1/phi = phi - 1 = 0.61803... Phi is the only positive number where its reciprocal equals itself minus one. This also means phi^2 = phi + 1, so a golden rectangle can be divided into a square and a smaller golden rectangle forever.

Who discovered the golden ratio?

Euclid of Alexandria described it in his Elements (~300 BC) as the extreme and mean ratio. The term golden ratio was coined by German mathematician Martin Ohm in 1835. The Greek letter \u03C6 was chosen in the early 20th century by mathematician Mark Barr.

When were Fibonacci numbers linked to \u03C6?

Leonardo of Pisa (Fibonacci) published his famous sequence in 1202 to model rabbit populations. Johannes Kepler explicitly noted in 1611 that the ratio of consecutive Fibonacci numbers approaches \u03C6.

Is the golden ratio really found in famous artworks and buildings?

Evidence is mixed and often overstated. Many claimed appearances in the Parthenon or Renaissance paintings require selectively choosing measurements. Mathematician George Markowsky published a thorough debunking in 1992, though genuine appearances in nature are well-documented.

Why is \u03C6 called the most irrational number?

\u03C6 = [1;1,1,1,...] in continued fraction form - all 1s - making it the slowest-converging continued fraction. Rational approximations converge more slowly to \u03C6 than to any other irrational number.

Is the golden ratio irrational?

Yes. \u03C6 satisfies x^2 = x + 1. The ancient Greeks knew it was irrational. Unlike \u03C0 and e which are transcendental, \u03C6 is algebraic - a root of a polynomial with integer coefficients.

Christoffer De Geer March 20, 2026

What is the Golden Ratio (φ)?

φ = (1 + √5) / 2 ≈ 1.61803…

φ² = φ + 1. Continued fraction: [1; 1, 1, 1, …]. Irrational and algebraic.

φ (phi) is the positive solution to x² = x + 1. This equation has a geometric meaning: if you divide a line segment so that the ratio of the whole to the longer part equals the ratio of the longer part to the shorter part, that ratio is φ. No other number has this self-similar property.

The golden division

Fibonacci ratios converge to φ

Table of Fibonacci ratios converging to phi
Fib pair	ratio	distance to φ
1, 1	1.000	0.618
2, 3	1.500	0.118
8, 13	1.625	0.007
55, 89	1.61818…	0.00015
to ∞	1.61803…	0

The golden ratio appears in the regular pentagon and pentagram, where diagonals intersect each other in the golden ratio. Every Fibonacci number divided by the previous one approaches φ. The continued fraction [1; 1, 1, 1, …] is the simplest infinite continued fraction: all 1s. This makes φ the hardest number to approximate by fractions, earning it the title "most irrational number".

The golden spiral: each square has a quarter-circle arc forming the nautilus curve

Cut a square from a golden rectangle. The remaining piece is another golden rectangle, smaller by factor 1/φ. Repeat forever. The arc traces the golden spiral seen in shells and galaxies.

φ satisfies φ² = φ + 1, so φ = 1 + 1/φ. Substituting repeatedly: φ = 1 + 1/(1 + 1/(1 + …)). This infinite continued fraction of all 1s is both the definition and the reason for its "most irrational" status. Computed to full precision: 1.61803398874989484820…

Fibonacci Sequence Golden Angle Silver Ratio Continued Fractions Irrational Numbers

The pentagon: every diagonal is exactly φ times the side

In a regular pentagon with side length 1, every diagonal has length φ ≈ 1.618. The diagonals also divide each other in the golden ratio. Draw all five diagonals and you get a pentagram: itself full of golden proportions.

Key facts about Golden Ratio φ

The golden ratio phi is approximately 1.61803398874989484820. It is the positive solution to x² = x + 1. Phi is irrational, algebraic, and the limiting ratio of consecutive Fibonacci numbers. It appears in the regular pentagon and icosahedron, in sunflower seed spirals, and in the proportions studied since ancient Greece. Its continued fraction [1; 1, 1, 1, ...] makes it the hardest real number to approximate with fractions, which is why phyllotaxis uses the golden angle derived from phi.

FAQ

What is 1/phi?

1/phi = phi - 1 = 0.61803... Phi is the only positive number where its reciprocal equals itself minus one. This also means phi^2 = phi + 1, so a golden rectangle can be divided into a square and a smaller golden rectangle forever.
Who discovered the golden ratio?

Euclid of Alexandria described it in his Elements (~300 BC) as the extreme and mean ratio. The term golden ratio was coined by German mathematician Martin Ohm in 1835. The Greek letter \u03C6 was chosen in the early 20th century by mathematician Mark Barr.
When were Fibonacci numbers linked to \u03C6?

Leonardo of Pisa (Fibonacci) published his famous sequence in 1202 to model rabbit populations. Johannes Kepler explicitly noted in 1611 that the ratio of consecutive Fibonacci numbers approaches \u03C6.
Is the golden ratio really found in famous artworks and buildings?

Evidence is mixed and often overstated. Many claimed appearances in the Parthenon or Renaissance paintings require selectively choosing measurements. Mathematician George Markowsky published a thorough debunking in 1992, though genuine appearances in nature are well-documented.
Why is \u03C6 called the most irrational number?

\u03C6 = [1;1,1,1,...] in continued fraction form - all 1s - making it the slowest-converging continued fraction. Rational approximations converge more slowly to \u03C6 than to any other irrational number.
Is the golden ratio irrational?

Yes. \u03C6 satisfies x^2 = x + 1. The ancient Greeks knew it was irrational. Unlike \u03C0 and e which are transcendental, \u03C6 is algebraic - a root of a polynomial with integer coefficients.

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