A friend wrote to say that he’d heard somewhere that “The human body has a mass that, more or less, is halfway between the mass of the proton (~1.672×10 2 kg) and the mass of the Sun (~1.988×10 3 kg). A value very close to the mass of an average human body is the geometric mean of those two values.”
Our physics color commentator, Rob Sheldon, responds,
Fractals have some very interesting properties, and although they were only named relatively recently (Mandelbrot etc), they are all over God’s creation: rocks, trees, ferns, galaxies etc. One of the advances of computer graphics in gaming is the use of fractals to create realistic nature scenes.
Okay, what is a fractal?
It is not just the Mandelbrot or Julia set, though that is what a mathematician will show you. It is best understood by example. Let me use 2D examples first, though Cantor did it in 1-D in the late 19th century. A square of side L has an area of L^2 and a perimeter of 4L. If one were to divide the area by the perimeter, the answer would be kL where k is some constant. We could do this for triangles, circles, elephant silhouettes and we’d always get the same answer kL where k might change depending on the shape. What doesn’t change is the exponent of L–> L^1.
Everyone thought this was totally normal until Koch’s snowflake curve. Take an equilateral triangle, and add little triangles on each side to make a star of David. Then add even tinier triangles on each of the 6 points to make a 18-pointed “snowflake”. Then add even tinier triangles to make a 48 pointed snowflake. You can keep this up forever, but the snowflake will never extend outside the circle drawn around the original triangle. So the area is not infinite, it converges to a finite number, 2/5 * sqrt(3) L^2.
On the other hand, the perimeter is getting longer and longer, and indeed never converges but after each iteration is 4/3 longer than the previous iteration. If we take (Area/Perimeter)n / (Area/Perimeter)(n+1) we can see how the ratio changes with each iteration “n”.
From Wikipedia where “n” is the iteration number, the area = kL^2[8-3(4/9)^n] whereas the perimeter = 3L(4/3)^n. Then the ratio has the L^2/L as we expected, but multiplied by some number. Since that number is growing by some fixed amount, we can take the logarithm and determine the exponent.
log(ratio) = log(8 – 3(4/9)^n) – log (4/3)^n ==> log(~8) – n log(4/3),
where 4/9 is smaller than one, and the more iterations the less it contributes, so in the limit it vanishes. That leaves the multiplicative exponent as log(4/3) or 1.26186, which is its “fractal dimension”. The length of the perimeter depends on how powerful a microscope you look at it. Navigating the coastline of England is longer by foot than by boat. And if you were an ant, even longer still.
This example applies to 3d shapes as well, where we could compare the volume and the surface area. For example, a tree can be drawn so the trunk bifurcates every L/n distance. And as it turns out, many natural objects possess this fractal property. That is to say, they do not resemble Minecraft at all.
But why is this important?
Glad you asked. If you lost your son in the woods, a search party would be advised to form a line and methodically beat through the woods. The area covered = length of line * distance travelled. But suppose you were by yourself, how do you go about it? This is the same problem ants have to solve when they look for food. It turns out a back-and-forth sweep is too slow, too inefficient. The ants use a fractal search algorithm with lots of twists and turns. And they avoid searching the same place twice by laying down a trail of pheromones. You’d have to use some GPS software or a large ball of twine, but with practice you could find your son twice as fast.
Or suppose you have photocells you want to maximize the solar power collected but you have limited amount of wire to connect them up, how should you arrange them to minimize the wire cost? Like a tree–using a fractal branching wiring pattern. Fractals connect areas with lengths, or volumes with areas using the least amount of material–they are an optimal solution.
So we are finally back to why neurons in the brain and galaxies in the universe are fractals–they are optimum solutions to connecting volumes with 1-D wires. We expect biology to have designed features that optimize the neuron to brain-matter, but why would galaxies do this?
To restate the obvious—because they are solving an optimizing problem, perhaps even the same optimizing problem in the brain—how to connect the matter of the universe together using a minimum amount of stuff. Now comes the fun part: Why? How?
I think the answer to both those questions is “Comets”. Comets carry information in the form of cyanobacteria, and therefore the universe is wired to transmit information efficiently. And the How is likewise comets because they form the nucleus of stars and galaxies–providing the gravitational instability that creates both out of distributed clouds of gas. Stepping back from the details, the universe, like our brains, has been designed for information flow because apparently the Designer likes information, (and beetles, said Haldane).
The physics of the universe and the physical characteristics of Earth determine what the maximum number of species will be for animals of different size scales. For an 8,000-mile-diameter Earth, the greatest theoretically possible number of animal species peaks for animals with adult body sizes equal to about one centimeter. The average adult body size for beetles is about one centimeter. Therefore, given God’s goal to pack Earth with as much life as possible and as diverse as possible, it should come as no surprise that there are so many species of beetles.
What’s the matter with beetles anyway? Look at this: