Karl: Tesselect Math
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Captain Oh My Captain
In order to understand numbers the way I am using them, you must be able and willing to follow through this illustration, which is a "personalization" of the "ordeal" I interpret numbers to be. I can't overemphasize how much this bizarre representation of a mathematical concept underlies my entire approach. It manages to satisfy my need for exact analysis and poetic digression simultaneously.
To start, imagine yourself the captain of a tiny spaceship. How tiny? As small as you can think about such things. The ship has one capacity, which is to fly in a straight line on a flat plane. You will recognize this immediately as the standard Euclidean arrangement, and if so, you should have no further problems connecting the significances of what happens to your captain. The captain has only one goal in mind, which is to bring his ship to a safe resting place after initiating his flight. What is safe? Safety is reached when the ship lands over any exact coordinate in the plane. You will recognize these places of safety as rational numbers, perhaps, in which case you are already well ahead of the game. Congratulations.
Your mission, should you choose to accept it, is to explore the space in your sector and find a place to call home, a place to stop your straight-line voyage through two dimensions. But the only real choice you can make is your initial direction. After that, you are committed to your pre-ordained flight plan, and your ship will simply keep flying along until it encounters a precise condition that signals it to stop. That condition is simply landing directly over the intersection of a grid coordinate.
The captain and his ship are extraordinarily well defined - well contained in a mathematical universe. His ship always starts from a single point, the origin, the (0,0) coordinate. He must always aim the ship somewhere between (1,0) directly to the right along the standard x axis (you should be imagining a piece of paper with a standard cartesian grid printed on it) and (0,1) directly up along the standard y axis. But wait, he has also been told that he needn't aim his ship north of the diagonal line, located at 45 degrees in the standard arc, made half way between the sides of his right angle sector.
His aiming point will thus have a slope between 0 (directly to the right) and 1 (up at an angle formed by going one unit right and one unit up). He is only going to explore the region between his starting point of (0,0) and the two edges of his designated sector, located at (1,0) and (1,1) on the plane. Bonus: Why is the rest of the right angle sector, or any other arc for that matter, omitted? Answer: Symmetry. It's structure mirrors the sector designated for exploration, so we can defer exploring it at least for the time being.
Now I hope you have the overall story thus far well digested. It is meant to be so very easy that someone who is terrible at math, like me, can really get a grip on it. The starting conditions, the players, their actions, are all presented as simply as possible, so that there can be few if any questions about what is going to happen next. The reason for this elaborately framed simplicity should also be dawning on you. We are about to get into very strange territory indeed, using a math that has been in place and under active scrutiny for millenia. Watch.
As the captain prepares to launch, he takes the one truly momentous action available to him: he takes aim. Now, exactly how he does this need not concern us, except to emphasize that his aiming mechanism is like a perfect swivel, allowing him unrestricted access to absolutely any directional angle possible between the slopes of 0 and 1. Another way to state this is that any angle whatsoever is as available as any other. In this aiming system, the captain does not labor under any pre-existing bias in favor of, for instance, aiming for slopes of 1/2, 2/3, or 7/23. He might hit those points, or he might miss them by just a hair. An angle represented by the fractional part of Pi (0.14159...), for instance, which we are pretty sure is irrational, is equally likely.
At this point, a digression. Recall that rational numbers have a definite ratio, which serves to define a point on the plane. The numerator can be taken as the x coordinate, and the denominator as the y coordinate. For instance, 1/1 has a slope of 1 and its point has the coordinates (1,1). Likewise 7/23 has a slope of 0.3043478260869565217391_, the 22-digit base-10 pattern repeats over and over, but the coordinate is still, of course (7,23). Irrational numbers, like Pi, do not repeat themselves, never end, and therefore do not have such a coordinate point - it simply doesn't exist, or we simply haven't found it yet.
Nevertheless, irrational numbers do have a direction (at a minimum they have a direction between two bordering rational directions). And as we have set up the scenario, it is as easy and as likely (perhaps more likely) for the captain to aim in an irrational direction as it is to aim in a rational direction, only not as safe. For if the captain happens to aim in an irrational direction, his ship will never land. It will keep flying in a straight path forever. And it may be added, if there was any doubt, that even reaching a very distant rational point might entail a journey so long as to wear creases in the forehead. Imagine, for instance, finding the point with slope like 3.48x10E64 / 7.02x10E64. Such number ranges represent the largest estimated counts we have, like all the atoms in all the stars.
Because my story intentionally solicits an emotional cast over an analytical theme, I want to emphasize here that such an outcome will make the captain very sad, if he is a home-body and pretending that when he finds rest he will also find his soul-mate, waiting for him there. Or perhaps he will be very happy to continue searching forever, for a rest that cannot ever be, however devoutly wished. If so, he is a very adventurous and dedicated explorer indeed. Either way, the captain cares deeply about such things as when and where his ship will stop - and so should you.
The act of aiming carries with it a sense of uncertainty. We may aim at a target, but we never know for sure whether we will hit it. Certainty comes with arrival. So it is with our ship and captain. He may try for a certain direction, or he may look at the multitude of coordinate points out there in his two dimensions and say to himself "I bet I can hit that one, over there." But he cannot be certain. Why not? Because he only has a grip on one point, not two. When you or I draw such a straight line on graph paper, we customarily place the ruler on the origin (0,0) and then pick another point, such as (7/23). Now, one could imagine our captain being so very sophisticated as to work out the perfect angle to take him from one point to another, but alas, he just can't. He must settle for a wee bit of uncertainty. And with uncertainty, a wee bit is a whole lot.
Let the voyage begin. The captain has taken aim, and his intention is to hit one of the closest points available, if he can. He aims more or less midway in the available arc of possible directions, generally in the direction of (1/2). He might land over it, and end his voyage almost before it began. But that would take a very lucky captain. Instead, he launches his ship, and flies right past it, closely on one side, and his ship is off and bound for the edges of the universe - did I mention that the flat plane extends to the outer edge of the universe?
It is still quite possible he will hit a rational coordinate and stop his voyage, so he is constantly looking out the ship's windows, watching points in the plane pass by, longingly thinking that the next one might be his resting point. One day, bored, he remembers that he has a surveying instrument on board - a specially equipped viewscope with rangefinders. This instrument allows him to measure the distance between his ship's location and two points (either both x's or both y's) that he is momentarily passing exactly between. When his ship forms the middle point in a straight line between two x's or two y's, then his rangefinder spits out a number that represents the ship's distance between these points as a unitless ratio. For instance, if his ship happens to be exactly half way between the two closest points, the instrument will read 0.5, if a third of the way, 0.33333_, and so on.
This may not seem very interesting, but since the captain has nothing else to do until his ship's distance from a point reaches 0 and stops, he starts to write down each ratio detected in a log book. The log tells him how close he has come to hitting a point in either direction. Since he is still smarting from the disappointment of missing his first, best chance to land home quickly, he takes to calling these measurements "defects," as they appear to him as records of the defects from his true aim and intended purpose. He has to reach 0 distance, no defects, in both directions to arrive precisely over a rational point and come to a stop, so he charts and analyzes the back and forth of his readings, hoping they will give him some clue as to whether his ship is on a course that will bring him home sooner, later, or never.
This might be the end of the story, but it isn't. The captain is not alone, and perhaps we can ultimately help him out. But first it might do well to pause and review some of the issues already at play. Here is a short list:
1. Complexity out of simplicity. We set up a very simple scenario that led to a rather complex account of a log of defects which may or may not help decide if we are lost temporarily or eternally.
2. Uncertainty. We found that uncertainty is a necessary condition, and it is more natural than certainty. We normally confine uncertainty - we pick two known (Cartesian) points, or we at least pick an exact (resolvable) angle, before heading out on a linear voyage. But if uncertainty reigns, satisfying these conditions becomes a contingency - we cannot know if we will stop, until we stop.
3. Time. Our captain probably has a clock. The defect log is merely a sequence of readings, and they are only able to be made at varying intervals. An alternative means of tracking position in this journey is to simply call out the time that has elapsed. Time is here defining spatial geometry in an interesting way.
