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