Logarithmic spirals are spirals found in nature that are self-similar, meaning parts of the spiral resemble the whole. They differ from Archimedean spirals in that the distance between curves increases geometrically rather than remaining constant. Jacob Bernoulli called them "miraculous spirals" because they increase in size while retaining their original shape. Logarithmic spirals can be expressed using the formula r=a.ebθ and are seen throughout nature in structures like shells, hurricanes, galaxies, and more, demonstrating mathematics' link to the physical world.

1. Logarithmic
Spirals
Francesca Farris

2. What are they?
Logarithmic spirals are spirals found in nature, unique
because they are self-similar. Self-similarity means that
a part of an object or image is the same as the whole.
Self-similarity in a fern
plant
Fractals, which we learned about in class,
are self-similar. The link here is to an
animated Mandelbrot sequence zoom. You
can see that as it zooms deeper and
deeper into the fractal set, the image stays
the same. Logarithmic spirals are also
seen in the animation.
Logarithmic
spiral

3. The Basics
The basic spiral is the
Archimedean spiral, in
which the distance
between the curves of the
spiral is constant, as seen
to the right.
In logarithmic spirals, the
distance between the
curves increases in
geometric size by a scale
factor, but the angle at
which each curve is
formed is constant and the
spiral retains its original
shape.
Archimedean spiral
Logarithmic spiral in nature

4. Spira Mirabilis
This fact, that logarithmic spirals have the unique quality of
increasing in size while retaining an unaltered shape, caused
Jacob Bernoulli, in his studies, to call them spira mirabilis
(“miraculous spiral”, in Latin).
Interestingly, Jacob Bernoulli was so fascinated by logarithmic
spirals that he wanted to have one put on his headstone, along
with the Latin quote “Eadem mutata resurgo” (“Although
changed, I shall arise the same”), which describes logarithmic
spirals very well. Ironically, an Archimedean spiral was placed on
his headstone by mistake.
Spira
mirabilis, as
seen in a
shell
Spira
mirabilis, as
seen in a
head of
Romanesco
broccoli

5. Polar Coordinates
Logarithmic spirals can be created on a polar coordinate
graphing system, rather than the Cartesian coordinate
system of graphing which we would use to graph normal
functions.
To graph polar functions, you would use a number that lies
along the x-axis, just like with the Cartesian system, as your
first point. But rather than using a number that lies along the
y-axis as your second point, you would use an angle to
determine where that point was.

6. Logarithmic Formula
In order to graph a logarithmic spiral (or any polar coordinates),
you must find the values of r and theta (r,θ), just like how you
would find the values for x and y (x,y) to graph a normal function.
Logarithmic curves are expressed using the formula r=a . ebθ,
where r is the radius, or distance from the center point (called the
pole), e is the base for the logarithm, a and b are constants, and θ
is the angle of the curve. You can use this formula, substituted with
values on a graph for a and b, to create a logarithmic spiral.
By increasing a, the distance of the curve from the pole on the
graph, you are widening the spiral, but by leaving θ at a constant,
you are keeping the angle the same; therefore, the spiral does not
change shape.

7. The Golden Spiral
In class we learned about the golden ratio and how it can
form a golden spiral, using the growth factor phi (ϕ). This
sort of spiral increases in size by a rate that follows the
Fibonacci sequence (1+0=1, 1+1=2, 2+1=3, 3+2=5, 5+3=8,
8+5=13, …). This spiral forms a golden rectangle, which is
an example of the golden ratio at work, as well as the
Fibonacci sequence; each square in the golden rectangle
increases in size based on the next number in the Fibonacci
sequence.

8. Logarithmic Spirals in Nature
The logarithmic spiral is a prime example of nature’s
perfection in its fundamental structure. These spirals can be
seen in many plants, animal shells, the path birds fly on to
spiral in on prey, the formation of hurricanes and whirlpools,
spiral galaxies (like the Milky Way), and many other things.
Logarithmic spiral as seen
in a whirlpool Logarithmic spiral as seen
in the galaxy

9. In Conclusion
The prevalence of so many logarithmic and other
similar spirals in nature can be taken as a philosophical
statement on the similarity of all things, and teaches us
that despite variations, there are some things that we
all share. This, among other things, is one example of
the link between mathematics and our tangible
existence.
Image designed by Alex Grey