What is the Golden Ratio? Phi Explained

The golden ratio, written as φ (phi), is approximately 1.61803398874.... It describes a proportion where the ratio of the whole to the larger part equals the ratio of the larger part to the smaller part.

The definition

If you divide a line into two parts so that the whole:large equals large:small, you get phi. Algebraically: φ = (1 + √5) / 2

Connection to Fibonacci

The Fibonacci sequence (1, 1, 2, 3, 5, 8, 13, 21...) converges to phi. Dividing each number by the previous one gets closer and closer to 1.61803...

Where phi appears

• Nautilus shells: the spiral grows at a rate related to phi
• Sunflower seeds: arranged in spirals whose counts are Fibonacci numbers
• Pentagons: diagonals divide each other in the golden ratio

Key facts

Fact	Value
Decimal value	1.61803398874...
Algebraic form	(1 + √5) / 2
Continued fraction	[1; 1, 1, 1, ...]
Named by	Mark Barr, early 20th century

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