## Why Phi? – a simple Venus-Mercury model

Posted: March 22, 2015 by oldbrew in Fibonacci, Phi, solar system dynamics
Tags: ,

The model is ~99.78% accurate

The model is in the diagram, so here’s the explanation.
Divide the orbit period of Venus by that of Mercury:
0.61519726 years / 0.2408467 years = 2.554310522

To get to whole numbers, round the result up to 2.56 then:
2.56 x 5 = 12.8
12.8 x 5 = 64
64 / 25 = 2.56

64 = 8² and 25 = 5²
Therefore the approximate ratio of Mercury:Venus orbit periods is 8²:5².
The number of conjunctions in the period is the difference in orbit numbers:
8² – 5² = 64 – 25 = 39 = 13 x 3

Phi link: 2,3,5,8, and 13 are all Fibonacci numbers.

2.554310522 / 2.56 = 0.99777755~ so the accuracy of the model is around 99.78%.

An even more accurate model would be:
626 Venus = 1599 Mercury.
1599 / 626 = 2.554313 i.e. almost the same as 2.554310522 = the true ratio.

Note that 1600 / 625 = 2.56 which is the same as 8² / 5².
So there’s one more Venus (626) and one less Mercury orbit (1599) in reality, every 385.11 years, compared to our model.

Footnote:
1600 = 8² x 5²
625 = 5² x 5²
(The common 5² is redundant in the ratio, leaving 8²:5²)

1. tchannon says:

The concept of “cogged” relationships, not random orbital parameters is fairly well accepted whilst restricted in publications to a very small subset of all bodies.

Also I think widely accepted is the transitional nature of the system with greater and lesser degrees of locked in. The system is probably also inherently unstable, chaotic, the whole lot doesn’t fit together.

Assuming the above is fair comment then what about bodies which are not locked but in transition?
Can these be identified?
How does varying degrees of orbital skew from circular fit in?
Does this mean that some “relationships” should be intermediate?

2. oldbrew says:

TC: the KAM theorem may cover part of your query, especially the bit about “sufficiently irrational” frequencies. I suspect that if the biggest bodies e.g. Jupiter and Saturn are relatively stable the rest have little option but to fall in line somehow – or leave, hence the Kirkwood gaps.

http://en.wikipedia.org/wiki/Kolmogorov%E2%80%93Arnold%E2%80%93Moser_theorem

They do claim to be able to (sometimes) forecast which bodies might be forced out of their current orbits, perhaps with a timescale too. Of course we won’t be around to check on the accuracy 😉

3. Ray Tomes says:

The ratio 2.55431 is nearer to 2.5555 than to 2.56, which is a ratio 23/9. Not a Fibonacci ratio.

4. oldbrew says:

New paper: ‘Chaos in navigation satellite orbits caused by the perturbed motion of the Moon’

We find that secular resonances, involving linear combinations of the frequencies of nodal and apsidal precession and the rate of regression of lunar nodes, occur in profusion so that the phase space is threaded by a devious stochastic web. As in all cases in the Solar System, chaos ensues where resonances overlap.’

http://arxiv-web2.library.cornell.edu/pdf/1503.02581v1.pdf

Looks interesting, only read the abstract so far.
Note: the lunar apsidal cycle of ~8.85 years is 5:2 with the mean solar Hale cycle (~22.1 years)

5. oldbrew says:

@ Ray Tomes

That’s true but we are dealing with ‘near-resonances’ in the solar system.
1599 Mercury / 626 Venus does match the required ratio, and the figures at the top are only a model.

I should add that the difference between these two planetary periods is only a fraction of one day (probably less than one third using mean values).

6. tallbloke says:

Simple and elegant OB, I like it.

Then when we consider that 5, 8 and 13 also apply to Earth-Venus orbits/conjunctions, it’s clear that something Fibonacci is going on. Hence my mnemonic, which is accurate to less than 0.5% error:

“In 2/3 of a Jupiter orbit Earth and Venus meet 5 times, as Earth orbits 8 times while Venus orbits 13 times, as venus is passed by Mercury 21 times while orbiting the Sun 34 times.”

7. oldbrew says:

Re 13 x 3 conjunctions: 3 Venus-Mercury is about the same period as the Chandler Wobble (~433.7 days), so it would be 13 CW.

‘Our best estimates, judging from comprehensive sets of Monte Carlo simulations, are P = 433.7 +/- 1.8 (lo) days’ [Furuya and Chao 1996].

8. tallbloke says:

Someone who no longer posts here as they are unable to debate civilly just popped by with a pseudonym to say 23Me/9V is close to half the length of an average solar cycle. (11.078yr), and that 23*3 Me / 14 = 433.56 days.

Which is true, but unlikely to be mechanistically relevant, as average length solar cycles very rarely occur.

9. oldbrew says:

As stated earlier: ‘1599 / 626 = 2.554313 i.e. almost the same as 2.554310522 = the true ratio.’

We’re just pointing out that adding 1 Mercury to and subtracting 1 Venus from this reality does give you 8²:5².
There are various similar examples of near resonances, including the one TB mentioned: 8 Earth = 13 Venus (99.97% true).

10. TB, OB,
Have you checked the “music”, chromatic and diatonic scales, or their reciprocals, to see something other than phi or chaos?

“(The common 5² is redundant in the ratio, leaving 8²:5²)”

Are you now giving significance to the ratio of (FIBS)^2 also? Not very Pythagorean for whole numbers!

11. tallbloke says:

The amount to which E-V is out of ratio means the precession of the conjunction-cycle fits into simple ratios with other planet pair conjunction-cycle precession periods. That’s not a coincidence. It’s a ‘cogging’ which minimises ‘interference’ between planets.

12. tallbloke says:

Will, we have noticed a lot of relationships where powers of phi come into play. We don’t yet know how this fits into the number space we’re investigating.

13. Bruckner8 says:

“Music” would be subjective. Harmonic Series is nothing more than rational numbers. And with the errors reported here in this “number theory,” one could find any rational number (ratio of two integers) within that error. It wouldn’t impress me in the least. I’ve spoken on this before.

14. Bruckner8 says: March 23, 2015 at 2:01 am

“Music” would be subjective. Harmonic Series is nothing more than rational numbers. ”

Indeed my whole point. Why do some combinations of those chromatic and diatonic scales, or their reciprocals, within a harmonic interval sound pleasant and others truly ugly!! What does the term steps actually mean? 🙂

15. oldbrew says:

A reminder of the graphic from: ‘Why Phi? – Earth’s secret neighbours’

16. oldbrew says: March 23, 2015 at 9:19 am

“A reminder of the graphic from”

OK OB, nice numbers,
Not trying to be antagonistic, but helpful. Here with pressure waves, large whales sound pleasant, large submarines sound ugly!

17. Bruckner8 says:

Will Janoschka says: March 23, 2015 at 4:03 am

“Indeed my whole point. Why do some combinations of those chromatic and diatonic scales, or their reciprocals, within a harmonic interval sound pleasant and others truly ugly!! What does the term steps actually mean? :)”

I’m not sure if this question is real or rhetorical, but here goes! A “Step” is man-made definition depending on context/temperament. But since we’re talking about Harmonic Series, every interval is pure (beatless, no interference), and thus nothing is considered “ugly,” unless we’re adding human subjectivity. (Heck, at one time, perfect 5ths (3/2 ratio) were considered “consonant” and perfect 4ths (4/3 ratio) were considered “dissonant!”)

From that definition, equal temperament (modern piano tuning) is all ugly, since NONE of the intervals are pure! All “steps” have been rendered equal, in a 12th root of 2 ratio! (2^(1/12)):1 (not a rational number)

Getting back to Harmonic Series, and the current discussion, the ratio 440/439 is a valid ratio, and thus is pure. But some humans can’t tell them apart, whereas qualified musicians might call it “out of tune,” since they want to hear it as a unision (1:1 ratio). But it’s as pure as any other rational-number ratio.

This is why harmonic series pattern-recognition doesn’t impress me in the least. We could find any 2 integers whose ratio is within a given error.

The work done by Roger and oldbrew is a lot more interesting because it involves patterns of a very specific subset of Harmonic Series: Those integers in the Fibonacci Series. 440/339 doesn’t exist in the F.S.

18. Bruckner8 says: March 24, 2015 at 7:23 pm

“From that definition, equal temperament (modern piano tuning) is all ugly, since NONE of the intervals are pure! All “steps” have been rendered equal, in a 12th root of 2 ratio! (2^(1/12)):1 (not a rational number)”

1.059463094-5 is pretty constant,with most folk not being able to distinguish a 1 hertz error in the second harmonic.

“Getting back to Harmonic Series, and the current discussion, the ratio 440/439 is a valid ratio, and thus is pure. But some humans can’t tell them apart, whereas qualified musicians might call it “out of tune,” since they want to hear it as a unision (1:1 ratio). But it’s as pure as any other rational-number ratio.”

I’m not sure if this question is real or rhetorical, but what scale of things can this solar system or galaxy sense “out of tune” and take correction. It likely is not in range of earthling hearing or feeling, but how about cycles/orbits per 10^6 years?

19. oldbrew says:

Another example – 34 Mars:8² Earth is 99.92% correct (‘near-resonance’).
34, 8 and 2 are Fibonacci numbers.

20. With PHI, 2X = 1+5^(.5) But what “means” 2X?, twice the time or twice the frequency?
“(Heck, at one time, perfect 5ths (3/2 ratio) were considered “consonant” and perfect 4ths (4/3 ratio) were considered “dissonant!”)”
How about 5/2, 5/3, 8/3, 8/2 and 8/5?

21. With PHI, 2X = 1+5^(.5) But what “means” 2X? If X= PI than 1/FIB(n) is the radian angle between seeds at distance (n) from the centre of a sunflower!

22. stenies2012 says:

Modified-Fibonacci-Dual-Lucas Method for Earthquake Prediction. in http://www.etek.org.cy/uploads/fck/detailed_agenda_2_1.pdf

23. Bruckner8 says:

Will Janoschka says: March 25, 2015 at 12:23 pm

“How about 5/2, 5/3, 8/3, 8/2 and 8/5?”

My point is that human subjectivity defines “consonant” and “dissonant,” and changes with time/culture. Yet the definition of Harmonic Series is always beatless, lacking interference, “pure.” From that definition, all rational numbers are “consonant, ” and thus it’s meaningless to judge one as “more consonant” than another.

“With PHI, 2X = 1+5^(.5) But what “means” 2X?, twice the time or twice the frequency?”

Ya got me. PHI is dimensionless. Musical intervals are dimensionless too. So I don’t understand the question.

24. oldbrew says:

stenies: re ‘Modified-Fibonacci-Dual-Lucas Method for Earthquake Prediction’

Every Lucas number is the sum of two Fibonacci numbers (e.g. 8 + 3 = 11, 13 + 5 = 18 etc.)
Every alternate Fibonacci number is the multiple of a Lucas and another Fibonacci number (21 = 3 x 7, 55 = 5 x 11 etc.)

25. oldbrew says:

Will J says: ‘Not very Pythagorean for whole numbers’

Think Kepler triangles (symbol = Phi = 1.6180339~)

26. Bruckner8 says:

to be clear, I’m using Harmonic Series as defined in music (but it is technically equivalent in mathematics). In music, the harmonic series of a fundamental frequency, f, are those frequencies with integer multiples of f: 2f, 3f, 4f, 5f, etc….Thus, any interval based on the harmonic series is a frequency ratio involving integers. f might not be an integer (I suppose it could be PHI Hz!), but any interval in the series will be rational, by definition (3f/2f = 3/2, no matter f)

Again, the frequency itself doesn’t even matter…the intervals (which are frequency ratios) *do* matter in the current discussion. All rational.

This is why I refer to the Harmonic Series as the Natural Numbers: 1,2,3,4,… because all of the rational numbers therein form the intervals of the musical harmonic series.

27. “Bruckner8 says: March 26, 2015 at 1:52 am
“Again, the frequency itself doesn’t even matter…the intervals (which are frequency ratios) *do* matter in the current discussion. All rational.”

You insist that frequency ratios must be rational (represented by the ratio if two integers) and whose inverse must also be rational. Why??

“This is why I refer to the Harmonic Series as the Natural Numbers: 1,2,3,4,… because all of the rational numbers therein form the intervals of the musical harmonic series.”

Those natural numbers are not the intervals they are the harmonics themselves, of frequencies (cycles/interval) that need not be, as you say, rational (commensurate)! Your 2^(-1/12) is irrational but can always be combined with all other incommensurate irrationals to form one irrational (and its irrational inverse) “interval” where everything else must be rational! This is always just a matter scale (overall size). In this mess of “frequencies” there will still be “chords” (multiple now integer frequencies) that must “appear” (subjective) concordant or discordant to whatever is. How do you determine which sets (now always rational) appear concordant. How do PI, PHI, e, all sq roots of non squares. combine to always integer “frequencies”? Why does this mater? The Fib’s and the Lucas’ series are interesting as they always span the whole “scale” with interesting integers that seem to have some meaning in this physical.

28. oldbrew says: March 25, 2015 at 10:58 pm

(Will J says: ‘Not very Pythagorean for whole numbers’)

“Think Kepler triangles (symbol = Phi = 1.6180339~)”

At least Pythagoras used “some” integers 3,4,5 and their multiples which is not very Fibonacci-an!
Do you see what size Fibonacci before Phi even starts to come close? You always use the littlest ones that have nothing to do with Phi. 🙂

29. tallbloke says:

Will J: “Do you see what size Fibonacci before Phi even starts to come close? You always use the littlest ones that have nothing to do with Phi.”

Depends what you mean by close. 13/8 is within 0.5%. Anyway, what we’ve found is that planet-pair ratios are close to simple whole number ratios, but not exact. My hypothesis is that exact resonant ratios lead to energy transfers which push the orbits away from exact ratios. But an attractor draws them back towards the exact ratio. A point of balance is found, at which the orbits stabilise.

The extent of the separation from the exact ratio is often greater than the difference between the ratio and phi or power of phi.

30. stenies2012 says:

correct
oldbrew.

31. oldbrew says:

Will J: ‘At least Pythagoras used “some” integers 3,4,5 and their multiples which is not very Fibonacci-an!’

3 and 5 are Fibonacci numbers, 4 is the square of a Fib. number. You won’t find any Pythagoreans with 3 Fibonacci numbers (but we have the Kepler triangle).

The next whole number P triangle is 5,12,13 – again, two Fibonacci numbers and the square of 12 is another one (144).

With 5² and 8² as in the Venus-Mercury model, the sum of these (25 + 64) is 89 – another Fibonacci number, so the hypotenuse of the P triangle would be the square root of 89.

32. oldbrew says:

stenies: as well as ‘Every alternate Fibonacci number is the multiple of a Lucas and another Fibonacci number’ – the other alternates are all multiples of two Fibonacci squares.

1² + 2² = 5, 2² + 3² = 13, 3² + 5² = 34, 5² + 8² = 89, 8² + 13² = 233 etc.

33. stenies2012 says:

oldbrew: Consecutive Fibonacci numbers are relatively prime

34. oldbrew says:

21,34 and 55 are consecutive but not primes?

35. Bruckner8 says:

Will Janoschka says: March 26, 2015 at 5:43 am

“You insist that frequency ratios must be rational (represented by the ratio if two integers) and whose inverse must also be rational. Why??”

I insist on showing that the Harmonic Series is often misused as some kind of mystical “Harmony of the Spheres,” when it’s merely all rational numbers…to which I say “so what?” It usually goes like this:

[Discoverer]: “oooh, look, the ratio of [these observed numbers, sometimes even having different units, lol] are close to 2/1, 3/2, 4/3, 5/4, X/(X-1) {etc}. I recognize those as the first X intervals of the musical harmonic series! EUREKA! Harmony of the Spheres!”

[me/skeptic] “so what? All rational numbers are in the harmonic series. Every single measurement you take; Every single observation you make, is a rational number within a given error….and thus the ratios are rational. If all ratios are rational, how can you be impressed by ‘harmony of the spheres?’ Furthermore, if the ratios ARE NOT RATIONAL, then they’re not based on the Harmonic Series! So quit claiming that the Harmonic Series is something magical.”

[Discoverer] “but, but, but *most* of my observations fall in that beautiful consonant range using only 1,2,3,5 and their multiples.”

[me/skeptic] “I’ll let the subjective “beautiful consonant” go. What about 7/5? I’m sure that ratio exists too. And 11/7. And 13/11. Looky here…I found some…and even 440/439…and 10^6/3^12, etc…those are all in the Harmonic Series as well”

[Discoverer] “They aren’t consonant.”

[me/skeptic] “says who? all intervals based on the Harmonic Series are beatless, resonant, have no interference. Have you actually listened to any of these ratios?”

[Discoverer] “Well, Mother Nature prefers simplicity of the octaves, 5ths, 4ths, 3rds.”

[me/skeptic] “My work is done here.” (beats head against wall)

36. Bruckner8 says:

Will Janoschka says: March 26, 2015 at 5:43 am

“The Fib’s and the Lucas’ series are interesting as they always span the whole “scale” with interesting integers that seem to have some meaning in this physical.”

Yes, and that’s why I’m more tolerant of the Talkshop’s work in this area. For if they can map all ratios “in their space” to a definable subset of the Natural Numbers, they’re on to something special! The Harmonic Series isn’t special.

If their ratios lead to irrational numbers, again, bad news for the magic of Harmonic Series.

Roger mentioned a “balance” that may be achieved within the exact ratios. That’s a very important concept as well, and tracking the +/- periods, as well as the +/- magnitudes will become part of that discovery. It will take many lifetimes, so it’s important they get their processes and predictions down, so others can track and compute in the future. They must be combing all historical positional data available for these patterns.

Unfortunately, even if everything boils down to Fib or Lucas, it doesn’t answer the question “why?” It only impresses us humans because we found a groovy pattern beyond the simplest Harmonic Series. i suppose we can still call it “Harmony of the Spheres” cuz it feels so good, as long as we don’t try to sell it as technically harmonic!

37. oldbrew says:

Bruckner says: ‘Unfortunately, even if everything boils down to Fib or Lucas, it doesn’t answer the question “why?” ‘

The KAM theorem has something related to that:

‘According to the KAM theorem (Kolmogorov, Arnold, Moser), the most stable periodic orbit of a dynamic system is that which has the Golden Ratio as a winding number. The more irrational the winding number is (the ratio of the resonance frequencies) the more stable is the periodic orbit. Since the Golden Ratio is the most irrational number, it follows that the orbit with the Golden Ratio as a winding number is the most stable. Therefore φ is the secret of the stability of most elementary particles. Vibration simulating particles which do not have a sufficient irrational winding number dissipate as fast as they are produced’

http://www.sacred-geometry.es/?q=en/content/phi-particle-physics

‘The methods introduced by Kolmogorov, Arnold, and Moser have developed into a large body of results related to quasi-periodic motions, now known as KAM theory.’

http://en.wikipedia.org/wiki/Kolmogorov%E2%80%93Arnold%E2%80%93Moser_theorem

38. tallbloke says:

Bruckner 8: it’s merely all rational numbers…to which I say “so what?”

I suggest a visit to Wikipedia’s page on orbital resonance as a first port of call.

39. oldbrew says:

Some serious stuff going on between a planet and a moon (Jupiter and Io) here:
‘Explosions of Jupiter’s aurora linked to extraordinary planet-moon interaction’ (note – not the Sun)

As Io circles around Jupiter and through the plasma torus, an enormous electrical current flows between them. Approximately 2 trillion watts of power is generated. The current follows the magnetic field lines to Jupiter’s surface where it creates lightning in the upper atmosphere. The first black and white Hubble Space Telescope image (top) shows the flux tube, where Io and Jupiter are linked by an electrical current of charged particles.’

http://www.planetaryexploration.net/jupiter/io/io_plasma_torus.html

40. oldbrew says: March 26, 2015 at 4:23 pm

(‘According to the KAM theorem (Kolmogorov, Arnold, Moser), the most stable periodic orbit of a dynamic system is that which has the Golden Ratio as a winding number. The more irrational the winding number is (the ratio of the resonance frequencies) the more stable is the periodic orbit. Since the Golden Ratio is the most irrational number, it follows that the orbit with the Golden Ratio as a winding number is the most stable. Therefore φ is the secret of the stability of most elementary particles. Vibration simulating particles which do not have a sufficient irrational winding number dissipate as fast as they are produced’)
http://www.sacred-geometry.es/?q=en/content/phi-particle-physics

Informative theoretical article, thank you! It does not say what a “winding number” may be! Is this like my expressed decreasing angular separation with increasing sunflower seed rows? Planetary orbits tend to circularize, but never make it. Why? What do “vibrating simulating particles” dissipate? The particles themselves? How and why are they produced? Is this science?
“the Golden Ratio is the most irrational number”?? How is root 5 “most” irrational, or even more irrational than PI? Keep up the good work, but, “are we having fun yet?” 🙂

41. oldbrew says:

Will J: are there any other infinite surds?