In geometry, a golden spiral is a logarithmic spiral whose growth factor is φ, the golden ratio. That is, a golden spiral gets wider (or further from its origin) by a factor of φ for every quarter turn it makes. A golden spiral with initial radius 1 has the following polar equation :
r
=
φ
θ
2
π
{\displaystyle r=\varphi ^{\theta {\frac {2}{\pi }}}\,}
The polar equation for a golden spiral is the same as for other logarithmic spirals, but with a special value of the growth factor b:
r
=
a
e
b
θ
{\displaystyle r=ae^{b\theta }\,}
or
θ
=
1
b
ln
(
r
/
a
)
,
{\displaystyle \theta ={\frac {1}{b}}\ln(r/a),}
with e being the base of natural logarithms, a being the initial radius of the spiral, and b such that when θ is a right angle (a quarter turn in either direction):
e
b
θ
r
i
g
h
t
=
φ
{\displaystyle e^{b\theta _{\mathrm { right } }}\,=\varphi }
Therefore, b is given by
b
=
ln
φ
θ
r
i
g
h
t
.
{\displaystyle b={\ln {\varphi } \over \theta _{\mathrm { right } }}.}
The numerical value of b depends on whether the right angle is measured as 90 degrees or as
π
2
{\displaystyle \textstyle {\frac {\pi }{2}}}
radians; and since the angle can be in either direction, it is easiest to write the formula for the absolute value of
b
{\displaystyle b}
(that is, b can also be the negative of this value):
|
b
|
=
ln
φ
90
≐
0.0053468
{\displaystyle |b|={\ln {\varphi } \over 90}\doteq 0.0053468\,}
for θ in degrees;
|
b
|
=
ln
φ
π
/
2
≐
0.3063489
{\displaystyle |b|={\ln {\varphi } \over \pi /2}\doteq 0.3063489\,}
for θ in radians A212225.
An alternate formula for a logarithmic and golden spiral is:
r
=
a
c
θ
{\displaystyle r=ac^{\theta }\,}
where the constant c is given by:
c
=
e
b
{\displaystyle c=e^{b}\,}
which for the golden spiral gives c values of:
c
=
φ
1
90
≐
1.0053611
{\displaystyle c=\varphi ^{\frac {1}{90}}\doteq 1.0053611}
if θ is measured in degrees, and
c
=
φ
2
π
≐
1.358456.
{\displaystyle c=\varphi ^{\frac {2}{\pi }}\doteq 1.358456.}
A212224
if θ is measured in radians.
