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The Length of the Golden Section Spiral - 2014.

Golden section spiral engraved and painted sandstone Golden section spiral diagram 1
30cm by 34cm.


The spiral is made from an infinite series of quarter circles with radii r1, r2, r3, r4, ...

So the length,

Golden section spiral equation 1



The sum of this infinite convergent geometric series is

Golden section spiral equation 2
So
Golden section spiral equation 3


Two semicircles with diameters of Φ & 1 also have a combined length of

Golden section spiral equation 4


So the finite green curve has the same length as the infinite red spiral:

Golden section spiral diagram 2




Other equal length spirals are possible:



The blue spiral is made from semicircles whose diameters decrease from 1 in the golden proportion:

Its length is
Golden section spiral equation 5


So :
Golden section spiral equation 3

Golden section spiral diagram 4




The purple spiral in this diagram is made from three quarter circles whose diameters decrease in the golden proportion. The initial diameter of two thirds makes the length of this spiral equal to all the other spirals.



Golden section spiral equation 5
Golden section spiral diagram 6




The orange spiral in this diagram is made from parts of circles decreased in the golden proportion; about 222 of arc. Again the diameters of these arcs decrease in the golden proportion. The initial diameter of a half of Φ makes the length of this spiral equal to all the other spirals.



Golden section spiral equation 8
Golden section spiral diagram 7




Golden section spiral equation 9



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