Posts Tagged ‘phi’

Why Phi? – the rainbow angle

Posted: September 3, 2017 by oldbrew in Maths, Measurement, Phi, weather

The rainbow angle [credit: Hong Kong Observatory]

The minimum deviation angle for the primary bow [of a rainbow] is 137.5° according to Wikipedia. This is known as the rainbow angle. A circle is 360 degrees, so the ratio of the rainbow angle to the circle is therefore the square of the golden ratio i.e. 137.5:360 = 1:2.61818~.
– – –
Hong Kong Observatory has some useful explanatory text and graphics (rounding 137.5 to 138 degrees) titled:
Why is the region outside the primary rainbow much darker than that inside the primary rainbow?
Written by : SIU Kai-chee (summer intern) and HUNG Fan-yiu

Let’s first look at Figure 1, which shows sun rays entering a water drop and going through refraction and reflection.

The ray (ray no. 1) passing through the centre goes directly backward on reflection, i.e. a change in direction of 180 degrees.

For ray no. 2, this angle becomes smaller, following the rules of refraction and reflection.

For the next (ray no. 3) the angle continues to decrease, so on and so forth. This trend does not continue for long, however.


The 1,100 year orbit of ‘DeeDee’

The solar system’s dwarf-planet population is about to increase by one, reports The far-flung object 2014 UZ224 — informally known as DeeDee, for “Distant Dwarf” — is about 395 miles wide (635 kilometers), new observations reveal.

That means the frigid object probably harbors enough mass to be shaped into a sphere by its own gravity, entitling it to “dwarf planet” status, researchers said.

Astronomers first spotted DeeDee in 2014 using the optical Blanco telescope at the Cerro Tololo Inter-American Observatory in Chile (though they didn’t announce the discovery until 2016).

[click on image to enlarge]

[click on image to enlarge]

Another one to add to the ‘how and why did they do that?’ list of ancient sites. Years of research lie ahead.

Imagine you are about to plan and construct a building that involves several complicated geometrical shapes, but you aren’t allowed to write down any numbers or notes as you do it. For most of us, this would be impossible.

Yet, new research from Arizona State University has revealed that the ancient Southwestern Pueblo people, who had no written language or written number system, were able to do just that – and used these skills to build sophisticated architectural complexes, reports

Dr. Sherry Towers, a professor with the ASU Simon A. Levin Mathematical, Computational and Modeling Sciences Center, uncovered these findings while spending several years studying the Sun Temple archaeological site in Mesa Verde National Park in Colorado, constructed around A.D. 1200.

“The site is known to have been an important focus of ceremony in the region for the ancestral Pueblo peoples, including solstice observations,” Towers says. “My original interest in the site involved looking at whether it was used for observing stars as well.”

However, as Towers delved deeper into the site’s layout and architecture, interesting patterns began to emerge.

Why Phi? – a lunar ratios model

Posted: January 8, 2017 by oldbrew in Cycles, modelling, moon, Phi
Tags: ,
Lunar ratios diagram

Lunar ratios diagram

The idea of this post is to try and show that the lunar apsidal and nodal cycles contain similar frequencies, one with the full moon cycle and the other with the quasi-biennial oscillation.

There are four periods in the diagram, one in each corner of the rectangle. For this model their values will be:

FMC = 411.78443 days
LAC = 3231.5 days
LNC = 6798.38 days
QBO = 866 days (derived from 2 Chandler wobbles @ 433 days each)
The QBO period is an assumption (see Footnote below) but the others can be calculated.

Why Phi? – modelling the solar cycle

Posted: August 27, 2016 by oldbrew in solar system dynamics
Tags: ,


We’re familiar with the idea of the solar cycle, e.g.:
‘The solar cycle or solar magnetic activity cycle is the nearly periodic 11-year change in the Sun’s activity (including changes in the levels of solar radiation and ejection of solar material) and appearance (changes in the number of sunspots, flares, and other manifestations).

They have been observed (by changes in the sun’s appearance and by changes seen on Earth, such as auroras) for centuries.’

Here we’ll try a bit of pattern-hunting, so to speak.


Pluto's non-standard orbit [credit: Wikipedia]

Pluto’s non-standard orbit [credit: Wikipedia]

‘Pluto’s orbital period is 248 Earth years. Its orbital characteristics are substantially different from those of the planets, which follow nearly circular orbits around the Sun close to a flat reference plane called the ecliptic. In contrast, Pluto’s orbit is moderately inclined relative to the ecliptic (over 17°) and moderately eccentric (elliptical). This eccentricity means a small region of Pluto’s orbit lies nearer the Sun than Neptune’s.’ – Wikipedia


Why Phi? – lunar eclipses at Stonehenge

Posted: February 19, 2016 by oldbrew in Celestial Mechanics, Cycles, moon, Phi
Tags: ,

Bluestone Horseshoe at Stonehenge - 19 Stones

Bluestone Horseshoe at Stonehenge – 19 Stones

Stonehenge Visitors Guide – under ‘Eclipse Cycles’ – says:

‘Now, it’s widely accepted that Stonehenge was used to predict eclipses. The inner “horseshoe” of 19 stones at the very heart of Stonehenge actually acted as a long-term calculator that could predict lunar eclipses. By moving one of Stonehenge’s markers along the 30 markers of the outer circle, it’s discovered that the cycle of the moon can be predicted. Moving this marker one lunar month at a time – as opposed to one lunar day the others were moved – made it possible for them to mark when a lunar eclipse was going to occur in the typical 47-month lunar eclipse cycle. The marker would go around the circle 38 times [2 x 19] and halfway through its next circle, on the 47th full moon, a lunar eclipse would occur.’


Combined precession cycle [credit: wikipedia]

Combined precession cycle [credit: wikipedia]

‘Because of apsidal precession the Earth’s argument of periapsis slowly increases; it takes about 112000 years for the ellipse to revolve once relative to the fixed stars. The Earth’s polar axis, and hence the solstices and equinoxes, precess with a period of about 26000 years in relation to the fixed stars. These two forms of ‘precession’ combine so that it takes about 21000 years for the ellipse to revolve once relative to the vernal equinox, that is, for the perihelion to return to the same date (given a calendar that tracks the seasons perfectly).’Wikipedia

Here we’ll fit the three precession cycles into one model and briefly examine its workings.


Earth's orbit [credit: NASA]

Earth’s orbit [credit: NASA]

We’ll assume the diagram is self-explanatory but if not, this should help (see opening paragraphs).

We’re looking at Aphelion minus Perihelion (A – P) distances of the giant planets.
Figures are given in units of a million kms. (lowest value first), using Jupiter as a baseline.


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)