ϕ and the Golden Section

ϕ is the golden number we have already introduced in the blog in a post about the precession.(http://earthmeasured.com/precession-equinoxes-flat-earth/#more-2182) We can state that a line is divided according to the golden section when we can find this proportion:

When these segments respect the above proportion, the ratio AC/CB will correspond to 1,6180339887…

Euclid and the golden number

Euclid was the first to describe this digit, which is also known as the golden number.  ϕ, like Pi, is obviously an irrational number. Euclid described this ratio only for geometrical purposes; he probably didn’t imagine this number could have important consequences in very different fields. Think, for instance to the disposition of leaves on a tree in botanic or to the description of galaxies in astronomy.

ϕ and the fractal world

Exactly like  Pi, ϕ can be expressed as a sum of many elements, a bigger one summed up with many other fractals. One way to express it is:

Interesting is the fact that, as you can deduce from this equation, ϕ is given by a series of fractions with many repeated 1. We have repeatedly noticed the fact that, when describing the Earth, repeated digits appear many and many times.

ϕ and the number 72

I have already highlighted the link between ϕ and the number 72, but, about this subject, I want to add something that is really interesting.

72° is the fifth part of the circle: 360°/72=5 and there is actually a link between ϕ and the pentagon. Let’s draw a pentagon inscribed in a circle. Draw then two diagonals of the pentagon, dividing thus the pentagon in three triangles.

ϕ and the pentagon

The ratio between the diagonal and the side of the pentagon AB/BD is ϕ again.

But when we divide the angle of 72° with a segment, we find the point C, and, again, we have AC/CB=ϕ.

The pentacle, or five-pointed star, often mentioned in connection with Baphomet, is thus connected to ϕ. Incredible is the fact that ϕ is in relation to the number 666 too. We can write -2*sin666=ϕ. We don’t want, of course, to link the golden number with the Beast of Revelation, but again with a number made up of repeated digits: 666.

Contiguous Fibonacci numbers

ϕ can be rationalized using the Fibonacci series 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987…. Consider, in fact, the ratio between contiguous Fibonacci numbers:

1/1=1,000000

2/1=2,000000

3/2=1,500000

5/3=1,666000

8/5=1,600000

13/8=1,625000

21/13=1,615385

34/21=1,619048

55/34=1,617647

89/55=1,618182

144/89=1,617978

233/144=1,618056

377/233=1,618026

610/377=1,618037

987/610=1,618033

These ratios get nearer and nearer to the golden number and this astonishing phenomenon was discovered by Kepler, the astronomer.

The number eleven in a Shiller’s tragedy

In Schiller’s tragedy “The Piccolomini”, the astrologer Seni states that eleven is the sin, (sic!!) because it is one more than Ten Commandments. The Fibonacci series has a beautiful property connected with the number 11, again a number with repeated digits. If you sum ten numbers of the series, the result is always perfectly divisible for eleven.

For example 1+1+2+3+5+8+13+21+34+55=143

143/11=13

ϕ and the phyllotaxis arrangement

An evidence of the fact that the Fibonacci series proves to be fit in describing the Earth lies in botanic and stays in relation with the phyllotaxis arrangement, i.e., the disposition of the leaves on trees, leaves and branches are arranged to maximize the exposition to the sun. On lime trees, leaves are ordered on two opposite sides, being the coefficient of phyllotaxis ½, that means that, with one turn around the stem, there are two leaves or branches. The beech has a coefficient 1/3, the apple 2/5 while there are cases of trees with a coefficient 3/8. All these ratios are made with alternated terms of Fibonacci series.

 

The golden rectangles and the fractals

Fibonacci is the link between the golden number and fractals, both being math instruments that are perfect to describe the nature of reality. Let’s consider in fact the logarithmic spiral. It can be obtained from a series of golden rectangles, one inside the other. They are obtained by subtracting a square to the rectangle, as you can see in the following picture. A golden rectangle has the property that the ratio between the sides is ϕ. The same spiral can be obtained from a golden triangle, the one with a 72° angle inside.

 

“Eadem mutato resurgo”

This spiral has a particular property: while growing it doesn’t change its shape. This property is called self similarity, and is the same we can find in fractals: parts of fractals are similar to the total.

That is exactly the property required by a lot of phenomena of natural growth. Think for example to the Nautilus which builds rooms increasingly greater. While the shell gets greater, the radius increases proportionally. As a result, the general shape remains always the same. You could say: “Eadem mutato resurgo”.

The shape of galaxies, Newton and the golden spiral

There are many natural shapes that are similar to the logarithmic spiral and many astronomers link the shape of galaxies to the golden spiral.

On the other hand, we could make a consideration of Newton’s law of gravity. It states that, by doubling the distance, the attraction force decreases, according to a factor 4. This is because the force diminishes with the square of the distance. Due to this law, in a globular system, planets’ orbits around the sun are assumed to have an elliptic shape. But, let’s suppose that the attraction force could diminish of factor 8 instead of 4. So, if, by any chance, the distance doubles, you should imagine a totally different universe. When the gravity decreases according to the cube of the distance, the planets’ orbits will consequently become logarithmic spirals. As a consequence, the Earth would collapse or it would depart from the sun. Newton’s laws, naturally, do not act in harmony with the flat earth math.

ϕ and the music octave scale

Harmony and proportions are obviously the basic elements in music too. It appears well established the fact that the Fibonacci sequence of numbers and the associated “golden ratio” are manifested in many works of art. These numbers also underlie certain musical intervals and compositions. The Fibonacci sequence is evident even in the musical structure of the octave scale. Moreover, the greatest of luthiers, Stradivarius, designed his violins around the golden ratio.

Thus, when approaching art, you can easily find that the theory of proportions has to be considered the rational basis for beauty. And, together with this,  math is exalted as the foundation of many different artistic activities and mainly of music.

 

 Proportions and the book of Job

This aesthetic of the proportions, while uniting grace and beauty, makes me remember a verse of Job. It refers to Leviathan and it recites:

I will not conceal his parts, nor his power, nor his comely proportion. (Job 41:12) King James Bible

 Leviathan is, in the book of Job, a poetic representation of the vault of the heavens. On the other hand, Behemoth is a representation of the earth. Their proportions are regulated according to an extremely refined aesthetical math: numbers of the great joy or repeated units, like 111 or 666; or even irrational numbers, like Pi or ϕ. And, of course, auto-similarity and fractals lay at the cornerstone.

 

Bye, bye, my reader.

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