The number e has many names – Euler’s number, the natural constant, the natural base, etc. You might ask, what on earth is natural about e? It’s irrational, and it’s got a complicated (but understandable!) definition. If we want to call any irrational number natural, we might as well call π natural—at least there’s a reaso…
Ok. Let’s hear e’s side of the story. Because as we shall see, traces of e can actually be found almost everywhere in our lives… If you’re in the northern hemisphere right now, you may have noticed that spring is on its way, and your room is filled with bees and moths and flies again. At night, they hang around your desk lamp. You wave them away, but they always come back to the source of light. You’ve probably heard of the saying “like a moth to a flame” and may have wondered, why are moths attracted to flames (or more modernly, lights)? Is it because they love light? Well, if they love light so much, why don’t they all fly up towards the sun?
In fact, moths and many other insects have evolved since ancient times to navigate using light rays from the sun, the moon, or other stars. These objects are far enough away that by the time they reach Earth, their light rays lie almost parallel to each other. The insects navigate by flying at a constant angle to these light rays, which enables them to travel in a straight line. And as we all know, the quickest way to get somewhere is to walk the straight line between there and where you are now.
This sounds like a great strategy to fly in a straight line! Good for them!
Well, humans came around and invented fire, and fire emits light rays too. But this time it’s different—the flames are closer now, and their rays are no longer “almost parallel.” So the poor little moths, still thinking that the light rays are from a distant star, end up flying like this:
Where in this example we’ve used 90º for the angle between the moth’s path and the light rays (of course depending on the destination, our little moth may fly with another angle). From the diagram we can see how the path spirals around the flame—well, sort of. The spiral is not smooth because we only have eight light rays here for reference. In reality, there are way more. As a result, our moth does indeed take a smooth spiral path around the source of light, like this:
This spiral has a name in mathematics: the logarithmic spiral. It has the property that the angle between each “ray” and the spiral is always the same, so it is also named the “equiangular spiral.” In polar coordinates, it can be written as a variation of r = eθ. But, what are polar coordinates, you might ask? Well, we know that in Cartesian coordinates (the common x-y coordinates we often use to plot functions), a point on a plane is defined with it’s x-value and y-value. Similarly in polar coordinates, a point on a plane is defined with a distance r and an angle θ:
And by completing the right-angled triangle in the diagram on the right, we can get the following relationship between the two coordinate systems:
x = r cos(θ)
y = r sin(θ)
The equation r = eθ tells us that all points on the logarithmic spiral share the same property that their distance from the origin is equal to exactly e to the power of its angle—pretty cool, right?
Moths’ flight path is not the only place where the logarithmic spiral emerges. It can be found almost everywhere in nature, from plants to shells to galaxies. Like how π is hidden among circles, e lurks among these logarithmic spirals that naturally occur in our universe. Do you think it deserves its name as the ‘natural base’ now?
You can read the first article in the series on ‘e and the Sheldon Mitosis’ here.