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LOGARITHMIC SPIRAL
Curve studied by Descartes and Torricelli in 1638, then
by Jacques Bernoulli (16541705).
Other names: equiangular spiral, Bernoulli spiral, spira mirabilis; the name "logarithmic spiral" was given by Varignon. Jacques Bernoulli had a logarithmic spiral engraved on its gravestone in the Basel Munster, with the epigraph: eadem mutata resurgo, "although changed (mutata), I arise (resurgo) the same (eadem)". Nonetheless, the engraver traced an Archimedean spiral... 
Polar equation: ,
k being the cotangent of the constant polar
tangential angle: .
Complex parametrization: . Transcendental curve. Curvilinear abscissa and intrinsic equation 2: , s being counted from the asymptotic point O. Radius of curvature: . Intrinsic equation 1: . Intrinsic equation 2: (with ). Pedal equation: . 
The logarithmic spiral can be defined as
 the curve the polar tangential angle
of which
remains constant (different from a right angle)
 the curve the curvature of which
is inversely proportional to the curvilinear abscissa
 the curve the radius of curvature
of which is inversely proportional to (and greater than) the radius vector
( with l
> 1)
Therefore, the logarithmic spirals with centre O
are the trajectories at angle.
of the pencil of lines issued from O.
The logarithmic spiral is also the stereographic projection from the south pole of the rhumb lines of the spheres with centre O, forming an angle with the meridians (since the stereographic projection is a conformal map).
Finally, it is the planar expansion of an helix of a cone of revolution.
The logarithmic spiral has an exceptional stability with respect to the classic geometrical transformations:
 any rotation with centre O and angle of the spiral amounts to a homothety with same centre and ratio , which in turn amounts to the identity if .
Rotation equals homothety! 
Homothety equals identity! 
 any inversion with centre O
amounts to a reflection about an axis passing by O.
 its evolute
is a logarithmic spiral with same centre and same angle(and,
besides, the limit of the nth evolute of any curve is a logarithmic
spiral).
 its caustics by reflexion or diffraction, when the light source is at O, are logarithmic spirals.
 the mating gear associated to a gear shaped like an logarithmic spiral is an isometric logarithmic spiral.
When a logarithmic spiral rolls on a line, the asymptotic point describes another line:
The logarithmic spiral is a solution to the three following physical problems:
1) The force centred on O that
makes a point in space describe a logarithmic spiral is proportional to
1/r^{3} (this force,
according to the Binet formula, is proportional to
which, here, is equal to ,
with u = 1/r).
2) the curve (called brachistochrone)
that minimises the travel time of a point moving freely along this curve,
when it is itself turning around a fixed centre O at constant speed,
in the case where the speed of the moving point vanishes at O, is
a logarithmic spiral.
3) A particle of mass m_{1} and charge q placed in a uniform magnetic field of intensity B with a speed v_{0} perpendicular to the field describes a logarithmic spiral with and .
see: perso.libertysurf.fr/hdehaan/mecanique/M6/M6_2/M6_2_cadre.htm
If u is a complex number different from 0, then
the image points of the geometric sequence
with common ratio u are located on an exact logarithmic spiral,
with parameter .
Opposite, u = 1+i/4. 

In the same way, any sequence points of polar coordinates
such that the
are in geometric progression with common ratio a and the
are in arithmetic progression with common difference b describes
a logarithmic spiral with k = ln a / b.
For example, on the opposite figure, the points with polar coordinates ((1,1)^{k } ; (k+2l)p/n) were traced, in blue if l is even, in red otherwise. These points are placed in a quincunx pattern at the intersection between concentric circles with radii in geometric progression and concurrent lines; but they are also located on the logarithmic spirals with equation , hence the nice visual effect. 

Here, the same spirals, but coloured in triangles in the style of this mosaic which adorned a Roman villa in Corinth in the 2nd century AD. 
Furthermore, as it could be noticed on the previous figures, the spirals and form two orthogonal lattices. 
Phyllotaxis
The scales of pine cones develop on a logarithmic spiral in such a way that the angular difference (in relation to the centre) between two consecutive scales is the golden angle equal to where is the golden ratio; the scales can then be classified into secondary spirals, the cardinality of a certain kind always being a Fibonacci number; here, there are 13 red spirals and 8 green ones. 


View of points of polar coordinates with r = 1 005, located on the spiral , for N ranging from 0 to 100, simulating the scales. The black angle connecting the first two scales is therefore equal to the golden angle. red secondary spirals and secondary green spirals appear, which rather models the sunflower florets. On the view secondary spirals. On the picture where the scales are indicated with the appearance number, we note that the kth secondary spiral is associated with is formed of the points satisfying N = k + a multiple of , for k ranging from 0 to ; for example, the first red spiral connects scales whose number is multiple of 21. The scales of the kth secondary spiral associated with are located on the logarithmic spiral of polar equation where is the integer closest to , ie , seen the links between the Fibonacci sequence and the golden ratio ( is the polar angle between two successive scales of a secondary spiral). Note: the angle between two red and green spirals that
intersect is constant but not related to the golden angle as we sometimes
see written, since it depends on the value of r!



If we increase the ratio r to 1.02, it is rather
the 8 =
red spirals associated with
and the 13 =
green spirals associated with ,
which are visible naturally, as for the pine cone; on the second figure
are drawn the 34 =
red spirals associated with
and the 55 =
green spirals associated with ,
again with r = 1.005.
These secondary spirals associated with the Fibonacci numbers appear visually because the fibonaccian multiples of the golden angle are close to the null angle (see wikipedia): we can see clearly in the figure where the scales are indicated with their number of appearance that the fibonaccian scales 3, 5, 8, 13, 21, 34, 55, 89 are close to the x axis. On this page, we can test different angles of rotation between two sunflower florets and note that the arrangement of the florets is optimal in the case of the golden angle.


If we consider a sequence of concurrent lines D_{1},
D_{2},
.... each of them forming an angle e with the next one, and so that
if, starting from M_{1} on D_{1},
M_{2}
on D_{2} is defined so that the angle
between D_{1} and M_{1}M_{2}
is equal to ,
and M_{3},
M_{4},
... are defined in the same way, then the moduli of the points M_{n}
are in geometric progression with common ratio
and their arguments are in arithmetic progression with common difference
e,
and thus are on a logarithmic spiral with ;
it can be checked that
goes indeed to cot
as goes
to 0, so that the spiral corresponding to goes
to the logarithmic spiral with k = cot .
here, y = 100°, e = p/10. 
here, y = 100°, e = p/50. 
REMARK: when = 90°, M_{i+1} is defined so that M_{i} is its orthogonal projection on D_{i} (careful not to mistake it with the Theodore spiral which is an approximate Archimedean spiral).
Another approximate construction consists in taking arcs
of circles with constant angles
and radii in geometric progression with common ratio ,
with tangential connection.
For a very beautiful special case, see the
golden
spiral.
The spiral traced on a cone which is projected on a logarithmic
spiral is the conical
helix.
See also the spiral of the rotating rod and the mutual pursuit curves.
Ceiling of a room in the Pavlovsk Palace in Saint Petersburg
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© Robert FERRÉOL 2019