The Golden Section

This page is a sub-page of our page on Euclidean Geometry.

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The interactive simulations on this page can be navigated with the Free Viewer
of the Graphing Calculator.

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Related sources of information:

An artist theory on the physics of ‘Time’ as a physical process. Quantum Atom Theory

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The golden section in mathematics:

The logarithmic spiral
Fibonacci numbers
Polygon spiral.svg

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The golden section in music:

Music and the Fibonacci Sequence and Phi
Nature’s smallest and most beautiful secret (The pythagorean comma is an audible rate that feeds back as phi (the golden section)
Natures smallest and most beautiful secret (Some slides for the ‘Small is Beautiful’ Festival at CAT 2015)
Golden Mean Music (by David Friddle)
• ‘STRINGULARITY’ and ‘PER-CAPITALISM’ – for a ‘Well Tempered Climate Accord’

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The golden section in art and design:

The golden ratio – design’s biggest myth

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/////// From Wikipedia: [ https://en.wikipedia.org/wiki/Logarithmic_spiral ]

A logarithmic spiral, equiangular spiral or growth spiral is a self-similar spiral curve which often appears in nature. The logarithmic spiral was first described by Descartes and later extensively investigated by Jacob Bernoulli, who called it Spira mirabilis, Latin for “miraculous spiral”. [...]

Although this curve had already been named by other mathematicians, the specific name (“miraculous” or “marvelous” spiral) was given to this curve by Jacob Bernoulli, because he was fascinated by one of its unique mathematical properties: the size of the spiral increases but its shape is unaltered with each successive curve, a property known as self-similarity.

Possibly as a result of this unique property, the spira mirabilis has evolved in nature, appearing in certain growing forms such as nautilus shells and sunflower heads. Jacob Bernoulli wanted such a spiral engraved on his headstone along with the phrase “Eadem mutata resurgo” (“Although changed, I shall arise the same.”), but, by error, an Archimedean spiral was placed there instead.

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