(It is the first time I use HTML for displaying math stuff, I hope it works with your browser. What is below is mostly an extra. It has finally 
not much to do with the picture, but inspired it. So maybe the picture could become a little bit more readable under some light.)

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Everybody knows what the divine or golden proportion is all about. Plenty of internet pages are available. The most relevant to the picture are this page, and especially this one.

When I see so much harmony, I can't help thinking that it has to go even deeper. How come the golden ratio can be "extracted" from the number 5? Well the answer is simple enough. Let us recall first the definition of φ.

Saying that φ = 1.6180339887498948482... is not what we can call a definition, no more than saying that π = 3.1415926535897932385... Of course, whatever its decimal expansion is, we rather define π as the circumference of any circle divided by its diameter. Now we have a definition of π.

But what is φ's? φ is defined as the ratio A/B satisfying the property "A+B is to A what A is to B". One form is 1.6180339887498948482... and another one is

which is the analytical solution of the equation

yielding the properties that φ = 1 + 1/φ and φ² = φ + 1. As a result, we get this fantastic identity:

which works no matter the basis (octal, decimal, etc).

Therefore, there is a strong relationship between φ and integers (0, 1, 2, ...). Moreover, φ has two different values, because a √ has two signs as there are two possible values for x satisfying x² = 5... If we take that into account, we obtain the "other" value of φ:

whose numerical value is

φ' = -0.6180339887498948482...

bringing back the divine expansion.

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At this point, these are all well-known facts discovered centuries ago. However, I feel something is still missing because we seem limited to perfectly exact properties. When I look at that, I am instantly searching for other links between the integers 1, 2 and 5. Then something pops in my mind: what would be the hypothenuse of a right triangle with sides 1 and 2?... you got it: √5. So all the numbers used to define φ are actually the sides of a right triangle.

On the other hand, φ can also be exactly defined as:

φ = 2 cos(π/5)

again only reusing the integers 2 and 5 together with π, the base of everything circular.

π/5 is actually the internal angle of the pentagram which reveals the golden ratio many times because it is made out of 5 golden triangles. Moreover the vertices lie all around a circle. Therefore we have the necessary information: there must be something to say about a circle and maybe a sphere of radius √5.

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Of course, one cannot expect exact facts when comparing integer and irrational numbers as they are not comparable in the first place. One can only expect strong similarities. As an example, the internal angle of the pentagram is 36 degrees (π/5). If we take the cosine of this angle, we get:

cos(π/5) = φ/2 = 0.8090169943749474241...

which is not and cannot be a rational number, unless we extract the meaning involving integers. Note that this is a risky process because it is sometimes possible to get some meaning out of purely random numbers. It is still possible to come across a highly improbable coincidence, but all in all it is, well... improbable.

In this case, the value 0.8090169943749474241... is very close to the rational 0.809017 which clearly reveals 3 integers separated by 0's: 8, 9, and 17. What is so special about these integers? Obviously 17 = 8 + 9, but there is more when we take into account the integers 1, 2, and 5 defining φ. The number 3 is introduced in a powerful way with 3 = 1+2, 5 = 2+3 and

17 = 2+3*5 = 8 + 9 = 2³ + 3²

providing a very strong relationship between these numbers, or... a pure coincidence.

I still consider such an approach as scientific, because there is no difference between such a fact, and empirical evidence. Suppose you find a fossil of a bone. At first, it is still possible that such a bone is actually only a rock having roughly the shape of a bone. Alone, a fossil is not a proof. The same can be said about the previous "property" about 17. But if you find many, many bones, and that these bones can be connected with each other revealing an entire skeleton, then we must consider that as a proof, not a random coincidence.

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Let us go back to our sphere of radius √5. A theory can still be scientific, without being purely mathematical. Suppose we "play the game" such as comparing the area and the circumference of a circle. These are not comparable because the units do not fit. The area of a circle is quadratic and the circumference is linear. Whatever the measured values, one cannot compare X km with Y km². Lengths, areas, and higher-dimensional volumes are not comparable with each other.

However let us play the game with a circle of radius √5. The difference between the area and the circumference of such a circle is:

π√5² - 2π√5 = 1.658333805867513406...

where we still only use the numbers 2, 5, and π. Then well, what do we get?... almost φ again! The numerical values are not that close (about 2.5%), but the digits are amazingly similar, at least for the first 10. If we take a closer look:

1.658333805867513406...
1.618033988749894848...

sometimes when the digits don't match vertically, such as the 0 (zero), they are still there but at a different location. In this case, if we consider it as a piece of evidence, and not a purely random improbable coincidence, we have the new information that it cannot be a mathematical or algebraic identity. It is either a pure coincidence or a purely divine relationship, not only a geometrical one that we used to call divine.

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There is another coincidence about a sphere of radius √5. If we get the surface area of such a sphere (4π√5² = 20π), it gives exactly 100 times the golden angle (π/5). Instead of subtracting them, let us do the comparison by dividing the volume of the sphere by its area:

which simply yields

but what is special about this number? Well, don't ask me why, but if we add 1 to it (i.e. considering the ratio (volume+area)/area), we get:

1 + √5/3 = 1.7453559924999298988...

where nothing obvious can be seen. But I recall that the sphere area (4π√5² = 20π) gives exactly 100 times the golden angle (π/5). However a surface is not an angle in the first place. Therefore, suppose we also consider this number (1.74...) to be an angle in radian, what would be the corresponding angle in degrees?

1.7453559924999298988... * 180/π = 100.00153211811294439...

that is well... almost exactly 100 degrees (within less than two 1000th of a percent).

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At this point, we have the choice. Either we consider all of this as doubtful, irrelevant, and even fallacious. OR... we simply keep looking for other such evidence (fossils) in order to get the very big picture. And as the theory gathers more and more empirical data, eventually it will be able to make predictions.