Pi-R-Squared spaceship

The Pi-R-Squared spaceship is a fictituous spaceship due to Dave Greene, supposedly consisting of two pi grandparents and two R-pentominoes, with a population of 20 in a 55 × 34 bounding box, moving at speed c/137.

It was claimed to have been "seen briefly in 1974 on a PDP-12 display by Tony Honcho Jarnow" (an anagram of "John Horton Conway"), arising from the ash of a random 128 × 80 soup, "just before an unfortunate power outage due to tripping on a power cable".

Generalisations

The idea is that the R-pentomino and the five-cell pi grandparent are each fairly common and therefore likely to pop up in general, regardless of what the pattern was like tens or hundreds of generations earlier (as long as it had not already adopted a stable/repeating configuration). If one of the many initial patterns that could lead to this also had two R-pentominoes and two pi grandparents (in the same configuration), then it would result in a spaceship. Near-miss examples of this include:

• In August 2013, Martin Grant discovered a near-miss spaceship, where an R-pentomino descendant moves at a speed of 14c/65 orthogonal with a dot spark.
• In December 2020, Tanner Jacobi found a mechanism where two blocks and two traffic light predecessors, aided by a line of gutter symmetry, happened to form from the same arrangement shifted by one cell 61 generations later minus one dot spark on each side due to their sheer commonness.
• An angel happens to form from a glide-reflected (or, because it has reflection symmetry, rotated) copy of itself with one extra cell after nineteen generations.
• Adding a cell to a block and honey farm predecessor interacting happens to result in the block and honey farm predecessor forming again (due to their commonness) with the same relative position, resulting in a near-miss glide-reflective 2c/120 spaceship.
• The two-glider octomino, which is a fairly common pattern, happens to form again from itself 39 generations later in a different position when placed on an 11 × 11 torus. An R-pentomino exhibits similar behavior on a 16 × 31 Klein bottle, appearing glide-reflected after 590 generations (although there is some other junk that dies eighteen generations later without interacting with the R-sequence).