Editing Golden ratio (section)

====Golden spiral====
{{main|Golden spiral}}
[[File:FakeRealLogSpiral.svg|thumb|The [[golden spiral]] (red) and its approximation by quarter-circles (green), with overlaps shown in yellow]]
[[File:Golden triangle and Fibonacci spiral.svg|175px|thumb|A [[logarithmic spiral]] whose radius grows by the golden ratio per {{math|108°}} of turn, surrounding nested golden isosceles triangles. This is a different spiral from the [[golden spiral]], which grows by the golden ratio per {{math|90°}} of turn.<ref name=loeb-varney />]]
[[Logarithmic spirals]] are [[self-similar]] spirals where distances covered per turn are in [[geometric progression]]. A logarithmic spiral whose radius increases by a factor of the golden ratio for each quarter-turn is called the [[golden spiral]]. These spirals can be approximated by quarter-circles that grow by the golden ratio,<ref name=quarter-circles /> or their approximations generated from Fibonacci numbers,<ref name=diedrichs /> often depicted inscribed within a spiraling pattern of squares growing in the same ratio. The exact logarithmic spiral form of the golden spiral can be described by the [[Polar coordinate system|polar equation]] with {{tmath|(r,\theta)}}:
<math display=block>r = \varphi^{2\theta/\pi}.</math>

Not all logarithmic spirals are connected to the golden ratio, and not all spirals that are connected to the golden ratio are the same shape as the golden spiral. For instance, a different logarithmic spiral, encasing a nested sequence of golden isosceles triangles, grows by the golden ratio for each {{tmath|108^\circ}} that it turns, instead of the {{tmath|90^\circ}} turning angle of the golden spiral.<ref name=loeb-varney /> Another variation, called the "better golden spiral", grows by the golden ratio for each half-turn, rather than each quarter-turn.<ref name=quarter-circles />