The Glider is the basic building block for Game of Life object constructions. They are also the most common naturally occuring moving object, and seventh most common overall. Because Gliders appear to move across the field, they can be timed and positioned to interact with other objects including other gliders. A normalized glider (SOF 3-1-011.) as shown, points to the Northwest (NW) direction, has a phase 0 and a location of [0,0].
For glider-glider collisions, in order to establish a framework in which to refelate to all other components, there is always one glider headed Southeast located cell [0,0] with phase 0. In the following illustrations, its path is shown in green. The prefered direction for the other gliders is either headed Southwest (aright angle collision) or Northwest (ahead-on collision).
Because in glider-glider collisions all the objects are moving, there are patterns which are identical if one of them is advanced by a multiple of the glider's 4 generation period. Relative to the target glider, it's as if the colliding glider moves the difference in their horizontal and vertical speeds of 1 cell per 4 generations. In the illustration, the the bottom row shows the top row advanced by 4 gens. On the left, the gliders have each moved one cell in their direction. On the right, the target glider position is held constant, so it appears as if the right glider has moved two cells toward the target. This number is always even, which means that there is a parity factor involved that depends on whether the two gliders are an even or odd number of cells apart.
The path of a colliding glider can be projected foward as if the target were not there. This is used to determine the location of the colliding glider relative to the target. The position can then be specified in relation to this pseudo-collision, by projecting back or forward a given number of periods. Not really necessary for collisions involving two gliders, it becomes important in the classification of three or more gliders.
Equivalence— Two collisiions are said to be equivalent when one can be transformed into the other by advancing the first by a number of generations. In the case of multiple gliders, it may also be necessary to tranform the pattern by rotation, reflection or phasing to show the equivalnce. The pattern which cannot be transformed into an equivalent pattern is definied to be the start of a collision.
For right angle collisions, the relative motion of the two gliders is as if the colliding glider moved two cells laterally toward the target glider. The parity is then simply if they are an even or odd number of cells apart.
There are 11 possible paths on which two gliders moving at right angles to each other can interact. The number of the path is the veritcal offset of the projected path of the colliding glider relative to the target glider.
Reversing the locations of the target and colliding gliders is equivalent to reversing the sign of the number used to specify the path. Each path can also be offset by one cell to account for the even or odd parity. Because of this, there end up being a total of 12 unique paths between two gliders at right angles. Since each colliding glider can be in any one of 4 phases, this leaves 48 possible unique collisions. It turns out that in some cases, the gliders merely pass by one another, resulting in 38 instead. Shown at right are the eight collisions that occur on Path 0.
Head on collisiions are a bit more complicated than the right angled collisions. Where advancing two right-angled gliders by 4 gens was equivalent to moving the colliding glider two cells to the West, for head-on, it's equivalent to moving 2 cells to the Northwest.Parity is determined by whether the are diagonally an even or odd number of cells apart.
The paths are numbered by subtracting the vertical offset from the horizontal, then adding 1: path = (h - v + 1). As with the right angled paths, there are 11 possible paths on which the gliders can interact, and again, because of symmetry, there are actually 5 pairs of identical paths.
Parity, however, is a bit more complicated because phase must be taken into account. Because of this, each path ends up with 5 distinct collisions, with the other three being duplicates.