Tuesday 25 January 2011

The First Fractal, The Second Curve

If we assume that the length of a circle's circumference is known (well, do you KNOW pi?), then it was known before 100 BC. So what was the next curve whose length was figured out, and when.. Ok, do that movie thing where you have the pages ripping off the calendar...keep waiting...more pages... lots more... and then in the least expected place.

After centuries of trying to figure out the length of the elliptic circumference, or the length of a parabola... and failing,... mathematicians had about decided that there just were not many curves that could be rectified (find the length).

Then, in 1645, Evangelista Torricelli (he's the guy who invented the barometer) was playing with a logarithmic spiral. Now the logarithmic spiral has several nice properties. First,it is equiangular, that is, if you draw a tangent to the spiral at any point, the angle between that tangent and a line drawn to the origin will always be the same angle. If you write the formula in polar form as
then b is the cotangent of the angle. As b gets close to zero, the angle approaches 90 degrees, and the spiral gets more circular. In fact when b=0, the equation produces a circle.

The log spiral also has the property that if you drew a ray from the origin in any direction, the intersections with the spiral would occur at distances from the origin that are in a geometric ratio; if it crosses at distances of one unit and two units, the next cross would be at four units, and then at eight, etc.

But it was this property that made it such an unlikely candidate to be the 2nd rectified curve because it also decreases toward the origin in proportional steps. So working the same curve I mentioned above back toward the origin it will cross the x-axis at 1/2, and then 1/4, and then...."Holy infinite sequence Batman, this goes on forever." But in spite of the fact that there are an infinite number of spirals around the origin, the distance from any point on the curve to the origin is finite.... read twice, that's right... finite.. and Torricelli not only figured that out, he put a number on it (and the children all say "Ooohhhh.") He did it by recognizing that, although the word would not be invented for over 300 years to come, the log spiral is a fractal. Any part of the curve is self-similar to any other. From this, Torricelli was able to determine that the infinite number of loops around the origin would simply add up to the distance from the tangent to the y-axis. It wasn't quite calculus...but we were sneaking up on it.

The log spiral so enchanted Jacques Bernoulli that he commissioned it to be put on his grave. [unfortunately it wasn't drawn correctly]. His Latin inscription said, "eadem mutata resurgo" which translates to something like "though changed, I will arise the same."

I got an email after I posted this reminding me that Descartes was the discoverer of the equiangular spiral. It seems it comes up when studying dynamics.
I have also been guided to a neat approach to drawing the curve at Wolfram's Mathworld page:

"The logarithmic spiral can be constructed from equally spaced rays by starting at a point along one ray, and drawing the perpendicular to a neighboring ray. As the number of rays approaches infinity, the sequence of segments approaches the smooth logarithmic spiral (Hilton et al. 1997, pp. 2-3). "

Mathworld also has a nice page to illustrate that the log spiral will show up in mutual pursuit problems.
Beautiful story about Toricelli and the barometer, probably false, but still cute: When he was using water filled barometers the tubes would have to be over thirty feet high. Torricelli had one stick through the roof of his house and he put a red floating ball in the water to make it more visible. Unfortunatly, his superstitious neighbors could see this, and became somewhat upset when the appearance and disappearance of the floating "devil" seemed to cause the weather to change... so they stormed his house and burnt it down. That was supposedly ONE Of the reasons he switched to a mercury barometer.

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