Questions

Which number can be found in

• the design and proportions of buildings
• computer graphics
• the geometric properties of pentagons and icosahedrons
• French politics

This number is “Phi” !

The design and proportions of buildings

I first heard about Phi, very pretentiously called “the golden number,” in the field of architecture.

I was a bit skeptical about the almost mystical use of this number, more or less convincingly, in many famous constructions like the Parthenon or the pyramids.

Phi has fascinating proportional properties that make it unique among numbers. These properties are often used in mathematics, art, and design to create aesthetically pleasing compositions.

1. Self-Similarity: Dividing a line into two parts such that the ratio of the whole line to the longer part is the same as the ratio of the longer part to the shorter part gives the golden ratio, $\mathrm{\Phi }$.

   $$\frac{a+b}{a}=\frac{a}{b}=\mathrm{\Phi }$$

2. Reciprocal Property: The reciprocal of $\mathrm{\Phi }$ is equal to $\mathrm{\Phi }-1$, which is approximately 0.618.

   $$\frac{1}{\mathrm{\Phi }}=\mathrm{\Phi }-1$$

3. Square Property: The square of $\mathrm{\Phi }$ is equal to $\mathrm{\Phi }+1$, which is approximately 2.618.

   $${\mathrm{\Phi }}^2=\mathrm{\Phi }+1$$

These proportional properties are why $\mathrm{\Phi }$ is often referred to as the “Divine Proportion” and why it appears in various natural and human-made structures.

In computer graphics!

The first surprise was learning that this very pretentious number is used in the field of computer graphics for harmoniously distributing points on a rectangle or a disc.

This number is unique in its ability to produce non-overlapping patterns, a corollary of the fact that it is the number with the slowest possible convergence in its continued fraction expansion.

This contributes to the absence of cycles and overlaps in the patterns.

That’s already super cool !

For this reason, it’s interesting to have this value memorized.

$$\mathrm{\Phi }=\frac{1+\sqrt{5}}{2}$$ $$\mathrm{\Phi }\approx 1.618034$$ $$\frac{1}{\mathrm{\Phi }}=\mathrm{\Phi }-1\approx 0.618034$$ $${\mathrm{\Phi }}^2=\mathrm{\Phi }+1\approx 2.618034$$

In 2D and 3D geometry!

Shane, a well-known Shadertoy user, recently shared an intuitive method to construct an icosahedron using three perpendicular golden rectangles.

A golden rectangle is a rectangle whose side lengths are in the golden ratio, $\mathrm{\Phi }$. This means that the ratio of the longer side to the shorter side is $\mathrm{\Phi }$. If the longer side of the rectangle is $a$ and the shorter side is $b$, then:

$$\frac{a}{b}=\mathrm{\Phi }$$

From the reading of Divine Proportion [Luca Pacioli 1509] we deduce that some mathematicians of that time knew the golden relationships in the icosahedron thoroughly !

One can construct a geometric figure with 2 dimensions a and b that fulfill the golden ratio relation $a/b=(a+b)/a$.

It turns out that this figure has by construction, angles of $\frac{\pi }{5}rad$, $\frac{\pi }{5}rad$, and $\frac{3\pi }{5}rad$.

• Pentagons and Icosahedrons use PI/5 rad (36°) for their constructions.
• This explains why Phi equals 2 * cos(π/5).

Construction using Desmos

By construction of these 2 isosceles triangles, $AC/AB=AB/AD=x$

Taking special case where $AB=1$ : $AC=x$

$$AD=AC-AB=x-1$$ $$AC/AB=AB/AD\iff x=1/(x-1)\iff x\ast (x-1)=1$$

We conclude that x is the positive solution of the equation $x^2=x+1$ and $x=\mathrm{\Phi }$

You can also observe the relation :

$$\mathrm{\Phi }=2\cdot \cos \left(\frac{\pi }{5}\right)$$

In French politics!

It’s worth noting that Phi is also the symbol used by the LFI (La France Insoumise), a French ultra-left party. This association is unfortunate, as it links such a remarkable mathematical constant to a political group that some may find disagreeable.

References

Golden ratio - Wikipedia

Golden rectangle - Wikipedia

GM Shaders Mini: Phi

almanach ou dictionnaire des nombres - curiosités et propriétés

ICOSAHEDRON AND Ф

The exact value of cos(π/5) (Geometric Proof)