User:Tropylium/TI engine/fat

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The secondary type Φ- of the TI engine can be extended downward in six ways so that it will regenerate itself. Of these "demisecondary" subtypes, only one can possibly be regenerated by either itself or by any other subtype:

. . . O . . .
. . O O O . .
. O O O O O .
O O . . . O O

(You may recognize this shape as generation 1 of queen bee, as a part of generation 9 of pi-heptomino, or a part of the 9th and 11th parents of honey farm.)

There are again 44 ways to extend this pattern one line down (out of 272 possibilities) in which this shape can regenerate itself, moved one cell forward in two generations. These can be analyzed in terms of three sections:

. . . . O . . . .
. . . O O O . . .
. . O O O O O . .
. O O . . . O O .
A A A B B B A A A

I will call the cells marked A the wings, and the cells marked B the core. A basic criterion for the engine to regenerate itself is that on both sides, at most one wing cell is present. If lateral core cells are present, no wing cells are necessary at all. Any number of core cells may be present. Some further limits on wing/core combinations however apply.

Observe moreover that, in any spaceship with a well-defined axis thruout, the left wing and the right wing will not affect one another's evolution over two generations.

These particular tertiary subtypes of the engine can therefore be named with three-letter codes according to their wing and core makeup.

  • Wing types: 0123 (where 1 = least lateral cell, 3 = most lateral cell)
  • Core types: 01234567 (I will use here left-to-right binary order: 000 100 010 110 001 101 011 111)
    • Subtypes with core 4 or 6 are reflections of subtypes with core 1 or 3 (respectively). E.g. 242 = 212; 361 = 133.

Transition stats

Subtypes sorted in order (left wing, right wing, core). Subtypes that can be generated by some other subtype are marked in red; subtypes that can regenerate themselves are marked in green. If a cell has two entries, the lower is the count of subtypes that emerge reflected from their listed orientation, the upper the count of those that emerge unreflected.

101 121 171 201 221 231 271 202 222 232 272 total
050 12 9 - - -
2
- - - - 1 25
070 1 5 - - - 1
2
- - - - 2 11
140
150
160
170
101 - 3 - - - 4
2
- - - - 9 18
111
121 - - - - - - - 3 - - - 3
131
151 13 9 - - 1
2
- 1 - - 1 31
171 - 1 - - - 1 - - - - 1 3
240
250
260
270
201 - 2 - - - -
7
- - - - 13
211
221
231 - - - -
4
- -
2
- 4 - -
4
- 14
241
251
261
271
202 - 3 1 - -
2
- -
2
- - - 3 11
212
222 3 1 - -
3
-
2
- - 1 3 - - 13
232
252
272 1 6 - -
2
-
12
- - 3 10 - - 34
350
370
311
331
351
371
312
332
352
372
313
333
353
373