Sweet Square Dancing Jesus: The Archimedes Palimpsest and Egyptian Fractions! :o

As far as I can tell this hasn't been widely publicized.

Only the part about the "method of exhaustion" has.
But part B of the Archimedes Palimpsest has solutions and proof
of Irrational Numbers in calculus with fewer steps and greater accuracy.

E.J. Dijksterhuis included Heiberg's view in the 1987 "Archimedes" biography published by Princeton Press, The discussion begins with an Archimedes Lemma: In Quadrature of the Parabola Archimedes proves the following proposition on the sum of a geometrical progression with a common ratio of 1/4.

Given a series of magnitudes, each of which is equal to four times the order of the next, all of the magnitudes and one-third of the least added together will exceed the greatest by one-third.

Let the magnitudes A, B, C, D, E be given such that

A + B + C + D + E + 1/3E = (4/3)A

Dijksterhuis wrote out the 1/4 geometric infinite series:

4A/3 = A + A/4 + A/16 + A/64 + ...

an infinite series.

Heiberg published a finite Egyptian fraction series side of the discussion, as Dijksterhuis wrote as:

4A/3 = A + A/4 + A/12

that proved the accuracy of a finite 1/4 geometric series method that followed Eudoxus that used the same tradition.

planet math: Archimedes

/mind blown

/reexamine own research

/find answers everywhere.

/run around in ellipses, run around in ellipses, fall down giggling




Like consider the Metatonic cycle.

Earth orbit 365.256363004
. Moon orbit 29.530589

19 years = approximately 235 full lunations.

But there is always that pesky non terminating fraction left over.
Let's use a derivative of the golden mean and see if we can generate
the synodic lunar orbit to any significant degree of accuracy.

365.256363004 * 19 = 6939.870897076


divided by (235 + (inverse of golden mean / 100))


6939.870897076
/235.006180339
= 29.530588885
= 29.530589 Moon orbit according to wikipedia


David Grouchy

S&F for the "Sweet square dancing Jesus" - that made me LOL.

Otherwise, I have no idea what you are talking about. I am right-brain dominant.

I also gave you a star for "Sweet Square Dancing Jesus",
same as last poster I have no clue as to what you are
writing about, I probably wouldn't know if you explained it.
Sorry.

Originally posted by AwakeinNM
Otherwise, I have no idea what you are talking about. I am right-brain dominant.

In the old method of math an answer was supposed to reduced to a single decimal form.

4/3 (four thirds) would be written as

1.333333 repeating infinitely

But all those non terminating fractions can be confusing to students of algebra.

Recursive fractions, on the other hand are things like

1 + 1/4 + 1/12

So that if either 1/4 or 1/12
appears on either side of the = sign in an algebraic equation it is easier to recognize it and eliminate.

More examples of Egyptian style recursive fractions compared to standard fractions.




[atsimg]http://files.abovetopsecret.com/images/member/aaf61e1c5435.png[/atsimg]




In all my years of studying astronomy and the solar system,
I never noticed the appearance of the inverse of the golden mean
as the remainder between exactly 19 years and 235 exact solar lunations.

This stuff is nitrous oxide
with ground effect lights at the car show.
Cause after a long night of square dancing, he says "pimp my ride."


David Grouchy

Originally posted by crazydaisy
I have no clue as to what you are
writing about, I probably wouldn't know if you explained it.
Sorry.

Incommensurate Numbers
are fractions that repeat forever, like Pi.

Archimedes knew how to solve that problem.

... and my mind is blown too.


David Grouchy

OP appears to be providing us plebs with information to the effect, in summary, that ancient peoples had mathematics and deep thought on a par or better than some of what we know today. However, OP has , fairly , refused to dumb down, and speak english, in the hopes that proper math whizzes will see the article and salivate... and perhaps go mad...

Myself, I am only just barely able to decipher this meaning from the article, and have no idea of the exact implications .

Originally posted by TrueBrit
However, OP has , fairly , refused to dumb down, and speak english,

Ah, that is easy enough to put into english.

In school we were taught to add and subtract.
And then there were the multiplication tables.

Did anyone study division tables?

That's all this is about.


David Grouchy


But the effect is like discovering we have a second arm,
when we have been living in a world that only uses one arm to do everything.
Both arms together are what really let the square dancers get swinging about to the beat!

reply to post by davidgrouchy


Please do not misunderstand me. I believe that you are better off avoiding dumbing the information down. Those who will be able to benifit from it will understand it, and those who wont benifit from having it wont have a clue what it means in either case. Such is the way of mathematics so far as I can judge. Oddly, mathematics has never seemed logical enough for me to follow, so I have no use for it. However, you show me an application of this which actualy benifits man, and I will go crazy as if you had just shown me a new way to build mortice locks (Im a locksmith) .

Originally posted by TrueBrit
I will go crazy as if you had just shown me a new way to build mortice locks (Im a locksmith) .

Yes,
Start with the spaces between the coils of the return spring,
and design the lock backwards from there.


David Grouchy

reply to post by davidgrouchy


Damn. LOL, wanna tell me how better to forecast hotel occupancy in a hotel that hasn't even opened yet, in a down economy?

This is a good thread. I don't fully understand the implications of this, as I have not spent much time perusing the math of alchemy.

Having said that, it is very obvious that there is something else tangible here: that you don't need multimillion dollar equipment to unravel the nature of reality. You just need to sit outside, observe nature, and meditate deeply on what you see.

Originally posted by bigfatfurrytexan

Damn. LOL, wanna tell me how better to forecast hotel occupancy in a hotel that hasn't even opened yet, in a down economy?

Ah, you got me there Texan.

I haven't done any work looking at
where the visitors are comming from
and how they get their information on where to stay.

Does someone at the airport or bus termial tell them?
Does the gps in their car tell them?
Do visitors stay somewhere traditional that their company/family has "always" used?
Are the people in town all excited about the place and "sending people" over?
It seems to be a question of expectations.

Add to that the variable of a "down economy" and I'm way out of my depth.
I have no idea how that affects the guests.

Hell it could be a question of having a big ole sign saying "free wi-fi".
I have no clue.


David Grouchy

reply to post by davidgrouchy


Yes to all of those, and other factors, too. Hotel forecasting is all about market analysis using cold hard numbers and gut feeling. And always hedging your bets because the forecasts can never account for real world happenings, like a freak ice storm.

I was just jesting with you, of course. Forecasting is forecasting, no matter where you are at. Who ever can extrapolate past data to create models of future data the best usually is copied. I have done it for three different industries, and it is always the same.

Just finished a preliminary analysis of the solar system.
Here are the Archimedes style formulas to get their orbital periods.


Mercury (9 +3/8)^2
Venus (14 +1/1)^2
Earth (19 +1/9)^2
Mars (26 +1/5)^2

Ceres (40 +1/1)^2

Jupiter (65 +4/5)^2
Saturn (104 +1/9)^2
Uranus (175 +1/2)^2
Neptune (245 +1/3)^2



out of 109,197.710156 total cummulative days for all orbital periods combined I get an error of 13.550595444444 days using the formulas above.
An error rate of 0.00012409230

Which compared to the Titus Bode law
an error rate of 0.88564294631

is a seven thousand fold improvement in accuracy.


/Grouch goes to make himself a sandwich

David Grouchy

reply to post by davidgrouchy


LIke isaid, i don't fully understand the implications here (and it looks like you are just starting to understand them properly)...

....but could this mathematical system be something that might bring a closing of the "as above, so below" maxim to the quantum theory/general relativity rift?

Any decimal representation can be rewritten as an infinite sum of fractions, this is not news at all.

3.1416 = 3 + 1/10 + 4/100 + 1/1000 + 6/10000

An arbitrary infinite sum sometimes equals a singular value that we can represent in other ways, but sometimes the infinite sum goes on an on, never converging on a concrete value.

en.wikipedia.org...(mathematics)


Why, when we write out a decimal representation of Pi, does it always have to end in "..." ? Because any decimal representation, and therefore any sum of infinite fractions, never exactly equals Pi itself. We can only ever get an approximation in this form.

en.wikipedia.org...

If you don't believe me, please be more specific about your representation of the golden ratio, or try to come up with a finite description of the value of Pi.

Originally posted by davidgrouchy

Just finished a preliminary analysis of the solar system.
Here are the Archimedes style formulas to get their orbital periods.


Mercury (9 +3/8)^2
Venus (14 +1/1)^2
Earth (19 +1/9)^2
Mars (26 +1/5)^2

Ceres (40 +1/1)^2

Jupiter (65 +4/5)^2
Saturn (104 +1/9)^2
Uranus (175 +1/2)^2
Neptune (245 +1/3)^2



out of 109,197.710156 total cummulative days for all orbital periods combined I get an error of 13.550595444444 days using the formulas above.
An error rate of 0.00012409230

Which compared to the Titus Bode law
an error rate of 0.88564294631

is a seven thousand fold improvement in accuracy.


/Grouch goes to make himself a sandwich

David Grouchy

I can provide you with a much more accurate way of determining the period of planetary orbits:

en.wikipedia.org...

Using Newtonian physics, our most exacting mathematical methods were able to determine the position of mercury so precisely that when we noticed a small discrepancy on the order of 3 arcseconds, it was only later explained by Einstein's theory of general relativity and is now seen as one of the slam-dunk confirmations of his theory. When we take general relativity into account, that discrepancy is predicted and explained by theory.

en.wikipedia.org...

Knowing the masses of the planets and sun, and the positions & velocities of the planets, we can use our modern physics to predict their positions at any arbitrary point in spacetime, past, present or future, to accuracies much greater than a single arcsecond.

Furthermore.

Let's look at your math.

The earth's orbital period is 365.242199 days (notice I include *units* here, which you should get in the habit of doing) (your number, I haven't checked it)

Let's call that value E. E=365.242199 days.
A moon orbit (your value) is 29.530589 days. M = 29.530589 days.

The golden ratio is commonly notated as the greek phi. Phi = 1.61803399... (this is not exact. Phi does not equal exactly 1.61803399. It goes on from there. We can only ever approximate it as a decimal expansion.)

The formula that you provide is :
E * 19 / (235 + (Phi^-1 / 100)) = M

We can greatly simplify this formula since it's mostly constants (dimensionless numbers)

E * 19/(235 + (Phi^-1)/100) = E * (19/2.35) * Phi = E * Phi * 19/2.35

Using your value for Phi, the above expression simplifies to just:
E * 0.0808489

Does E * 0.0808489 equal M?

Well, strictly speaking, no. E * 0.0808489 equals 29.5294.

M equals 29.530589. Again, these are your numbers.

So, strictly speaking, your math is wrong. On top of that, all of the quantities given above, whether the orbital periods of the Earth or of the Moon, or the decimal representation of the golden ratio, could be given to greater accuracy (more decimal places.) This is because they cannot be simply represented as a single ratio of two whole numbers (a fraction)

For instance, instead of saying Phi =~ 1.61803399, we could get more accurate and say

Phi = 1.61803398874

Strictly speaking, this would change our answers. And Phi can be expanded even beyond that, infinitely so. Here's a webpage showing Phi to 2000 decimal places:

www.maths.surrey.ac.uk...


So what, exactly, were you trying to show, again? That the earth's period, multiplied by an arbitrary number of your choosing, is close to the moon's period? I could have done the same thing in less steps and to greater accuracy... It doesn't prove anything physically.

Furthermore. The "magic" of your equation basically comes from introducing the arbitrary numbers 19 and 235, and the inverse of Phi divided by 100. What is the logic in introducing these numbers? How did you come across these numbers? Is there a general logic you could apply here to extend your system to other examples of orbital mechanics?

That is, what if I wanted to compare the orbital period of Mars to its moon's, Phobos's orbital period? Could I just take Orbit_mars * 19 / (235 + Phi^-1 / 100 ) and get Phobos's orbital period?

Not even close! Using your above formula,
Mars * 19 / (235 + (Phi^-1)/100) = 55.5409, which is a far cry from 0.31891023, the approximate value of Phobo's orbital period.

Which means we'd have to come up with a whole new set of arbitrary numbers to make the formula work.


This is not how physics gets done. If I wanted to, I could come up with a formula relating phobo's orbital period to the number of days bush spent in office:
Phobos (days) = 1.09143*10^-4 * Bush_years_in_office (days)

What if I wanted to shoehorn the golden ratio in there somewhere? No problem:

Phobos =~ Bush / Phi * 0.00017659731

Originally posted by wirehead
Furthermore.

Let's look at your math.


. . .

Using your value for Phi, the above expression simplifies to just:
E * 0.0808489

That's not my value for Phi.

For the golden mean I used the formula itself, non terminating fractions included.

(1 + square root of (5) ) divided by 2


David Grouchy

Originally posted by wirehead

So what, exactly, were you trying to show, again? That the earth's period, multiplied by an arbitrary number of your choosing, is close to the moon's period? I could have done the same thing in less steps and to greater accuracy... It doesn't prove anything physically.

Now that is a fair question!

I have no arbitrary number of my choosing, magic or otherwise.

I have a systematic method of analysis.
One that I can input any data for orbital periods into
and get simplified formulas out as our result.

For instance, notice that in your posts there is a different period given for the orbital period of the Earth than the one I listed in my previous posts. I just used the current periods as listed on Wikipedia for each body. As you may know these change. Both from year to year, and as more accurate observations are made.

But either way,
I could also run the analysis using any numbers anyone wishes.

It's "The Method" as described by Archimedes that I'm using,
and that I find so spectacular.

I can show my procedures and each step of my work,
but that's not what I think I am being asked about here.

More like I am being cautioned against believing that any
single formula that is output by the method is special in it's own right,
or has any tangible representation of reality itself.

And on that note you and I are in complete agreement.

I analysed from powers 1 to 5, numerators and denominators of single digit,
and found that the formulas listed above for each planet and asteroid
are the most accurate of any possible combination within that set.

Additionally the analysis isn't complete.
It only contains one fraction in each formula.
Not the full recursive set as used by the Egyptians.

I expect the results to be an order of magnitude more accurate, if not near perfect,
once complete.


David Grouchy