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The Amazing Golden Ratio (Phi)

The Golden ratio is maybe the greatest number, lets explore why?

The golden ratio can be found in both the Hexagon and the Pentagon.

Phi in Hexagon and Pentagon.

Phi (golden section) triangle and the 345 triangle have a unique relationship.

phi and the 345 triangle

And in order to understand phi we must look in detail at this relationship. Below showing the most important angles related to the 345 triangle.

345 symmetry

This relationship becomes very sophisticated very quickly.

phi related angles

Below we can see the complicated world of phi, showing relationships to many interesting numbers and angles.

6 angles phi

Yes it is complicated.

The amazing world of phi

Both the first (GOLD) and fourth (NICKEL) metallic mean is related to Phi.

Metallic means

The metallic means using cosmic geometry. Again phi triangle is mean 1 and 4.


Here we can see that there is also a relationship between the golden ratio and the number sequences (Fibonacci and Lucas).

Number Sequences (Fibonacci, Lucas and Pell)

We can also relate phi to Binet’s formulas, as shown below.

Binet’s formulas

It is because of its unique fractal nature. (incommensurability, phase conjugation, harmonic convergence) Only Phi has this fractal nature.

Incommensurability (fractals)

Below we can see the golden Spiral, along with the golden rectangle. Light and Magnets both share this fractal nature.

Phase Conjugation

Here we can see the symmetry of the magnetic flux lines. (ferro-cell image)

Symmetry of magnetic flux lines

Light and Magnetism both use the magical ratio.

Golden Ratio and Visible Light

The colours of the visible light spectrum and Phi. This electro-magnetic universe has a favourite number and it appears to be a golden.

A geometrical figure, where each part has the same character as the whole.

Harmonic Convergence

They are seen in things like snowflakes in which patterns recur at different scales.

If you know of any interesting phi relationships, please comment.

Art by Ken Wheeler, Dan Winter, Clay Taylor and me.


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