Logarithmic Spirals

The logarithmic spiral has many interesting properties, and shows an interesting connection between φ and e. One way to make a logarithmic spiral: take a golden rectangle, and then put another golden rectangle in it, so that the width of the large rectangle is the length of the small rectangle. Now add another rectangle in the smaller rectangle, and keep adding rectangles for as long as you can. If you inscribe this series of rectangles with a spiral, then that spiral will be a logarithmic spiral. The general equation for logarithmic spirals is r=e^θ, and this golden spiral (made with golden rectangles) is a transformation on it.

Polar Graphing

r=e^θ is a polar equation used to graph the basic logarithmic spiral described above. Polar equations and polar graphing are just another way to describe where points are in the x-y plane. The normal graphing that you are probably most familiar with describes where points are by their horizontal and vertical distance from the origin. Points are given by their x and y coordinants (x,y). Sometimes it's easier to use polar coordinants instead; they make circles and spirals really simple to describe.

Polar graphing uses r and θ instead of x and y to describe a point's position on the graph. Imagine a straight line between the origin (0,0) and the point you want to describe. The radius r is the length of that line. θ ("theta") is the angle (measured counterclockwise) between the positive x-axis and that straight line we imagined. It is measured in radians.

To graph a polar equation like r=e^θ we start with θ equal to zero. From the equation, if we plug in 0 for θ we get a radius of 1. So, our starting point in the (x,y) coordinant system is (0,1). From that point we draw a line by increasing our angle theta and letting it determine the distance r our graphed line should be from the origin. The above equation is special because a given increase in theta always causes the same ratio of increase in r.

Jakob Bernoulli’s Favorite

The Bernoullis were a family of mathematicians and scientists who were active for 150 years. Johann, his brother Jakob, and Jakob’s son Daniel were the most prominent among the family, which was famous for the feuding between its members. Jakob Bernoulli did a lot of work on different curves, but his favorite was the logarithmic spiral. In his own words:

“Since this marvelous spiral, by such a singular and wonderful peculiarity … always produces a spiral similar to itself, however it may be Involved or evolved, or reflected of refracted… it may be used as a symbol, either of fortitude and constancy in adversity, or of the human body, which after all its changes, even after death, will be restored to its perfect self.”

Jakob Bernoulli wanted his tombstone to be engraved with a logarithmic spiral. Unfortunately, he only got an Archimedean spiral, which is a spiral that increases by a constant difference, not a constant ratio.

Spira Mirabilis

This spiral has many interesting properties, which made Jakob Bernoulli call it 'Spira Mirabilis’ (the marvelous spiral).

The general equation of these spirals is r=e^aθ, where a is a constant. The equation we looked at earlier, r=e^θ, is just a Spira Mirabilis with a=1.

The Logarithmic Spiral is also sometimes called the equiangular spiral, because any tangent to the curve will make the same angle with a line drawn from that tangential point to the origin. The only other curves that do this are circles, where the angle is always π/2 (90º).

Some basic transformations: if a is positive then the spiral will start at (0,1) and grow outwards as we increase θ. If a is negative, the spiral will get smaller and smaller as θ increases, and spiral into the origin. If a=0, then the exponent of e is always 0, and consequently, r is always 1 regardless of θ. This is the equation of the unit circle.

Some Nifty Properties

What Bernoulli was talking about in the above quote was how logarithmic spirals hardly change under different geometric transformations. For instance, you can take the inverse of a logarithmic spiral, such that the new spiral when multiplied with your old spiral would give simply the number 1. This inverse is the same as the old curve except that the sign of ‘a’ has changed. The only difference is that a growing spiral changes to one that spirals inward and vice versa. The inverse is still a logarithmic spiral.

Other transformations which use the existing curve to create a new curve include the evolute, involute, caustic, and pedal. These transormations all act nicely on logarithmic spirals, creating an equal spiral to the original. You can explore these transformations for yourself at this ‘Famous Curves Index’.

Make Your Very Own Logarithmic Spiral(s)

In a perfectly square room, get four people to stand in each corner, facing the center. At a signal make them all start walking towards the person to their right, at the same speed. Each person will start walking along a wall, but as the person they are aiming at moves, everyone's course will gradually start veer off the initially straight path and spiral towards the center. If everything works out, then everyone will meet in the exact center of the room, having traveled in a logarithmic spiral from their corner. This experiment works with any regular polygon, adjusting the number of participants to the number of sides of the polygon. It is sometimes known as the four bug problem, the mice problem, the beetle problem, or the four dog problem, but generally people are easier to control. See an animation.