Posts Tagged ‘phi’

Io, Europa and Ganymede - three of Jupiter's four Galilean moons

Io, Europa and Ganymede – three of Jupiter’s four Galilean moons

The resonance of three of the four Galilean moons of Jupiter is well-known. Or is it?

We’re usually told there’s a 1:2:4 orbital ratio between Ganymede, Europa and Io, but while this is not far from the truth, a closer look shows something else.


Phi and the Great Pyramid of Khufu

Posted: November 19, 2015 by oldbrew in Maths, Measurement, Phi
Great Pyramid of Giza from a 19th-century stereopticon card photo [credit: Wikipedia]

Great Pyramid of Giza from a 19th-century stereopticon card photo [credit: Wikipedia]

Let’s have a look at some numbers for the Great Pyramid.

Source: Building the Great Pyramid (aka Cheops)
Copyright 2006 Franz Löhner and Teresa Zuberbühler

Dimensions as designed (in Egyptian royal cubits):
Length: 440
Height: 280
Slope: 356

Original dimensions as built (a,h and c in the pyramid diagram below):
Length: 230.36m (half = 115.18m)
Height: 146.59m
Slope: 186.42m


lunar_TYTallbloke writes: Stuart ‘Oldbrew’ has been getting his calculator warm to discover the congruences in various aspects of the Lunar orbit around Earth, and its relationship to Earth-Moon orbit around the Sun. Emerging from this study are some useful insights into longer periods, such as the ‘precession of the equinoxes‘.

Some matching periods of lunar numbers:
86105 tropical months (TM) @ 27.321582 days = 2352524.8 days
85377 anomalistic months (AM) @ 27.55455 days = 2352524.8 days
79664 synodic months (SM) @ 29.530589 days = 2352524.8 days

These identical values are used in the chart on the right (top row). The second row numbers are the difference between the numbers in the first row (TM – AM and AM – SM).
The derivation of the third row number (6441) is shown on the chart itself [click on the chart to enlarge it].

The period of 6441 tropical years (6440.75 sidereal years) is one quarter of the Earth’s ‘precession of the equinox’.
Multiplying by 4: 25764 tropical years = 25763 sidereal years.
The difference of 1 is due to precession.


Neptune (top), Uranus, Saturn, Jupiter (bottom)

Neptune (top), Uranus, Saturn, Jupiter (bottom)

Continuing our long-term series researching Fibonacci and/or Phi based ratios in planetary conjunction periods, it’s time for a look at the inner- and outer-most gas giants of our solar system: Jupiter and Neptune.

Initial analysis shows the period of 14 Jupiter orbits is close to that of one Neptune orbit of the Sun, and even closer to the period of 13 (14 less 1) Jupiter-Neptune (J-N) conjunctions.

It also turns out that there’s a multiple of 13 J-N that equates to a whole number of Earth orbits:
Jupiter-Neptune(J-N) average conjunction period = 12.782793 years
221 J-N = ~2825 years (2824.9972y)
(221 = 13 x 17)

But this period is not a whole number of either Jupiter or Neptune orbits.
This is resolved by multiplying by a factor of 7.


From the top: Neptune, Uranus, Saturn, Jupiter [image credit: NASA/JPL]

From the top: Neptune, Uranus, Saturn, Jupiter
[image credit: NASA/JPL]

Continuing our quest to understand more about planetary frequencies, we turn to links between the largest planet Jupiter and the two ‘outer’ giant planets, Uranus and Neptune.

This model is based on a match of synodic periods, which is found to be:
22 Uranus-Neptune (U-N) = 273 Jupiter-Uranus (J-U) = 295 Jupiter-Neptune (J-N)

The period is just under 3771 years (~3770.93y).
To find a link to Fibonacci numbers we can look first at Jupiter-Uranus:
273 J-U = 13 x 21 (13 and 21 are Fibonacci numbers)


See main post for details [image credit: Wikipedia / WolfmanSF]

See main post for details [image credit: Wikipedia / WolfmanSF]

In this extract from Wikipedia we’ve highlighted the relevant part in bold, so without more ado:

Styx, Nix, and Hydra are in a 3-body orbital resonance with orbital periods in a ratio of 18:22:33. The ratios are exact when orbital precession is taken into account. This means that in a recurring cycle there are 11 orbits of Styx for every 9 of Nix and 6 of Hydra. Nix and Hydra are in a simple 2:3 resonance. The ratios of synodic periods are such that there are 5 Styx–Hydra conjunctions and 3 Nix–Hydra conjunctions for every 2 conjunctions of Styx and Nix.


[credit: F.Espenak / NASA]

[credit: F.Espenak / NASA]

NASA: 2004 AND 2012 TRANSITS OF VENUS – Introduction says:
‘Transits of Venus across the disk of the Sun are among the rarest of planetary alignments. Indeed, only six such events have occurred since the invention of the telescope (1631, 1639, 1761, 1769, 1874 and 1882). The next two transits of Venus will occur on 2004 June 08 and 2012 June 06.’

Obviously there are three pairs of transits (eight years apart per pair) shown in the brackets, plus the fourth pair that occurred in 2004 and 2012. The model we use here is structured as per this graphic:

This is where it gets a little bit tricky perhaps.


[image credit:]

[image credit:]

Only two questions are needed here:

(1) What is the period of a Jupiter(J)-Saturn(S)-Earth(E) (JSE) triple conjunction?
JSE = 21 J-S or 382 J-E or 403 S-E conjunctions (21+382 = 403) in 417.166 years (as an average or mean value).

(2) What is the period of a Jupiter(J)-Saturn(S)-Venus(V) (JSV) triple conjunction?
JSV = 13 J-S or 398 J-V or 411 S-V conjunctions (13+398 = 411) in 258.245 years (as an average or mean value).

Since JSV = 13 J-S and JSE = 21 J-S, the ratio of JSV:JSE is 13:21 exactly (in theory).

As these are consecutive Fibonacci numbers, the ratio is almost 1:Phi or the golden ratio.
Golden ratio: relationship to Fibonacci sequence


Jupiter dominates the solar system

Jupiter dominates the solar system

By far the two largest bodies in our solar system are Jupiter and Saturn. In terms of angular momentum: ‘That of Jupiter contributes the bulk of the Solar System’s angular momentum, 60.3%. Then comes Saturn at 24.5%, Neptune at 7.9%, and Uranus at 5.3%’ (source), leaving only 2% for everything else. Jupiter and Saturn together account for nearly 85% of the total.

The data tell us that for every 21 Jupiter-Saturn (J-S) conjunctions there are 382 Jupiter-Earth (J-E) conjunctions and 403 Saturn-Earth (S-E) conjunctions (21 + 382 = 403).

Since one J-S conjunction moves 117.14703 degrees retrograde from the position of the previous one, the movement of 21 will be 21 x 117.14703 = 2460.0876, or 2460 degrees as a round number.

The nearest multiple of a full rotation of 360 degrees to 2460 is 2520 (= 7 x 360).
Therefore 21 J-S has a net movement of almost 60 degrees (2520 – 2460) from its start position.


Click on image to enlarge

Click on image to enlarge

The Mars-Earth model is based on 34 Mars orbits. This equates to 64 years, which is 8². Since Venus makes 13 orbits of Earth in 8 years, we can easily add it to the model.
2,3,5,8,13 and 34 are Fibonacci numbers.

The story doesn’t end there, because as the diagram shows this results in a 3:4:7 relationship between the 3 sets of synodic periods. This was analysed in detail in a paper by astrophysicist Ian Wilson, featured at the Talkshop in 2013:

Ian Wilson: Connecting the Planetary Periodicities to Changes in the Earth’s Length of Day


The model is ~99.78% accurate

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.

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

18 Inex cycles = 521 years [click to enlarge]

18 Inex cycles = 521 years
[click to enlarge]

In the wake of today’s solar eclipse and following an earlier post on the same topic, we have another perspective on the 521 year period that corresponds exactly to 18 Inex eclipse cycles.

An Inex corresponds to:
358 lunations (synodic months) = 28.94444 years
388.50011 draconic months
30.50011 eclipse years

This means two Inex = 716 synodic months (358×2) and 777 draconic months (388.5×2).
This period will also be 61 eclipse or draconic years (777 – 716 or 30.5 x 2).

Each number in the diagram (below the top line) is derived from the numbers above it. Note that 18 Inex is the same period as 28 lunar nodal cycles. Both periods end at the lunar node they started at.

We can build on this, first by looking at data from a well-known science paper by Keeling & Whorf titled:
‘The 1,800-year oceanic tidal cycle: A possible cause of rapid climate change’


Kepler Space Telescope [NASA]

Kepler Space Telescope [NASA]

A very interesting report of a new science paper has appeared in the New Scientist:

‘William Ditto and his colleagues at the University of Hawaii, Manoa, compared the two strongest oscillations, or tones, made by the variable star KIC 5520878, using observations by NASA’s Kepler space telescope. They noticed that dividing the frequency of the secondary note by that of the primary, or lowest, note gives a value near the “golden ratio” – a number that shows up often in art and nature and is close to 1.618′

So is it real or did they perhaps just imagine it?
Let’s start with the abstract :

‘The unprecedented light curves of the Kepler space telescope document how the brightness of some stars pulsates at primary and secondary frequencies whose ratios are near the golden mean, the most irrational number.’


What is a Saros? Quoting Wikipedia:
‘One saros period after an eclipse, the Sun, Earth, and Moon return to approximately the same relative geometry, a near straight line, and a nearly identical eclipse will occur’

‘It takes between 1226 and 1550 years for the members of a saros series to traverse the Earth’s surface from north to south (or vice-versa)’

Only a few lines to go … (more…)

1685 Toro and 1866 Sisyphus may be names you haven’t heard of but they’re
orbiting the Sun in the neighbourhood of our planet. What are they and what
exactly are they doing?

They are known as Apollo asteroids (see footnote) – two of several dozen in fact.

‘They are Earth-crosser asteroids that have orbital semi-major axes greater
than that of the Earth (more than 1 AU) but perihelion distances less than
the Earth’s aphelion distance (which is 1.017 AU).’

What they are doing is orbiting the Earth in interesting synodic relationships
with it. Toro completes 5 orbits of the Sun every 8 Earth years/orbits while 5
Sisyphus orbits take 13 Earth years/orbits, thus 8 Sisyphus = 13 Toro orbits
(as very close approximations). On a longer time scale the figures are:
825 Toro = 1319 Earth orbits (825:1320 = 5:8) and
100 Sisyphus = 163 Toro orbits (100:162.5 = 8:13).




Barbury Crop Circle represents Pi

Update: I made a dumb algebra mis-step – back to the drawing board. :)

I believe I’ve found a new way to calculate the value of Pi. Before anyone starts shouting at me, the value I’ve arrived at is Pi, not some new number I’m claiming to be the circumference of a circle divided by its diameter.

So, what is the equation I’ve come up with which can calculate the value of Pi?

Here it is:


Many other people have noticed Phi relationships in the solar system in the past, from Kepler onwards, and there are several websites which cover this interesting topic. But up until now, so far as I know,  no-one has been able to find a single simple scheme linking all the planets and the Sun into a harmonious whole system described by the basic Fibonacci series. A couple of weeks ago while I was on holiday, I had a few long ‘brainstorming sessions’ with Tim Cullen, and decided to roll my sleeves up and get the calculator hot to test my ideas. What I discovered is laid out below in the style of a simple ‘paper’. Encouraged by an opinion from a PhD astrophysicist that this is “a remarkable discovery”, I will be rewriting this for submission to a journal with the more speculative elements removed and some extra number theory added to give it a sporting chance of acceptance. For now, this post establishes the basics, but there is much more I have discovered, and I will be using some of that extra material in more posts soon.


Relations between the Fibonacci Series and Solar System Orbits

Roger Tattersall – February 13 2013


The linear recurrence equation: an = an-1 + an-2 with the starting conditions: a1 = a2 = 1 generates the familiar Fibonacci series: 1,1,2,3,5,8,13… This paper will use the first twenty terms of the sequence to demonstrate a close match between the Fibonacci series and the dynamic relationships between all the planets, and two dwarf planets in the Solar System. The average error across the twenty eight data points is demonstrated to be under 2.75%. The scientific implication of the result is discussed.


Since it was noticed that five synodic conjunctions occur as Earth orbits the Sun eight times while Venus orbits thirteen times, many attempts have been made to connect the Fibonacci series and it’s convergent ‘golden ratio’ of 1.618:1 to the structure of the solar system. Most of these attempts have concentrated on the radial distances or semi-major axes of the planet’s orbits, in the style of Bode’s Law, and have foundered in the inner solar system.