Thursday, May 24, 2018

Considering Mathematical Dimension

                                                                                                written 12 May, 2018
                                                                                                published 24 May 2018

            I was fortunate in my college experience.  I took calculus in high school, so the three years of math, as part of my engineering degree, were upper level subjects.  I retain little of the rigor of those subjects now, but still appreciate the concepts presented.  One of my favorite classes was "n" dimensional geometry, a study of the relationships between dimensions.  The dynamic between two specific dimensions is repeated between any other pair, without limit.
            Edwin Abbott wrote "Flatland" in 1884 to illuminate this kind of dimensional perspective.  Flatland is a two-dimensional world, inhabited by intelligent polygons and lines.  One of them, Square, dreams of going to a one-dimensional world of intelligent points, where he unsuccessfully tries to describe his world to them.  When he awakens, he is visited by Sphere, an intelligent being from a three-dimensional world.  Square can only see Sphere as a circle of varying sizes, as Sphere intersects Flatland.  At each level, the inhabitants doubt the reality of the higher dimensional being, as do four-dimensional materialists of today.
            I am particularly fascinated by the relationship between two-dimensional waves on the ocean surface, where the three-dimensional volumes of the ocean and atmosphere intersect.  We experience the world as four-dimensional space/time, so the relationship between two and three dimensions is within our perspective.  Each individual ocean wave peak has a unique location, distinct from any other wave peak, yet all arising from, and remaining part of, the same ocean.  This simultaneous individuality and unity is visually obvious and represents a shift in dimensional perspective.  It helps us imagine dimensional perspective shifts into higher dimensions.
            We are used to thinking of dimensions in terms of integer numbers, but fractal geometry introduces non-integer dimensionality.  A straight line is one-dimensional, but picture a very wiggly line, never crossing itself, drawn on a piece of paper.  If it wiggles enough, it could almost cover the entire surface.  This can be thought of as, say, a 1.8-dimensional object, more than a one-dimensional line, but less than a two-dimensional plane.  Similarly, a convoluted ocean surface could be thought of as, say, 2.3-dimensional, more than a flat plane, but less than a volume.  In this way, complexity within a given dimensionality begins to extend into the qualities of the next higher dimension.
            Some consider the brain as a consciousness transceiver, able to send and receive conscious thought, rather than containing them.  If we assume, for a moment, that consciousness is a quality of a higher dimension than material matter, the complexity of the human brain pushing into this higher dimension might explain how it can host self-awareness.
            Patterns in higher dimensions show up in the real world in many situations.  Complex non-linear systems with multiple feedback loops, like weather systems, are described as chaotic, sensitive to initial conditions, and difficult to predict.  However, they are quantifiable by stable patterns called strange attractors, existing in higher dimensions.
            British biologist Rupert Sheldrake proposes that higher dimensional patterns, which he calls morphogenetic fields, direct the growth of organisms as they shift from the few initially identical cells immediately after fertilization, to an array of differentiated cells constituting a mature, multi-celled organism.
            At the galactic scale, we have known for decades that black holes are areas of such intense gravitation that the shape of space/time is distorted, and energy and matter disappear once they cross an "event horizon".  The laws of physics break down in the singularity that is the center of a black hole.  Some speculate that space/time is ripped, opening a hole through higher dimensions.  Recent theoretical work suggests there may be reciprocal white holes, where energy is entering our universe.  The original Big Bang is considered one of these events.  Science fiction has long proposed warp speed travel over galactic distances utilizing wormholes, and folding space/time in a higher dimension.
            In a closed system, energy dissipates over time, for example, a cellphone battery running down.  But if a system is not closed, new energy can come into the system at any point, recharging the battery.  A system closed at one level of dimensions is open to higher levels.   We don't really know if our world is closed or open.