# MATH

MANDELBROT SET

This is a journey into a place called the Mandelbrot Set, which is a mindscape. It’s a place anyone in the universe can explore. Aliens in another galaxy can visit the same exotic places in the Mandelbrot Set that we do. The red arrow indicates where I’m going to zoom in.

To understand what the Mandelbrot Set is, I have to explain complex numbers from second year algebra. That name doesn’t mean the numbers are complicated, it just means they are numbers which have two parts, and when the two parts are multiplied, they have to be handled separately just like a binomial. For instance, 2 + 3i is a complex number. If I multiply it by itself (square it), I end up with 4+ 6i + 6i + 9i^2. Now to continue I have to explain what i is. i is defined as the square root of -1. The normal operations of algebra don’t give us the square roots of negative numbers, because the squares of negative numbers are always positive. But in Algebra Deluxe we have defined i, by fiat, as the square root of -1 so we don’t end up quadratic equations with only one solution, or even none. In an unfortunate choice of math jargon i is called an imaginary number but that doesn’t mean it’s pure fantasy. For instance, i^i = e(-π/2)

So now that we know i is the square root of -1, it follows that i^2 is simply -1. In my squared complex number above, the solution becomes 4 + 12i – 9, or -5 + 12i. Now what good are complex numbers? They extend the one-dimensional number line of real numbers to the two-dimensional Argand Plane. The real part of my complex numbers (+2 in the first case, -5 in the second case) represent the horizontal position of a point. The “imaginary” part of my complex number (+3 in the first case, +12 in the second case) represent the vertical position of a point. So here’s what we do to generate the Mandelbrot set. Each point on the screen is a complex number which is multiplied by itself repeatedly. Each solution is a new point. If the solutions remain on the screen after, say, 100 iterations, that point is assigned a dark color. If the solutions fly off the screen, they are assigned lighter colors based on how soon they depart. And the result is a wonderful fractal universe that (with the proper software, such as Xaos) you can dive into real time with your mouse, as it smoothly zooms in (or out if you hold down the right-clicker instead).

This universe is like Mount Everest. We didn’t make it. God didn’t even make it. It’s just “there”, and it has always been there, like the distribution of primes. You can go down and down, forever and ever, and the funny thing is, sometimes you find little baby copies of the whole Mandelbrot Set down there, but they are never exact copies, only endless variations of the theme. It’s really a mind blower.

Evidence for evolution exhibit 13451A

The four genera of great apes, namely humans, chimpanzees, orangutans and gorillas,  all need an external source of vitamin C.  At other points in evolutionary history, bats and guinea pigs also lost this vitamin C-producing  gene. Yet, many other mammals don’t need supplemental vitamin C in their diet because they possess a functioning copy of the relevant gene and are able to produce it on their own.  That is why your dog or cat gets by just fine without orange juice.  The most satisfying explanation for these observations is descent with modification from a common ancestor

The Three Body Problem

I’m attempting to write a science fiction novel, Terminal Cruise, and it’s of the hard SF variety, ala the Three B’s, Greg Bear, David Brin, and Greg Benford.   So if I write about rockets, I have to become something of an amateur rocket scientist.   It’s not like Star Wars where the characters just get in a ship and fly from Tatooine to Alderaan.  My characters ride along ascent ellipses just like real astronauts do.

The Two Body Problem has an exact solution.  If only the Earth and the Moon existed in the whole universe, then we could calculate their position at any time in the future from a simple equation.  But add a third body, let us say a space station, and no general solution exists.  We can only approximate the answer numerically, running what is, in effect, a mathematical simulation of the problem.

But in 1772,   Joseph-Louis Lagrange discovered that the restricted coplanar three-body problem has five solutions if the mass of the largest body is at least 24 times greater than the mass of the second-largest body, and the mass of the third body is negligible compared to the other two. This works in the case of satellites in the Earth-Moon system, because the Earth is 81.3 times more massive than the Moon, and satellites are but a flea compared to both of them.

But the equation is a quintic, with a term that is taken to the fifth power.  Quintics are too complex to solve for their roots.  Cubics and quartics can be solved if you have plenty of scratch paper and no distractions.  Quadratics and linear equations are solved by high-school kids.  What this means is that as a practical matter we still need to use numeric methods to solve for the locations of the Lagrangian Points.  We can get close, but never exactly there.

To simplify the math, the 384,400 kilometers between the centers of the Earth and Moon are defined as one Distance Unit (DU). So the positions of L4 and L5 are simply 1.0 DU from both the Earth and the Moon, forming two equilateral triangles, as shown here in this contour map which combines the gravity of the two bodies plus the centrifugal force developed by the monthly rotation of the system.

As the map indicates, L4 and L5 are like bean-shaped valleys. If a satellite is parked there, the sun will perturb it into a bean-shaped “halo orbit” that circles close around the valley floor, and it will stay there forever. These are good places to park multi-trillion dollar artificial space colonies without losing them. Hundreds of asteroids have been found to exist in the corresponding Sun-Jupiter L4 and L5 points.

Lagrange Points L1, L2, and L3 all lie on the line that passes through the Earth and Moon. From L3, the Earth will always block the view of the Moon, so the joke goes that L3 is a great place to build an orbiting hospital to treat people prone to becoming werewolves. L2 is behind the Moon, so it is a great place to build a radio telescope immune to any interference from the noisy radio chatter of the Earth. L1 is between the Earth and Moon, so it is a great place to build a communications relay. The map shows these three points to be in “saddles” or mountain passes, so if a spacecraft runs out of station-keeping propellant and is perturbed away from them, it will drift to regions unknown.

To find the approximate location of these points we create a chart with the origin (0,0) at the center of gravity between the Earth and Moon, and we define a value μ to be the mass of the Moon relative to the mass of the whole system of the Earth plus Moon.

The position of the Earth is at – μ DU, which means the origin point (0,0) is about 1,000 miles below the surface of the Earth. The position of the Moon is at 1 – μ DU.

The position of L3 is -1 + (7/12) μ + (1127/20736) μ^3 + (7889/248832) μ ^4 = -0.992912 DU

To find the next two points we define a new value z = (μ/3)^1/3

The position of L2 is 1 – μ + z + (1/3)z^2 – (1/9)z^3 + (50/81)z^4

= 1.155669 DU

The position of L1 is 1 – μ – z + (1/3)z^2 + (1/9)z^3 – (58/81)z^4

= 0.836005 DU

The things we do for art.