The Golden Ratio ɸ is a beautiful irrational number that occurs in many patterns in nature. ɸ has many elegant and curious mathematical properties. This visualization (inspired by this video) visualizes ɸ and many other ratios using a distribution of points.

1.6180

Continued Fraction Expansion:

`1`

`+`

`1`

`1`

`+`

`1`

`1`

`+`

`1`

`1`

`+`

`1`

`1`

`+`

`...`

'Spokes' approximation: 5 or 3 or 2

You may observe that rational ratios (e.g. 0.1, 0.25, 0.5) display visible 'spokes' while irrational ratios (e.g. ɸ and √2) have a more randomized distribution of points. The degree of irrationality can be observed visually by trying to identify these spokes. In the case of ℼ, we can visually observe 7 spokes. We say that ℼ is an irrational number that can be closely approximated by a rational number 7^{1}.

To gain an understanding for this approximation, we can use the continued fraction expansion^{2} to represent irrational numbers as a recursive expansion of fractions. The fraction denominators (highlighted in yellow) can be recursively computed. These denominators are good candidates for approximating irrational numbers. If you visually inspect the spokes and approximation candidates, you will notice that rational numbers have exactly one candidate while highly irrational numbers will include more candidates for approximations.^{3}

Note that ɸ and √2 have interesting and infinite expansions (their recursive denominators are 1 and 2 respectively). ɸ has the most efficient expansion since it has the smallest possible recursive denominator (i.e. 1). Feel free to input a specific ratio and pausing the animation to study its continued fraction expression.

This is a familiar fact in math that 22/7 is a good approximation for ℼ.

↩https://en.wikipedia.org/wiki/Continued_fraction

↩This is not an exact and formal proof, but it is a good intuition for an approximation.

↩