Pythagorus theorem and Pothayanar theorem (tamil content) Someone translate it

by Kayalvizhi Sat Sep 08, 2012 12:24 am

Though not exact, this formula gives good results in a few cases I tested.

by Guest Tue Sep 11, 2012 12:38 am

8, 15, 17 - FAIL.

"Silly formula" was the response I got from someone when I tried to brag about the advanced civilization that I come from. Damn.

by Guest Tue Sep 11, 2012 12:41 am

blabberwock wrote:...when I tried to brag about the advanced civilization that I come from. Damn.

by Kayalvizhi Tue Sep 11, 2012 2:18 am

blabberwock wrote:8, 15, 17 - FAIL.

"Silly formula" was the response I got from someone when I tried to brag about the advanced civilization that I come from. Damn.

What are you talking about?

(15 - 15/8 ) + 8/2 = 13.125 + 4 = 17.125. Very close to the exact answer 17.

I said in my original post "Though not exact, this formula gives good results in a few cases I tested." True in this example too

by Guest Tue Sep 11, 2012 3:03 am

Kayalvizhi wrote:I said in my original post "Though not exact, this formula gives good results in a few cases I tested." True in this example too

this formula is purely by trial and error. many such formulas can be devised by studying loose relationships between the sides to get close but inexact results. there is no formal proof for this equation and that is all that matters in mathematics!

by Guest Tue Sep 11, 2012 3:19 am

Kayalvizhi wrote:

I said in my original post "Though not exact, this formula gives good results in a few cases I tested." True in this example too

16, 63, 65?

by Guest Tue Sep 11, 2012 3:35 am

160, 630, 650 (pythagoras)
160, 630, 631 (pothayanar)

error = 19

larger the numbers, larger the margin of error.

by Guest Tue Sep 11, 2012 4:04 am

Did more analysis on this (the pride of my civilization is at stake).

It seems that for certain ratios of a and b, this formula yields reasonable results.

More on this later perhaps.

by Guest Tue Sep 11, 2012 4:36 am

blabberwock wrote:Did more analysis on this (the pride of my civilization is at stake).

lolu

by MaxEntropy_Man Tue Sep 11, 2012 6:46 am

what is being asserted is that

Sqrt[(1+b^2/a^2)] ~ [(7/ Cool

+(b/2a)]

if it works, some kind of power series approximation is at work. didn't have the time to work it out yet.

edited post kapasi's suggestion:
Sqrt[(1+b^2/a^2)] ~ [(7/8 )+(b/2a)]

by MaxEntropy_Man Tue Sep 11, 2012 6:48 am

i typed seven divided by eight and gave me a smilie! what on earth is going on?

by Guest Tue Sep 11, 2012 6:54 am

"8 )" (w/o spaces) is a shortcut for that smilie. use a space between 8 and ).

by Jeremiah Mburuburu Tue Sep 11, 2012 11:35 am

blabberwock wrote:Did more analysis on this (the pride of my civilization is at stake)...

given your initial reaction to this topic, no one should put you in charge of redeeming the pride of your civilization; focus on extricating f from m. you might also bear in mind that approximations make little or no sense to ten-year-olds.

by Jeremiah Mburuburu Tue Sep 11, 2012 11:41 am

Huzefa Kapasi wrote:160, 630, 650 (pythagoras)
160, 630, 631 (pothayanar)

error = 19

larger the numbers, larger the margin of error.

note that the percent error in this case is (only) about 3%.

by Marathadi-Saamiyaar Tue Sep 11, 2012 11:44 am

Jeremiah Mburuburu wrote:
Huzefa Kapasi wrote:160, 630, 650 (pythagoras)
160, 630, 631 (pothayanar)

error = 19

larger the numbers, larger the margin of error.

note that the percent error in this case is (only) about 3%.

Agree... i noticed the error is within the allowable limits for most designs and experiments.

OTOH, I am sure Swami BOTHAIyanaar could have done better if he was simply pOthaiyanaar.

Is Pythogarus a reincarnation of pOthaiyanaar? Strange the name sounds similar. Has it been included in "Vedic Mathematics"?

by Idéfix Tue Sep 11, 2012 11:47 am

When did he come up with this formula?

by Guest Tue Sep 11, 2012 12:32 pm

panini press wrote:When did he come up with this formula?

2000 BC. (as i cursorily read on the net -- i may be wrong.) the formula is encrusted in a tamil poem of that era, probably sangam lore. pothayanar was a poet.

by Hellsangel Tue Sep 11, 2012 12:40 pm

Jeremiah Mburuburu wrote:
blabberwock wrote:Did more analysis on this (the pride of my civilization is at stake)...
given your initial reaction to this topic, no one should put you in charge of redeeming the pride of your civilization; focus on extricating f from m. you might also bear in mind that approximations make little or no sense to ten-year-olds.

The old dog's sarcasm-meter(notice the hyphen) - FAIL

by Guest Tue Sep 11, 2012 12:45 pm

Jeremiah Mburuburu wrote:
Huzefa Kapasi wrote:160, 630, 650 (pythagoras)
160, 630, 631 (pothayanar)

error = 19

larger the numbers, larger the margin of error.

note that the percent error in this case is (only) about 3%.

can you correct the 3% error without using squares and their roots? if the error is 3% can we not revise the formula to adjust it by 3%? speak up man!

by MaxEntropy_Man Tue Sep 11, 2012 1:18 pm

i think i figured a little bit of this out. first on the poem itself. nILAm = longer altitude (a), kundRam = shorter altitude (b), and karNam = hypotenuse (c).

so bascially what the poem asserts is:

c = Sqrt[a^2+b^2] ~ (a-a/8 )+ b/2

first rewrite the LHS in the above as,

a*Sqrt[1+(b/a)^2]

setting x = b/a, the exercise becomes one of obtaining the taylor series expansion of (1+x^2)^(1/2) near zero.

because i am lazy, i'll do this using wolfram alpha instead of by hand:

http://www.wolframalpha.com/input/?i=%281%2Bx^2%29^%281%2F2%29

wolfram gives the series as:

1+ x^2/2 - x^4/8 + O(x^5)

when x ~1, the first three terms give 1 + x^2/2 - x^4/8. the smaller x is than 1, the better this approximation works.

pOthayanAr's formula it seems to me crudely replaces x^2 and x^4 by just x which works better the closer x approaches 1. if you use the result with the first three terms obtained from the power series though, the error will grow as x^5.

by MaxEntropy_Man Tue Sep 11, 2012 1:25 pm

q to KV: when did pOthayanAr live?

by Marathadi-Saamiyaar Tue Sep 11, 2012 1:29 pm

MaxEntropy_Man wrote:i think i figured a little bit of this out. first on the poem itself. nILAm = longer altitude (a), kundRam = shorter altitude (b), and karNam = hypotenuse (c).

so bascially what the poem asserts is:

c = Sqrt[a^2+b^2] ~ (a-a/8 )+ b/2

first rewrite the LHS in the above as,

a*Sqrt[1+(b/a)^2]

setting x = b/a, the exercise becomes one of obtaining the taylor series expansion of (1+x^2)^(1/2) near zero.

......

pOthayanAr's formula it seems to me crudely replaces x^2 and x^4 by just x which works better the closer x approaches 1. if you use the result with the first three terms obtained from the power series though, the error will grow as x^5.

Yeah....exactly what I thought.... Wink

by MaxEntropy_Man Tue Sep 11, 2012 1:53 pm

i hope this is not someone with a knowledge of basic calculus and taylor series expansions fobbing off something as an ancient poem. however, i am not discounting the possibility that it is indeed an ancient poem because there is documented evidence (documented by prof rajiv sarada and others and discussed in this group by PP and others) that ancient malayALi mathematicians had known about power series. it is not a wild stretch from there to thinking about exchanges between malayALis and thamizhars who were basically the same people a long time ago.

by MaxEntropy_Man Tue Sep 11, 2012 2:04 pm

on second thoughts someone on the interwebs claimed that pOthayanAr lived in 2000 B.C. indian history often sounds like a fantasy especially because a lot this stuff is also related to cultural pride. that would put pO'r pre the first sangam. while that in itself sounds completely made up, the language used in the poem sounds quite modern. for example, there is no way i'd understand any of the sangam era poetry without the aid of some scholar's commentary. however, the language in this poem is very easy to understand because it sounds relatively modern, which is odd considering the claim that this is even older than the first sangam. something is definitely fishy. KV please clarify.

by Idéfix Tue Sep 11, 2012 2:13 pm

MaxEntropy_Man wrote:i hope this is not someone with a knowledge of basic calculus and taylor series expansions fobbing off something as an ancient poem. however, i am not discounting the possibility that it is indeed an ancient poem because there is documented evidence (documented by prof rajiv sarada and others and discussed in this group by PP and others) that ancient malayALi mathematicians had known about power series. it is not a wild stretch from there to thinking about exchanges between malayALis and thamizhars who were basically the same people a long time ago.

Yes, and that school was at its prime ~600 years ago. Before Newton and Leibniz, but nowhere as old as 2,000 BCE.

by Jeremiah Mburuburu Tue Sep 11, 2012 4:32 pm

Kayalvizhi wrote:

Though not exact, this formula gives good results in a few cases I tested.

it looks as if this approximation of the hypotenuse's length is within 10% of the correct value for all values of b/a from 1 to 19, where b = the larger leg's length, and a = the smaller leg's length. that's quite a wide interval. note that when "b/a" is less than 1, "larger leg's length" should be redefined. thus, the expression is useful for ratios of leg-lengths from 1/19 to 19/1.

moreover, the relative error is less than 3% for b/a values from 1 to 4; and less than 1% for b/a values between 1.18 and 2.85.

for example, for leg-lengths of 327.3 and 763.7, where b/a is about 2.33, the exact (pythagorean) hypotenuse length is about 830.880, while the approximate value according to the expression given in your post is 831.888, with a relative error of 0.121%.

thank you for your post.

by Jeremiah Mburuburu Tue Sep 11, 2012 4:36 pm

Huzefa Kapasi wrote:
blabberwock wrote:...when I tried to brag about the advanced civilization that I come from. Damn.

a moral of this story: someone is not right or sensible merely because s/he is equipped with The Thing.

by Jeremiah Mburuburu Tue Sep 11, 2012 4:41 pm

Huzefa Kapasi wrote:
panini press wrote:When did he come up with this formula?
2000 BC. (as i cursorily read on the net -- i may be wrong.) the formula is encrusted in a tamil poem of that era, probably sangam lore. pothayanar was a poet.

hahaha @"encrusted!" did you mean "embedded?"

by Hellsangel Tue Sep 11, 2012 4:43 pm

Jeremiah Mburuburu wrote:
Huzefa Kapasi wrote:
blabberwock wrote:...when I tried to brag about the advanced civilization that I come from. Damn.

a moral of this story: someone is not right or sensible merely because s/he is equipped with The Thing.

Another moral of the story: The old dog is as petty as ever.

by Hellsangel Tue Sep 11, 2012 5:10 pm

Yet another moral: Damn Tamils. Always claiming credit for other's work

by MaxEntropy_Man Tue Sep 11, 2012 5:22 pm

here is a plot showing two approximations to the function Sqrt[1+x^2], pOthayanAr and yours truly's taylor series expansion from which methinks pOthayanAr is derived. as you can see, close to 1, the agreement between pOthayanAr and your's truly is quite good.

Pythagorus theorem and Pothayanar theorem (tamil content) Someone translate it Pothay10

Pythagorus theorem and Pothayanar theorem (tamil content) Someone translate it Pothay10

by MaxEntropy_Man Tue Sep 11, 2012 5:31 pm

so the pome is a modern reconstruction of baudhayana (aka pOthayanAr)? i tend to believe that. the strongest evidence being i can read it and understand it without the aid of a commentary unlike ancient texts.

that maybe so, but b.wock shouldn't have been so dismissive only based on her dislike for the fish-eyed one.

by Idéfix Tue Sep 11, 2012 5:46 pm

MaxEntropy_Man wrote:so the pome is a modern reconstruction of baudhayana (aka pOthayanAr)? i tend to believe that. the strongest evidence being i can read it and understand it without the aid of a commentary unlike ancient texts.

that maybe so, but b.wock shouldn't have been so dismissive only based on her dislike for the fish-eyed one.

It could be a translation of Baudhayana's work. Perhaps the translation is from a few centuries ago, so you can read it unassisted.

by Guest Tue Sep 11, 2012 5:50 pm

can someone explain the baudhayana aka pothayanar part?

by MaxEntropy_Man Tue Sep 11, 2012 6:08 pm

Hellsangel wrote:Yet another moral: Damn Tamils. Always claiming credit for other's work

i re-read the thread carefully. nowhere did KV claim that the original author of said concept was tamil. that's a conclusion arrived at entirely by others including yours truly who was misled by subsequent posts. one could in principle read the poem as a modern reconstruction in tamil of baudhayana's original idea.

eta: so yet another moral is not to jump to conclusions too fast.

by Guest Tue Sep 11, 2012 6:12 pm

yours truly wants her questions answered too. the aka part of the whole thing.

by MaxEntropy_Man Tue Sep 11, 2012 6:15 pm

Natalia Romanova wrote:yours truly wants her questions answered too. the aka part of the whole thing.

which part of aka do you not understand? pOthaynAr is a thamizhization of baudayana.

by Guest Tue Sep 11, 2012 6:18 pm

MaxEntropy_Man wrote:
Natalia Romanova wrote:yours truly wants her questions answered too. the aka part of the whole thing.

which part of aka do you not understand? pOthaynAr is a thamizhization of baudayana.

really? how am i supposed to understand it anyway, so not intuitive. Didn't understand tarun to dharun either, and that was a simpler example. do u guys do it often? thamizhization? is that what made the famous dosai and sambar?

by Hellsangel Tue Sep 11, 2012 6:21 pm

MaxEntropy_Man wrote:
Hellsangel wrote:Yet another moral: Damn Tamils. Always claiming credit for other's work

i re-read the thread carefully. nowhere did KV claim that the original author of said concept was tamil. that's a conclusion arrived at entirely by others including yours truly who was misled by subsequent posts. one could in principle read the poem as a modern reconstruction in tamil of baudhayana's original idea.

eta: so yet another moral is not to jump to conclusions too fast.

And you, Il Professore, are not gullible enough to think Trollus Fishyeyeus will post anything not related to Tamil Glory.

by MaxEntropy_Man Tue Sep 11, 2012 6:22 pm

Natalia Romanova wrote:
MaxEntropy_Man wrote:
Natalia Romanova wrote:yours truly wants her questions answered too. the aka part of the whole thing.

which part of aka do you not understand? pOthaynAr is a thamizhization of baudayana.

really? how am i supposed to understand it anyway, so not intuitive. Didn't understand tarun to dharun either, and that was a simpler example. do u guys do it often? thamizhization? is that what made the famous dosai and sambar?

this is an interesting math discussion and i'm going to leave it that way. if you want to restart the usual trite nitpicking start another thread.

by Hellsangel Tue Sep 11, 2012 6:22 pm

MaxEntropy_Man wrote:
Natalia Romanova wrote:
MaxEntropy_Man wrote:
Natalia Romanova wrote:yours truly wants her questions answered too. the aka part of the whole thing.

which part of aka do you not understand? pOthaynAr is a thamizhization of baudayana.

really? how am i supposed to understand it anyway, so not intuitive. Didn't understand tarun to dharun either, and that was a simpler example. do u guys do it often? thamizhization? is that what made the famous dosai and sambar?

this is an interesting math discussion and i'm going to leave it that way. if you want to restart the usual trite nitpicking start another thread.

No no no! The nitpicking is for the old dog.

by Idéfix Tue Sep 11, 2012 6:37 pm

MaxEntropy_Man wrote:nowhere did KV claim that the original author of said concept was tamil.

Based on everything KV has ever posted, it was reasonable for others to assume that she was claiming this as a Tamil concept. I doubt KV posted this for its mathematical interest, and I am reasonably confident she did not know that this was originally written in Sanskrit. What she posted actually demonstrates the cultural links between Tamil Nadu and the rest of India -- something she'd rather minimize rather than highlight.

by Guest Tue Sep 11, 2012 7:00 pm

KV, you shouldn't have done this to me.

After I found out that the formula yields reasonable results for certain ratios of b and a, I had gone on and on about the greatness of our civilization and managed to make the "silly formula" person eat humble idli. And now I find that it is after all just a translation of a Hindian finding! Damn.

I am now too depressed to bother posting what I found out more about the accuracy of this formula.

by Guest Tue Sep 11, 2012 7:15 pm

Natalia Romanova wrote:can someone explain the baudhayana aka pothayanar part?

Nats, in tamil I think they have the same alphabet for 'pa' and 'ba' and also 'da' and 'tha' so there's always a confusion wrt these pronunciations. So, 'bau'/bo in baudhayana becomes 'po' and 'dha'yana becomes 'tha'yana. So, baudhayana is confused as pothayana. the 'r' at the end is there as a respect, i guess. Some tamils may read/pronounce it as 'bodhayana' (don't know if they can spell 'bau' in tamil).

Also noticed that kayal is confused on how 'Chidambaram' is pronounced. She calls him 'Chitamparam'. Not her fault. It's because of the limited letters in the language and using the same letter for different sounds. It must be confusing.

by Guest Tue Sep 11, 2012 7:19 pm

thanks kinns... max professes love for language and semantics. could have simply said the same.

by Guest Tue Sep 11, 2012 7:24 pm

lol

(19:21:27) seven: teddy bear in tamilnadu is daddy bear?
(19:21:30) seven: or daddy peaR?
(19:21:52) kinnera: pa and ba are the same
(19:22:02) seven: ahan
(19:22:04) kinnera: daddy can be paddy

by Guest Tue Sep 11, 2012 7:29 pm

Natalia Romanova wrote:lol

(19:21:27) seven: teddy bear in tamilnadu is daddy bear?
(19:21:30) seven: or daddy peaR?
(19:21:52) kinnera: pa and ba are the same
(19:22:02) seven: ahan
(19:22:04) kinnera: daddy can be paddy

and pear can be bear, i guess Razz

by MaxEntropy_Man Tue Sep 11, 2012 7:36 pm

Natalia Romanova wrote:thanks kinns... max professes love for language and semantics. could have simply said the same.

the reason i didn't was because i'm already all posted out about that. there's a saying tamil for something like this -- to grind the flour that has already been ground. short summary -- there is a certain minimalism in the tamil alphabet. however, there is an internal consistency in tamil phonetic rules which can be used to check against phonetic mistakes. not knowing them and internalizing them well often leads to bizarre results of the chitamparam kind and the ensuing puerile mirth-making by non-tamilians. there is nothing more to say.

by Guest Tue Sep 11, 2012 7:47 pm

MaxEntropy_Man wrote:
Natalia Romanova wrote:thanks kinns... max professes love for language and semantics. could have simply said the same.

the reason i didn't was because i'm already all posted out about that. there's a saying tamil for something like this -- to grind the flour that has already been ground. short summary -- there is a certain minimalism in the tamil alphabet. however, there is an internal consistency in tamil phonetic rules which can be used to check against phonetic mistakes. not knowing them and internalizing them well often leads to bizarre results of the chitamparam kind and the ensuing puerile mirth-making by non-tamilians. there is nothing more to say.

ya, up until then i thot those are two different people. wonder how u didn't catch the unusual sounding name (for a tamil) before. Or is the name common now?

by MaxEntropy_Man Tue Sep 11, 2012 8:03 pm

i stand corrected! on second thoughts, i misstated the baudayana approximation to pythagoras. i.e. baudayana's approximation to Sqrt[1+x^2] is NOT 1+(x/2-x/8 ), i.e. 1+3x/8, but 7/8 + x/2 which is much much better and over a wider range of x!

see here (the blue is the pythagorean function Sqrt[1+x^2] and the magenta is baudayana (7/8+x/2):

Pythagorus theorem and Pothayanar theorem (tamil content) Someone translate it Pothay11

Pythagorus theorem and Pothayanar theorem (tamil content) Someone translate it Pothay11

now i am truly mystified. i don't think it comes from a power series as i had originally and mistakenly imagined.

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