This is my playthrough of the tutorial. In the next video, I'll dive into the game and attempt the challenge! If you enjoyed this, you can find more 5D Chess content (and participate in future events!) on my stream here: / samettheturk
Horocycles in hyperbolic geometry are like parables in Euclidean geometry, they are tangent to an ideal point at infinity. Except they also have a constant radius of curvature.
Equidistants in hyperbolic geometry are also sometimes called hypercycles, although this becomes confusing in higher dimensions if we talk about hyperspheres (compared to horospheres). They are comparable to hyperbolas in Euclidean geometry, because they have a line asymptote (their axis). Except they have a constant radius of curvature. And instead of the two asymptotes of a Euclidean hyperbola (one for each ideal point), the hyperbolic equidistant only has but one line asymptote that is corresponding to the line through both ideal points, but countless asymptotes through either one of them, which is the same as with parallel convergent lines. Really, equidistants have a center in a hyperideal point, beyond infinity, coinciding with the center of a line, always beyond infinity of course. Similarly circles have an axis, which is a line beyond infinity, consisting only of hyperideal points, being the same as the axis of their central point. This is best seen in the Klein Beltrami projection of hyperbolic space, although we can always calculate such things, regardless of projection (though sometimes perhaps these coordinates may become imaginary rather than real or something). The center of a line is called its pole, and the axis of a point is called its polar. Pole and polar are dual. Also, the axis of a horocycle is a line tangential to infinity, touching it at the ideal center of the horocycle and otherwise only consisting of hyperideal points. For the purpose of telling whether lines or other curves cross or not, ideal and hyperideal points usually "don't count". And hyperideal lines are seldom used, except as helping constructions for certain symmetrical hyperbolic patterns. The "exterior" of the circle at infinity bounding the "interior" hyperbolic plane, is something i like to call a co-hyperbolic plane. It has the topology of an unbounded Moebius strip, and a metric that is locally Minkowsky, rather than locally Euclidean, as is the case for ordinary hyperbolic plane (and for elliptic/spherical plane). Similar things hold for higher dimensional co-hyperbolic space compared to hyperbolic space. All hyperideal lines are closed curves with a finite distance around them (although the sign may be imaginary). Thus angles of vertices and lengths of edges work oppositely in co-hyperbolic geometry compared to how they work in ordinary hyperbolic geometry.
Actually, in order to algorithmically find the grail of a camelot without cheating, you need something called "dead orbs" that you can place on the ground, and then pick up again. This allows you to keep track of the direction you want to be moving, despite holonomy and enemies, and more importantly it allows you to construct cordas of the table circle, then find the midpoint of the last corda, then create the perpendicular corda to that one, then find the midpoint of this corda, and repeat again, until you find the center and the grail. This is a safe method that always works. There are faster ways, but these rely on keeping track of how a camelot was generated and from which direction you encountered it first, and this technique is mainly for speedrunners.
The color thing isn't due to the fineness of the grid, the same point applies even on a continuous, untiled plane. Imagine you have a color metric that gives you distances between colors, and imagine that at point 0, you start with some color. Now, you want to smoothly color in the rest of the plane starting from point 0. Let's say that your smoothness constraint is that when moving 1 unit on the plane, you move at most 1 unit in the color space. So you pick two directions 90 degrees apart, and start coloring from point 0. But which directions in the color space do you pick for the two directions on the plane? Well, imagine we change color at the maximum rate. 1 unit out in each direction. Since the directions were at 90 degrees, the two new points are sqrt(2) from each other, which means that their distance in the color space can't be more than that either. So either we must move slower than our maximum rate sometimes to allow for "filling" to be done at the maximum rate, or we must constrain which directions we pick in the color space, e.g. when our plane directions are 90 degrees apart, the color space directions must also be 90 degrees (or less) apart. So it's a choice between very uneven and slow color change, or limiting ourselves to a small subset of possible colors. On a hyperbolic plane, things are much different. Two points on a circle 90 degrees apart are not r×sqrt(2) apart, they're slightly less than 2×r apart, with the shortest line between them going close to the center of the circle. And if we're going close to our starting point, we're going close to our starting color anyway. All other paths are much longer and thus allow for much more color change. And as you go further from your starting point, the amount of new additional space between your starting directions grows exponentially, allowing you to branch out your coloring directions every so often at a much greater rate than on a euclidean plane, so you can reach a larger number of colors faster.
