Logarithmic Spirals
Francesca Farris
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
The BasicsThe 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
Spira MirabilisThis 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
Polar CoordinatesLogarithmic 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.
Logarithmic FormulaIn 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.
The Golden SpiralIn 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.
Logarithmic Spirals in NatureThe 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
In ConclusionThe 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