publish.obsidian.md/uncarved/3+Resources/Public/Unit...

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

#122

Post by pcdoodle »

MrBuddyCasino wrote: post #70
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! On Seymour Cray: Another favorite pastime was digging a tunnel under his home; he attributed the secret of his success to "visits by elves" while he worked in the tunnel: "While I'm digging in the tunnel, the elves will often come to me with solutions to my problem." [0] [0] https://en.wikipedia.org/wiki/Seymour_Cray

That wiki was a fun read! It's fun to speculate about the elves and the tunnels.

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

#123

Post by bandrami »

The formula wasn't why people cared at the time; it had been empirically known for centuries. What caused such a stir was he was the first person to prove that for "most" right triangles there are no rational numbers P, Q such that PA = QC for leg A and hypotenuse C. That was what was earth-shaking.

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

#124

Post by dotancohen »

tapotatonumber9 wrote: post #30
loganc2342 wrote: post #18

> The Pythagorean Theorem is arguably the most famous statement in mathematics, and the fourth most beautiful equation. Just out of curiosity, which equations are considered the top three “most beautiful?”

The most delicious is: “(-1)^0.5 2^3 Σ π.”

You do realize that the volume of a Pizza with radius z and height a is:

  Pi ⋅ z ⋅ z ⋅ a

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

#125

Post by lynguist »

The mainstream is and was _enamored_ with Ancient Greece.

Our Western culture made Ancient Greek into the vocabulary root of our sciences.

We have a lineage of philosophy from Ancient Greece to the 19th century.

Only in the 19th century with archeology (again a neo-word made from Ancient Greek roots – it suggests to the mainstream that the Ancient Greek had a concept of archaeology, which they obviously didn’t have) we saw the truth: History goes thousands of years deeper, the origin of everything is thousands of years older.

Only 30 years ago the capital city of Hattuša was discovered; and only in the 20th century we gained an understanding of the multiple levels of the historic city of Troy.

Only recently we understand that "it didn’t start with Ancient Greece", but the mainstream still follows the tradition of medieval grammar schools and doesn’t look beyond Ancient Greece.

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

#126

Post by rcxdude »

andrewflnr wrote: post #120
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.

I've personally worked on a project where we used a 3-4-5 right triangle to lay it out on the ground. Straight lines alone do not get you right angles.

You can get right angles with a straightedge and compass, both of which you can make on a construction site with string and pegs in the ground. It's just an instance of bisecting the angle. Which is more convenient in practice is situational.

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

#127

Post by orf »

freework wrote: post #115
orf wrote: post #78

Earlier quoted context omitted.

… because those “squiggles” are just “words” in a “language” you can’t “read”? And if you could read it, you would find it contains a lot of relevant things, concluding with: > … 1.414213, which is nothing other than the decimal value of the square root of 2, accurate to the nearest one hundred thousandth. You might then think to yourself: > The conclusion is inescapable. The Babylonians knew the relation between the…

Here is a drawling of what they think this tablet says: https://commons.wikimedia.org/wiki/File:YBC_7289_sketch.svg It's just a bunch of numbers scribbled onto a tablet. For all we know it could just be some guy writing down the number of sheep he is willing to sell to his neighbor or something. To say this tablet proves the Mesopotamian knew about Pythagorean's theorem is quite a stretch. To the people who want to b…

Imagine for a moment that the people who created the tablet used a different number system than us, and also imagine that we knew that number system and could convert it.

Then those “bunch of numbers” becomes something else entirely. Specifically, they become a bunch of numbers that highly relate to the Pythagorean theorem.

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

#128

Post by darkerside »

dr_dshiv wrote: post #66
somenameforme wrote: post #39

Earlier quoted context omitted.

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…

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?

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

#129

Post by loa_in_ »

ssener2001 wrote: post #84
fsckboy wrote: post #33

Earlier quoted context omitted.

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…

I agree.Obtaining this formula by observing is more difficult than obtaining it by proving it. Besides, it is an issue that is already needed in field work. If you have pen, paper and a good number system, it shouldn't be too difficult to find the formula with shapes.

I imagine back then things like number systems and methods of proving were just made up as they go.

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

#130

Post by ChrisMarshallNY »

MrBuddyCasino wrote: post #70
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! On Seymour Cray: Another favorite pastime was digging a tunnel under his home; he attributed the secret of his success to "visits by elves" while he worked in the tunnel: "While I'm digging in the tunnel, the elves will often come to me with solutions to my problem." [0] [0] https://en.wikipedia.org/wiki/Seymour_Cray

That sounds like he was being figurative. He probably meant that he got ideas while working.

Every morning, I get up at 5, and take a 5K walk. During that time, I tend to "triage" the day ahead, and often solve problems that were vexing me, the night before.

Part of my walk is around a local high school track. There is a small flock of killdeer birds, that hang out there, and I guess they give me the ideas I have, as they often come to me, at that point in my walk.

I enjoyed this part:

> One story has it that when Cray was asked by management to provide detailed one-year and five-year plans for his next machine, he simply wrote, "Five-year goal: Build the biggest computer in the world. One year goal: One-fifth of the above." And another time, when expected to write a multi-page detailed status report for the company executives, Cray's two sentence report read: "Activity is progressing satisfactorily as outlined under the June plan. There have been no significant changes or deviations from the June plan."

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