Golden ratio

The irrational number (1 + √5)/2, approximately 1.618, that divides a segment so the whole-to-larger-part ratio equals the larger-to-smaller-part ratio.

Golden section: A division of a line segment into parts whose whole-to-larger ratio equals the larger-to-smaller ratio. This geometric division is the proportion that defines the number.
Euclid's Elements: A mathematical treatise by Euclid that systematically presents geometry and number theory. Its construction of division in extreme and mean ratio gives the classical geometric definition.
Euclid: An ancient Greek mathematician whose Elements became a foundational geometry textbook. Euclid described the defining division as a geometric construction, not a named number.
Golden-angle phyllotaxis: A plant-growth pattern in which successive organs are separated by an angle near 137.5 degrees. The angle is derived from the golden ratio and helps distribute leaves or seeds.
Silver ratio: The quadratic irrational number 1 + √2, approximately 2.414, associated with the proportions of regular octagons. It is another named metallic ratio, but has a different defining equation and geometry.
Irrational number: A real number that cannot be expressed as a ratio of two integers. The golden ratio has a nonterminating, nonrepeating decimal expansion.
Quadratic formula: A formula giving the roots of a quadratic equation from its coefficients. Applying it to x² − x − 1 = 0 yields (1 + √5)/2.
Pythagoreans: An ancient Greek philosophical community associated with mathematics, music, and number symbolism. Pentagonal geometry linked their mathematical tradition to the proportion.
Pentagram: A five-pointed star polygon formed by extending the sides of a regular pentagon. Its intersecting line segments repeatedly form golden-ratio divisions.
Plastic number: The real solution of x³ = x + 1, approximately 1.325. It is a prominent irrational constant associated with the Padovan sequence rather than Fibonacci ratios.

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