Adventures in Geometrical Intuition
31 October 2010
Over the past few days I’ve posted several strictly theoretical pieces that have touched on geometrical intuition and what I have elsewhere called thinking against the grain — A Question for Philosophically Inclined Mathematicians, Fractals and the Banach-Tarski Paradox, and A visceral feeling for epsilon zero.
Not long previously, in my post commemorating the passing of Benoît Mandelbrot, I discussed the rehabilitation of geometrical intuition in the wake of Mandelbrot’s work. The late nineteenth and early twentieth century work in the foundations of mathematics largely made the progress that it did by consciously forswearing geometrical intuition and seeking instead logically rigorous foundations that made no appeal to our ability to visualize or conceive particular spatial relationships. Mandelbrot said that, “The eye had been banished out of science. The eye had been excommunicated.” He was right, but the logically motivated foundationalists were right also: we are misled by geometrical intuition at least as often as we are led rightly by it.
Geometrical intuition, while it suffered during a period of relative neglect, was never entirely banished, never excommunicated to the extent of being beyond rehabilitation. Even Gödel, who formulated his paradoxical theorems employing the formal machinery of arithmetization, therefore deeply indebted to the implicit critique of geometrical intuition, wrote: “I only wanted to show that an innate Euclidean geometrical intuition which refers to reality and is a priori valid is logically possible and compatible with the existence of non-Euclidean geometry and with relativity theory.” (Collected Papers, Vol. III, p. 255) This is, of course, to damn geometrical intuition by way of faint praise, but being damned by faint praise is not the same as being condemned (or excommunicated). Geometrical intuition was down, but not out.
As Gödel observed, even non-Euclidean geometries are compatible with Euclidean geometrical intuition. When non-Euclidean geometries were first formulated by Bolyai, Lobachevski, and Riemann (I suppose I should mention Gauss too), they were interpreted as a death-blow to geometrical intuition, but it became apparent as these discoveries were integrated into the body of mathematical knowledge that what the non-Euclidean geometries had done was not to falsify geometrical intuition by way of counter-example, but to extend geometrical intuition through further (and unexpected) examples. The development of mathematics here exhibits not Aristotelian logic but Hegelian dialectical logic: Euclidean geometry was the thesis, non-Euclidean geometry was the antithesis, and contemporary geometry, incorporating all of these discoveries, is the synthesis.
Bertrand Russell, who was central in the philosophical struggle to find rigorous logical formulations for mathematical theories that had previously rested on geometrical intuition, wrote: “A logical theory may be tested by its capacity for dealing with puzzles, and it is a wholesome plan, in thinking about logic, to stock the mind with as many puzzles as possible, since these serve much the same purpose as is served by experiments in physical science.” (from the famous “On Denoting” paper) Though Russell thought of this as a test of logical theories, it is also a wholesome plan to stock the mind with counter-intuitive geometrical examples. Non-Euclidean geometry greatly contributed to the expansion and extrapolation of geometrical intuition by providing novel examples toward which intuition can expand.
In the interest of offering exercises and examples for geometrical intuition, In Fractals and the Banach-Tarski Paradox I suggested the construction of a fractal by raising a cube on each side of a cube. I realized that if instead of raising a cube we sink a cube inside, it would make for an interesting pattern. With a cube of the length of 3, six cubes indented into this cube, each of length 1, would meet the other interior cubes at a single line.
If we continue this iteration the smaller cubes inside (in the same proportion) would continue to meet along a single line. Iterated to infinity, I suspect that this would look interesting. I’m sure it’s already been done, but I don’t know the literature well enough to cite its previous incarnations.
The two dimensional version of this fractal looks like a square version of the well-known Sierpinski triangle, and the pattern of fractal division is quite similar.
One particularly interesting counter-intuitive curiosity is the ability to construct a figure of infinite length starting with an area of finite volume. If we take a finite square, cut it in half, and put the halves end-to-end, and then cut one of the halves again, and again put them end-to-end, and iterate this process to infinity (as with a fractal construction, though this is not a fractal), we take the original finite volume and stretch it out to an infinite length.
With a little cleverness we can make this infinite line constructed from a finite volume extend infinitely in both directions by cutting up the square and distributing it differently. Notice that, with these constructions, the area remains exactly the same, unlike Banach-Tarski constructions in which additional space is “extracted” from a mathematical continuum (which could be of any dimension).
Thinking of these above two constructions, it occurred to me that we might construct an interesting fractal from the second infinite line of finite area. This is unusual, because fractals usually aren’t constructed from rearranging areas in quite this way, but it is doable. We could take the middle third of each segment, cut it into three pieces, and assemble a “U” shaped construction in the middle of the segment. This process can be iterated with every segment, and the result would be a line that is infinite two over: it would be infinite in extent, and it would be infinite between any two arbitrary points. This constitutes another sense in which we might construct an infinite fractal.
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