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Math

Australian Mathematician Discovers Applied Geometry Engraved on 3,700-year-old Tablet (theguardian.com) 81

Posted by msmash from the how-about-that dept.
An Australian mathematician has discovered what may be the oldest known example of applied geometry, on a 3,700-year-old Babylonian clay tablet. Known as Si.427, the tablet bears a field plan measuring the boundaries of some land. From a report: The tablet dates from the Old Babylonian period between 1900 and 1600 BCE and was discovered in the late 19th century in what is now Iraq. It had been housed in the Istanbul Archaeological Museum before Dr Daniel Mansfield from the University of New South Wales tracked it down. Mansfield and Norman Wildberger, an associate professor at UNSW, had previously identified another Babylonian tablet as containing the world's oldest and most accurate trigonometric table. At the time, they speculated the tablet was likely to have had some practical use, possibly in surveying or construction. That tablet, Plimpton 322, described right-angle triangles using Pythagorean triples: three whole numbers in which the sum of the squares of the first two equals the square of the third -- for example, 3^2 + 4^2 = 5^2.
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Australian Mathematician Discovers Applied Geometry Engraved on 3,700-year-old Tablet More

Australian Mathematician Discovers Applied Geometry Engraved on 3,700-year-old Tablet

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  • Archeology is a fine job (Score:5, Funny)

    by nospam007 ( 722110 ) * on Wednesday August 04, 2021 @12:32PM (#61656009)

    After long years of studies you get to examine a tablet from 3600 years ago containing somebody's math homework.

    • Beats being a SysAdmin and examining somebody's programming homework.

      They failed.

      • But does this mean Pythagorean theorem was understood by people before Pythagorus

        • "But does this mean Pythagorean theorem was understood by people before Pythagorus"
          Indeed! The bastards!

          • Re: (Score:2)

            by DarkOx ( 621550 )

            I don't know the answer to that question but this discovery of a list of triples does not prove that.

            Just because some knew that 3 units by 4 units with a 5 units side makes a right triangle does not mean they know why. They may simple have worked it out empirically by measurement and made note of it. You can't really conclude from that they understand the relationship is the sum of the square of the two sides equals the square of the hypotenuse or they had the ability to set it up as an equation and solve

            • The Babylonians probably did know sq(3) + sq(4) = sq(5). But why do the sums of the squares matter? Why is that the key to right angles?

              The proof is very neat, and probably the first non-trivial geometric proof to be invented. But probably not until about 500BC

              Incidentally, my daughter's text book had a horrible proof of this that was quite unintuitive.

          • No. The Pythagorean Theorem applies to right triangles of any angles (meaning the non-right angle angles), not just a handful which happen to have integer ratios of legs and hypotenuse. And it's a theorem, not a table.

        • Re:Archeology is a fine job (Score:5, Funny)

          by jabuzz ( 182671 ) on Wednesday August 04, 2021 @01:12PM (#61656153) Homepage

          Yes, but we have known that for centuries.

          • Then why do we still call it Pythagorean theory? We stopped calling Pluto a planet. It should be changed.

            • Re: (Score:2)

              by Opyros ( 1153335 )
              There are a lot of theorems which are named for the wrong person [wikipedia.org].

            • He was attibuted historically to methods pertaining to it and it somehow stuck, despite other matematicians also being named at the same time. The exact reasons are probably lost in the mists of time but likely due to poor understanding of the origins. Remember that much of knowledge transfer was an oral tradition at this time so poor accuracy is to be expected.

              More importantly, what would a renaming accomplish? And what should we rename it to that makes sense?

              No, that would only cause years, if not decades

        • Re:Archeology is a fine job (Score:5, Interesting)

          by Aighearach ( 97333 ) on Wednesday August 04, 2021 @01:34PM (#61656219)

          Having a table is something you can do without the theorem. You don't even need math, you just need to be able to measure. You only need the ability to measure, and an understanding the proportions will scale.

          Same for the "trigonometric" tables. It doesn't imply any understanding of trigonometry. It shows that the problem of measurement was understood to be useful and repeatable. And indeed, without that understanding the math would never have been invented.

          • Re: (Score:1)

            by rtb61 ( 674572 )

            It showed much more than that. It showed the accepted legality of it. Put into clay, accepted by the justice system of the day, the land defined by the clay tablet using uniform measurement and calculation system in association with language. Applied mathematics and the legal description of land. Recognised by that society as a whole. Transfer of title.

            There would have been systems of justice and fraud all associated with those clay tablets, rather than might equals right. I own the land because I can pay

            • It doesn't show any of that. And that's really a wildly stupid claim.

              You go from having one example of something, to "recognized by that society as a whole."

              So if I hit you over the head with a clay tablet, no crime, the legitimacy is recognized by society as a whole. Because there is an example of it.

        • Almost certainly. the Pythagorean theorem is very easy to derive. It's just named after him because there's a relatively solid continuity from the roots of Western Philosophy to the modern world.

        • But does this mean Pythagorean theorem was understood by people before Pythagorus

          No, it means that Pythagoras discovered time travel!

          But he could not fit the time travel theorem in the margins of the paper he was writing . . .

        • It is known among anthropologists and archaeologists that Pythagoras was taught in Egypt before returning to Greece, so yes, it was known prior.

          • Huh? You think he couldn't come up with a new idea on his own?

            And how is it that anthropologists--who study people, particularly those in "primitive" tribes--discovered facts about Pythagoras' travels? It doesn't take an archaeologist, either; the story of his time in Egypt is contained in later Greek writings, not in any archaeological evidence. And those stories, which date from around a century after Pythagoras (https://talesoftimesforgotten.com/2021/03/24/did-pythagoras-study-philosophy-in-egypt/), m

        • But does this mean Pythagorean theorem was understood by people before Pythagorus

          Yes, for 1500 years before. Try reading an actual history book.

          • Try citing one. So-called Pythagorean Triples were known much earlier; the Pythagorean Theorem, possibly a little earlier, but not 1500 years.

        • Re: (Score:2)

          by chill ( 34294 )

          From the article:

          Though Plimpton 322 and Si.427 both use Pythagorean triples, they predate the Greek mathematician Pythagoras by more than 1,000 years.

          Wikipedia says...

          Pythagoras is traditionally thought to have received most of his education in Ancient Egypt, the Neo-Babylonian Empire, the Achaemenid Empire, and Crete. Modern scholarship has shown that the culture of Archaic Greece was heavily influenced by those of Levantine and Mesopotamian cultures. Like many other important Greek thinkers, Pythagoras was said to have studied in Egypt.

          Why is it called "Pythagorean Triples" if it was known for so long before Pythagoras? Probably because Pythagoras was well published, and from the Western perspective a lot of knowledge went Greece --> Rome (huge empire) --> West.

          • It could also just be that Pythagoras publication is the only work to still survive to this day. Usually when a culture is conquered, the first thing the conquering culture does is wipe out the history of the conquered. Often that includes burning libraries. Islam did this quite a bit in the middle east, and Christianity did the same to the Mayans, Aztecs, and others.

            • Usually when a culture is conquered, the first thing the conquering culture does is wipe out the history of the conquered.

              Your assumptions are going to be so fucked if you ever find out about the Abbasid Caliphate's House of Wisdom [wikipedia.org].

          • Why is it called "Pythagorean Triples" if it was known for so long before Pythagoras?

            Because this tablet was found like a hundred years ago, and it's not even signed. They're not going to rename Pythagorean triples, especially if they don't know *what* you should call them instead. "Anonymous Babylonian's triples"? Doesn't really have the same ring to it.

    • That sounds way cooler than anything I did this past year.

      • "That sounds way cooler than anything I did this past year."

        It is! I mean, how cool is that?

        Better still would be if on the backside there'd be: 'Teacher Ugh is a moron!'

    • Re: (Score:1)

      by Tablizer ( 95088 )

      4000 years from now an archeologist may find my homework and write an article: "Pre-Fusion People Were Dumbasses".

  • Probably still contested in courts back then.

    • Probably still contested in courts back then.

      Royal: "This land is my land"

      Courtier: "This land is your land"

      Underling: "This land was made for you and me!"

      Monarch: "Off with his head!"

  • The Babylonians used a base 60 number system – similar to how we keep time today – which made working with prime numbers larger than five difficult.

    Can anyone tell me why this is so ?

    • Re:Base number 60 makes prime numbers 5 hard (Score:4, Interesting)

      by alvinrod ( 889928 ) on Wednesday August 04, 2021 @01:02PM (#61656115)
      60 is a good base because it's evenly divisible by a lot of small numbers. You can get up to 7 before you have to worry about any kinds of fractions. That makes it particularly easy to divide quantities up among small groups or for simple everyday problems. It's similar to why the international system has weird measures like 12 inches per foot. Makes it easy to express a half, a third, and a forth of a foot as an integer value of inches. It's also really easy to count to 60 on your fingers. Spend a little time looking at your hands to see if you can figure out how they did it. It's really neat the first time you see it.

      • Re: (Score:1)

        by Tablizer ( 95088 )

        I think the question was about "which made working with prime numbers larger than five difficult", not about using base 60 itself.

        By the way, base 12 is the ideal in my opinion. It divides by 2, 3, 4, and 6. Using base 60 gets you "5" also, but at the expense of "too many digits". The cost of obtaining 5 is not worth it.

        Base 10 sucks. It was chosen because it made finger-based-counting easier. Mother Nature/God should have given us 6 digits on each hand. That and the scrotum are bigly mistakes (Other animal

        • Re:Base number 60 makes prime numbers 5 hard (Score:4, Informative)

          by alvinrod ( 889928 ) on Wednesday August 04, 2021 @03:46PM (#61656603)
          Base 12 counting is easy on fingers. Look at your fingers and notice that your four non-thumb fingers have two joints that segment each finger into three parts. You have four such fingers for a total of twelve distinct segments which you can position the thumb of the same hand onto to keep count. The other hand can keep count of twelves either using fingers or a similar system for a true base-12 system.

          • Re: (Score:1)

            by Tablizer ( 95088 )

            Ancient accountants would get Carpel Tunnel doing that

            • Is a pretty good approximation for angles below 45 degrees. Pilots used to use it all the time for wind vectors. So maybe base 60 helped them?

          • Thanks, I once read a reference to that but it wasn't clear how it was done, notably I using the thumb to indicate/hold is neat. Note that with 2 hands, it's easy to count to 99, fingers on one hand are digits, thumb is 5, other hand fingers are 10 and that thumb is 50 (maybe like an abacus). Works great in the decimal system. I used it a lot as a cash strapped student to know if my groceries were going to break the budget before getting to the cash register, to avoid the embarrassment of having to put back

        • Somewhere I read a memorandum from God to Himself, which being translated said something like "Shortened Adam's jaw. Remember to remove the back tooth."

        • BTW, there are societies (the Mayans are perhaps the most famous, although the Celts may have done so too) that use base 20. You can guess why. I don't know that it offers any advantages over base 10 when it comes to fractions, though.

          • The mayans had a mixed system. Alternating between 5 and 20.

            • Re: (Score:1)

              by Tablizer ( 95088 )

              > The mayans had a mixed system. Alternating between 5 and 20.

              That made Armageddon prophesies hard to schedule. We gotta know when to run up our credit cards.

      • the international system has weird measures like 12 inches per foot.

        Can you point me to the version number of the Systeme Internationale which includes either the foot or the inch? Of the 15 countries on 4 continents that I've worked, the only ones who used feet were Americans. Even their neighbours don't use feet. Even the technical pinnacle of America doesn't use feet and inches.

      • divisible by not just small numbers -- all the small primes: 60 = 2^2 * 3 * 5. ie Factorizing n gives a complete left sequence (from 2 to 5); we can extend this to the next bigger as 60*7=420. Don't know what they call it - i guess some kinda highly composite number. Reason why a minute has 60 seconds (an hour, 60 mins).

    • I've heard that base 12 was common in the Middle East because people would hand-count by running their thumb over the three joints (or phlanges) of each finger (rather than simply extending each finger).

      Whether or not this is true, it is still interesting that people count in different ways (see also Inglorious Bastards).

      • Hard to say. The Romans used base 10 and base 12 for fractions.
        Attempts at determining why may be us trying to impose our culture and way of thinking upon them.

        But ya, I agree. It's pretty cool.

    • Which part are you asking about? Why they used base 60, or why it makes prime numbers difficult?

    • Re:Base number 60 makes prime numbers 5 hard (Score:4, Informative)

      by idji ( 984038 ) on Wednesday August 04, 2021 @02:07PM (#61656347)
      60 has 10 factors: 30,20,15,12,10,6,5,4,3 and 2. This means can do a lot of division in base 60 without dealing with decimals.
      10 has 2 factors: 5 and 2.
      60/3=20, but 10/3 is 3.33, which the Babylonians couldn't handle.
      Base 60 handles the primes 2, 3 and 5 well, but Base 10 only handles primes 2 and 5 well. Base 60 can't handle 7, 11, 13 etc well (neither can Base 10). This made working with prime numbers larger than 5 difficult because you could not divide by them (until we invented decimals millenia later...)

    • Simple answer: it isn't difficult :P

  • It's on a Babylonian tablet (Score:3)

    by 93 Escort Wagon ( 326346 ) on Wednesday August 04, 2021 @12:48PM (#61656065)

    The tablet is Babylonian, and it was discovered in Iraq. Why is the nationality of the mathematician important? The article doesn't say "crikey!" even once.

    • haha, are you joking or serious?

      The civilization a mathematician was a part of is certainly important, it shows level of their accomplishments and learning.

      Of course, Babylon, the city that became capitol of empires twice, is just south of Baghdad, so yeah your Babylonian archaeological finds will often turn up in Iraq, who'd have thunk it?

    • There is no need for math to be involved in constructing a table of measurements. What it shows is that they understood the problem that the math would later solve, and they had an empirical solution.

      "Applied geometry" in this case doesn't mean the math field, it means applied, as in used for construction, and literal geometry, geo-measuring.

    • Re: (Score:2)

      by hey! ( 33014 )

      If you can tell from context that this tablet was written by a Babylonian, that tells you the Babylonian civilization at this time possessed this knowledge. Many other Near Eastern civilizations were at some point in history important centers of mathematics, like Egypt or Greece.

      This is a baby step towards discovering the early history of mathematics. We may someday be able to trace the origins of the *very* advanced mathematics we see from Archaic Period Greeks. When I learned about geometry some forty y

      • Right, agreed on all points - but why does it matter that the mathematician was Australian?

        There were only two mathematicians mentioned in the article - the Australian dude, and Pythagoras. Obviously I wasn't talking about Pythagoras.

        • Re: (Score:3)

          by hey! ( 33014 )

          Oh, I assumed you were talking about the *other* end of the stick. No, the nationality of the researcher doesn't matter, but it might be nice to acknowledge his institution, UNSW, especially as the same team earlier discovered the earliest trigonometric table -- also cuneiform and quite neat and exact in the sexigesimal system.

    • The only thing that the article tells us is that the tablet was written using the cuneiform writing system. That doesn't say which language it was written in, nor anything about the nationality of the tablet writer (or the intended tablet reader - not necessarily the same person).

      The cuneiform writing system was used to write at least 4 languages, as different nationalities conquered Mesopotamia, in the same way that with my QWERTY ("Latin" writing system) keyboard, I can write in Spanish, French and Germa

      • If you want an R2L writing system, there are other choices besides the Arabic languages: Persian/ Farsi, Urdu, Pashto, western Punjabi, all written with a Perso-Arabic script; or Syriac or Hebrew or Dhivehi, each written in their own script (the Syriac script resembles Arabic, and Thaana, the script used for Dhivehi, derives in part from the Arabic script).

        • ... none of which languages have significant involvement with the industry I work in. (The Swahili is also work-related, and the French, Spanish and German have all proved useful at work in the past.)

  • and most people still can't tell you what a hypotenuse is.

  • 3700-years-old Tablet (Score:3)

    by grumpy-cowboy ( 4342983 ) on Wednesday August 04, 2021 @01:29PM (#61656205)

    Can I run Linux on it?

  • More Triplets mentioned (Score:4, Informative)

    by 140Mandak262Jamuna ( 970587 ) on Wednesday August 04, 2021 @03:23PM (#61656541) Journal

    Si.427 contains three Pythagorean triples: 3, 4, 5; 8, 15, 17; and 5, 12, 13.

    This is the easiest right angle definition system for the ancients to use. Take a rope and make a loop 12 units long, mark off 3 and 4 units from the knot, stretch the loop into a triangle, you have a right angle measuring device. Can be constructed of any size, larger loops are more accurate. Using these rectangles they can laydown marker stones. Accurate land ownership records can be maintained. Quite ingenious.

    Egypt with annual Nile flooding erasing all boundary markers should have invented this system even earlier. Absence of evidence is not evidence of absence.

  • The folks here aren't very fond of religion and rightfully so as it applies to information, especially as it applies to hidden information of the past. The deeper we dig, the more stuff keeps appearing that we were told all along ancient peoples didn't have the ability to produce or the brain pan to understand. Here we are living in the 5th age thinking we're the shit, the bees knees, top of the heap. Others came before us at least as "advanced" as we are.
  • I assume the next logical step will be for Todd Howard to develop a way to run Skyrim on it.

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