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

Originally posted by wirehead

Using Newtonian physics, our most exacting mathematical methods were able to determine the position of mercury so precisely that [color=salmon]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.

Yes, I heard that one too.
But has anyone actually ever looked into it.
To see if there is something to the method,
to use it as an aid to doing one's own research?

I did.
It's propaganda.
But don't take my word for it.

Take Encyclopedia Britanica's word on the reality of the situation.

The Actual Motion of The Moon

In the late 19th century, Hill developed methods with which results of given accuracy could be achieved with much reduced labor. These methods have been effective in the discussion of the general prblem of three bodies (wikipedia: Three Body Problem) as well as in the practical solution of the lunar problem. The two principal features were (1) the introduction of the value of the ratio of the mean motions of the Earth and Moon from the outset to avoid convergence difficulties and (2) the use of a rotating rectangular coordinate system. Hill developed a rather elegant method of determining the motion of perigee that was also applied by John Couch Adams to the motion of the node. Hill's methods were used by Brown to develop the lunar theory that has been generally used to compute the coordinates of the Moon since 1923, although the theoretical motions of node and perigee were still unsatisfactory and were replaced by the observed values.

Brown's series contain about 1,500 terms, five times as many as in Hansen's Theory. The evaluation of these series is very laborious, and Brown, like some of his predecessors, prepared tables for the simplification of this work. They filled 650 quarto pages, and [color=gold]one man working alone could extract the coordinates just fast enough to keep up with the real Moon. The advent of the digital electronic computer has relieved this burden and also made feasible the direct evaluation of the series.

Encyclopedia Britanica, Inc
Volume 12
Page 417
( c ) 1980

Try not to be so dismissive.
Some of us are actually trying to advance science.
But just because the text books tend to act like all the
interesting problems have already been solved, doesn't
mean that there isn't a galaxy sized opening for improvements.


David Grouchy

The trouble with these math threads is that people like me can never tell if these guys are arguing or not.

Originally posted by davidgrouchy

Yes, I heard that one too.
But has anyone actually ever looked into it.
To see if there is something to the method,
to use it as an aid to doing one's own research?

I did.
It's propaganda.
But don't take my word for it.

Take Encyclopedia Britanica's word on the reality of the situation.

The Actual Motion of The Moon

In the late 19th century, Hill developed methods with which results of given accuracy could be achieved with much reduced labor. These methods have been effective in the discussion of the general prblem of three bodies (wikipedia: Three Body Problem) as well as in the practical solution of the lunar problem. The two principal features were (1) the introduction of the value of the ratio of the mean motions of the Earth and Moon from the outset to avoid convergence difficulties and (2) the use of a rotating rectangular coordinate system. Hill developed a rather elegant method of determining the motion of perigee that was also applied by John Couch Adams to the motion of the node. Hill's methods were used by Brown to develop the lunar theory that has been generally used to compute the coordinates of the Moon since 1923, although the theoretical motions of node and perigee were still unsatisfactory and were replaced by the observed values.

Brown's series contain about 1,500 terms, five times as many as in Hansen's Theory. The evaluation of these series is very laborious, and Brown, like some of his predecessors, prepared tables for the simplification of this work. They filled 650 quarto pages, and [color=gold]one man working alone could extract the coordinates just fast enough to keep up with the real Moon. The advent of the digital electronic computer has relieved this burden and also made feasible the direct evaluation of the series.

Encyclopedia Britanica, Inc
Volume 12
Page 417
( c ) 1980

This is referring to the motion of the moon, not of mercury... Newtonian physics is mostly fine for predicting the position of the moon, though as the article states, some of the aspects of the moon's orbit were better determined ad-hoc by observation.

General relativity steps in in more extreme cases where relativistic effects become apparent. Mercury, being closer to the sun, is a more accurate detector of gravitational forces.

Originally posted by davidgrouchy
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.

Please, elaborate on this system. What are you trying to show in your equations? I caution that I've not read any of your other threads. Just break it down for me simply.

What is the relationship between orbits that you are attempting to show? Are you trying to predict orbital periods? Draw connections between orbital periods of different bodies?

You say you can "run the analysis using any numbers anyone wishes"

What anaylsis, exactly? You're recasting individual numbers in different forms? Pairs of numbers? What exactly is the point of your thread?


Edit: If you can "run this analysis" on any pair of numbers, perhaps it would be simpler if you could run the analysis on the two numbers 2, 5 as a demonstration?

Originally posted by wirehead
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

Equivalent information to this is already included within the quote in the opening post.
Perhaps some more study should be spent on this part.

that proved the accuracy of a [color=gold] finite

1/4 geometric series method that followed Eudoxus that used the same tradition.

Yes, I agree the method of exhaustion
is well known to both mathematicians and historians of math.

Determining finite solutions
with recursive fractions is a different method though.


David Grouchy

Originally posted by AwakeinNM
The trouble with these math threads is that people like me can never tell if these guys are arguing or not.

That's particularly funny.

Argument is a name given to elements within math.
For instance in Propositional calculus, there are two basic concepts.
1). Closure under operations, and 2) Argument.

So, he hehe he, it's appropriate to say in a math discussion there is almost always an argument.

I guess the difference is that math heads can always agree which 'argument' is correct in the end,
and they tend to to be argumentative, per se.


David Grouchy

Originally posted by wirehead

Edit: If you can "run this analysis" on any pair of numbers, perhaps it would be simpler if you could run the analysis on the two numbers 2, 5 as a demonstration?

That would be "uninteresting" as neither of those are irrational numbers.

Originally posted by wirehead

Please, elaborate on this system. What are you trying to show in your equations? I caution that I've not read any of your other threads. Just break it down for me simply.

What is the relationship between orbits that you are attempting to show? Are you trying to predict orbital periods? Draw connections between orbital periods of different bodies?

The ecliptical orbits of the bodies of the solar system, are well modled using Newtonian physics, some special cases more accurately with Einstein (Newtonian being considered a subset of Relativity now), but the fundamental question still remains.

How did the orbits orgininate in the first place.
For that we get answers that contain a lot of hand waving
like acretion, capture, cooling, big bank, etcetera, etcetera...

Isn't it interesting, though, that there are no orbital periods that are
comensurate with any other. That they are all irrational when compared
to each other. One would think that by pure chance, at least one orbital body
would have a whole number relationship with at least one other orbital body.
But they don't. The solar system has no comon denominator.
What we have is the theory of gravity.
And this works great.

But so far it only works to define the predicted orbit (apparently born like athena
fully formed from the brow of Zeus) and detect new bodies by the perturbations of
that predicted orbit.

Using Archimedes method B with Egyptian recursive fractions it becomes possible
to analyse the relationships between the irrational numbers themselves.
And with such tools unexamined trends in the solar system become evident.

For instance:




Preliminary math
===========================

Orbital period / Rotation Period =

Take Luna
Orbital period = 27.32166 earth days
Rotation Period = 27.32166

27.32166 / 27.32166 = 1
===========================


How many bodies in our solar system have a 1:1 ratio of Orbital Period to Rotation Period?
This should be impossible, even once, considering that all orbits are eliptical.
The current theory is called Tidal Locking.
But people tend to only be aware of
our own Moon's tidal locking,
and the math looks good
in that one case.

But how many cases are there?

Well here is the list.

(7 Orbiting Jupiter)
Metis = 1
Adrastea = 1
Thebe = 1
Io = 1
Europa = 1
Ganymede = 1
Callisto = 1

(12 Orbiting Saturn)
Pan = 1
Atlas = 1
Prometheus = 1
Pandora = 1
Epimetheus = 1
Janus = 1
Mimas = 1
Enceladus = 1
Tethys = 1
Dione = 1
Rhea = 1
titan = 1

(7 Orbiting Neptune)
Naiad = 1
Thalassa = 1
Despina = 1
Galatea = 1
Larissa = 1
Proteus = 1
Triton = -1 (retrograde rotation)

(16 Orbiting Neptune)
Cordelia = 1
Ophelia = 1
Bianca = 1
Cressida = 1
Desdemona = 1
Juliet = 1
Portia = 1
Rosalind = 1
Cupid = 1
Perdita = 1
Belinda = 1
Puck = 1
Mab = 1
Miranda = 1
Ariel = 1
Umbriel = 1

(1 Orbiting Earth)
Luna = 1

(1 Orbiting Mars)
Phobos = 1

(1 Orbiting Pluto)
Charon = 1

(1 Orbiting Kalliope)
Linus = 1

case? = 46

There are 46 "moons" that always show the same face to the body they orbit.
. [color=gold] This is a major trend and fundamental property of our solar system.

Is so-called "tidal locking" sufficient to explain how this happens with eliptical orbits. Eliptical orbits should make the ratio of orbit to side facing the orbited body impossible. Are all of the moons at the same distance to mass ratio as all the others exhibiting so-called tidal locking?


David Grouchy

reply to post by davidgrouchy

I was not at all aware that this was the case with all the other planets' moons in our solar system. I must look into this further. Thanks for that information.

reply to post by davidgrouchy

The solar system has no comon denominator

Great discussion and thanks especially for the tidal lock information - amazing. I do however, have to leave before developing an anuerysm but my passing thoughts if you don't mind:

All the 1:1 ratio's seems 'relatively' common...

Also, 42, is 'relatively' close to 46.... hmmm

Originally posted by davidgrouchy
That would be "uninteresting" as neither of those are irrational numbers.

Okay, so now I'm starting to understand more about what you're claiming!

The ecliptical orbits of the bodies of the solar system, are well modled using Newtonian physics, some special cases more accurately with Einstein (Newtonian being considered a subset of Relativity now), but the fundamental question still remains.

How did the orbits orgininate in the first place.
For that we get answers that contain a lot of hand waving
like acretion, capture, cooling, big bank, etcetera, etcetera...

Isn't it interesting, though, that there are no orbital periods that are
comensurate with any other. That they are all irrational when compared
to each other. One would think that by pure chance, at least one orbital body
would have a whole number relationship with at least one other orbital body.
But they don't.

I would not expect such a thing. There's no a priori reason to expect it, especially given that we know that stable orbits can exist at any distance from the sun; if we pick any two random rational numbers, then, the odds are greatly against that they should be whole multiples of one another.

The solar system has no comon denominator.
What we have is the theory of gravity.
And this works great.

But so far it only works to define the predicted orbit (apparently born like athena
fully formed from the brow of Zeus) and detect new bodies by the perturbations of
that predicted orbit.

Of course, a theory of gravity is not a theory of origins of planets. These address two very different questions.

Using Archimedes method B with Egyptian recursive fractions it becomes possible
to analyse the relationships between the irrational numbers themselves.
And with such tools unexamined trends in the solar system become evident.

For instance:

Preliminary math
===========================

Orbital period / Rotation Period =

Take Luna
Orbital period = 27.32166 earth days
Rotation Period = 27.32166

27.32166 / 27.32166 = 1
===========================

You will find this relationship with any body in orbit that is tidally locked to its larger parter. In fact, it's pretty much the definition thereof. All this demonstrates is that the moon rotates such that one side always faces the earth. Any time this is the case with two bodies, you will find this relationship.

How many bodies in our solar system have a 1:1 ratio of Orbital Period to Rotation Period?
This should be impossible, even once, considering that all orbits are eliptical.
The current theory is called Tidal Locking.
But people tend to only be aware of
our own Moon's tidal locking,
and the math looks good
in that one case.

But how many cases are there?

Well here is the list

...

There are 46 "moons" that always show the same face to the body they orbit.
. [color=gold] This is a major trend and fundamental property of our solar system.

In fact, it's a trend, and result of, orbital mechanics. I know you're skeptical of it, but believe me, it works out if you do the math. It's quite well-understood and uncontroversial in physics. Tidal forces result from the relative distance between the far side and near side of an orbiting body giving rise to a difference in force on those faces (one side being pulled more strongly than the other,) resulting in the heavier half gradually coming to rest facing the host body. I could go into more detail if you wish.

Is so-called "tidal locking" sufficient to explain how this happens with eliptical orbits. Eliptical orbits should make the ratio of orbit to side facing the orbited body impossible. Are all of the moons at the same distance to mass ratio as all the others exhibiting so-called tidal locking?

I assume you realize that moons, being much smaller in mass relative to their host planets, and much closer, than the planets are to the sun, have much closer to circular orbits than the planets themselves have around the sun. You're right, though, it would be harder to have tidally-locked bodies in elliptical orbits, but still not impossible, and given enough time and an orbit that doesn't decay, we'd assume any such pair of bodies in orbit would end up tidally locked.