Euclidian geometry attempts to translate natural form into permutations of lines, but the universe is in fact fractal. To use the classic example, the coastline of, well, anywhere, is of INFINITE length- even though that is impossible, it is true. It simply depends upon the scale at which you measure it. The smaller the scale, the longer the line, until you are wrapping around the molecules of the rocks. For more detailed information, go read Rudy Rucker, George Cantor, or James Gleick's excellent book "Chaos", from which I have, um, borrowed several examples. The connection between fractals and Mars, I'll give you here. It may cause your brain to itch a little.
There is something mathematicians call a Koch
line, or Koch snowflake. Draw a triangle ( I could show you, but I want you to do this in your mind), and then add another triangle, one third as big, on each side. Voilą. A Star of David. Now, on each exposed side of those add-ons, do the same thing. Repeat. You could keep adding ever smaller triangles forever, without any bumping into each other. Nature does just that, with one difference- the unit isn't really a triangle.
Now consider the letter A - it represents a particular specific sound, but no two people actually pronounce it exactly the same way. We have a fixed alphabet, and a fixed mathematical nomenclature, by convention and agreement only. Some languages, especially the older ones, don't have fixed-letter vowels at all. And the Martian mathematics doesn't rely on our Euclidian geometry of lines and fixed solids. How could that possibly work, you ask? Going back to Nature, and the fractal, one needs only to look at clouds to understand. Clouds are not, as Mandelbrot points out, failed spheres, but rather growth patterns of infinite branching. From a sufficient distance, the variations smooth out until we see a shape, but it would also be fair to call it a perceptual illusion. The attempt by mathematicians like Mandelbrot to translate these processes into conventions that fit into equations has led to the concept of self-similarity, which is a virtual pattern of symmetry, instead of a fixed measurement. As above, so below, the same rules of relationship apply (where have I heard that before?), and there are myriads of sets of those rules interacting, depending on the perspective of the observer. So...it is possible to define and describe something purely in terms of its predictable patterns of interaction. "Interaction" in this context becomes a thing, a nodal point of similarity/symmetry, and even a few such points can define an object . The process is subjective, yet still accurate, rigorous and consistent. Go ahead, scratch. I'll wait.
We are conditioned to look at the world as being built up from the small to the large, multiplying the inside (building block) toward the outside (boundary). But that "outside" is a perpetual guesstimate. Moreover, if you measure a mile using a six inch pencil as a unit, errors will inevitably creep in, even though the math, on paper, is perfect. The Martian saw the world as a shaman does, from the outside to the inside, resonant and symmetrical, and symmetry does not suffer the "prejudice of scale" that static measurement does. Bigger is not only not better, it is irrelevant. A comparison can be drawn with an observation made by quantum physicist Richard Feynman about gravity. He noted that gravity can be considered a "push" instead of a "pull" with equal validity. The whole universe is pushing you away, and you erroneously perceive the resistance of the ground against your feet as some characteristic of your own mass.
Let's look at some examples of the Martian world view. Here is a building complex in the Cydonia region. Note the profile of a face in the main building outline. Also note that the groups of small towers are not, in fact alike, but all different in shape...yet similar.
