### Making switch-engines

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**July 2nd, 2014, 7:53 am**I'm interested in minimal ways (in various senses) to make switch-engines. I don't care how messy the result is, as my main interest in this is in its implications for Sparse Life (infinite, random, arbitrarily low-density Game of Life fields). This post includes some of the results I know of; I'd be pleased to hear of others, particularly but not exclusively improvements in some sense. Paul Callahan and I did some work on this in 1997 (most of the work was Paul's), and I've continued on and off ever since. Paul undertook a comprehensive survey of all patterns of up to 9 cells, and found none with infinite growth (I intend to replicate this search and extend to 10 cells in the near future); and found several infinite growth patterns starting with 10 cells - all switch-engines.

Eight cells can get a switch-engine going, but its exhaust catches up and destroys it. Here are the best-known pattern, and two others, which take either one or two more steps to reach the same phase. There are quite a few others, but all quite close to one of these.

x = 4, y = 6, rule = B3/S23

o$2o$o2bo2$bobo$2bo!

x = 5, y = 7, rule = B3/S23

o$bo$2bo$bo$o$2b3o!

x = 4, y = 7, rule = B3/S23

o$o$b2o$3bo$3bo$3bo$3bo!

There are various ways of adding a blinker or preblock to any of these to produce an 11-cell pattern with infinite growth (either a block-laying or glider-stream switch-engine can be produced), but as noted above, Paul found some 10-cell examples, reproduced below:

Block-layer:

x = 8, y = 6, rule = B3/S23

5bobo$5bo$3bo$bobo$2obo$bo!

Another block-layer:

x = 15, y = 58, rule = B3/S23

13bo$12b2o$13b2o52$bo$o$b2o$3bo!

And a glider-streamer:

x = 18, y = 35, rule = B3/S23

b2o$2o$bo29$16bo$15bobo$15bo$14bo!

This one (another block-layer) was, I think, found by Tom Rokicki:

x = 59, y = 29, rule = B3/S23

57bo$56b2o$57b2o24$o$2o$obo!

The last 3 of these 4 could be altered in trivial ways by substituting different r-pentomino predecessors (there are 5 5-cell 1-step predecessors of the classic r-pentomino, and 3 5-cell 2-step predecessors, and of course all 4 could be rotated and reflected, but aside from those variants, does anyone know any more?

Paul also looked for infinite-growth patterns with small bounding boxes, and I think found one in a 5-by-5 box, which I don't have, and over 40 in 8-by-8 boxes (all switch-engines), which I won't post unless anyone particularly wants them. I think he also produced various interactions of 3 synchronised gliders from the same direction

colliding with a block to produce a surviving switch-engine, but again, I don't have these. I do have the following pattern creating a block-layer, also starting with three gliders and a block, but with one glider coming from the NW and the other two from the SE, and in which there can be an arbitrarily long gap between the first and second gliders and the remaining pair:

x = 337, y = 264, rule = B3/S23

bo$2bo$3o24$22b2o$22b2o214$271b2o$271bobo$271bo18$334b3o$334bo$335bo!

A blinker can be substituted for the block:

x = 337, y = 267, rule = B3/S23

bo$2bo$3o24$20b3o218$271b2o$271bobo$271bo18$334b3o$334bo$335bo!

Also, this can easily be converted to a 4-glider construction of a

block-layer, like this:

x = 343, y = 270, rule = B3/S23

bo$2bo$3o7$5b2o$5bobo$5bo236$277b2o$277bobo$277bo18$340b3o$340bo$341bo!

or this:

x = 344, y = 262, rule = B3/S23

9b3o$9bo$10bo2$3o$2bo$bo233$278b2o$278bobo$278bo18$341b3o$341bo$342bo!

or this:

x = 346, y = 276, rule = B3/S23

9bobo$9b2o$10bo$bo$2bo$3o248$280b2o$280bobo$280bo18$343b3o$343bo$344bo!

Moving on, I have found various interactions between a single glider and three small still lifes or oscillators (many derived from Paul Callahan's 8-by-8 starting points), all of which start by creating some successor to the r-pentomino, and in all of which a blinker or block simply tames the exhaust.

Glider, two beehives and a blinker => block-layer:

x = 20, y = 8, rule = B3/S23

2bo$obo16bo$b2o16bo$6b2o11bo$5bo2bo$6b2o8b2o$15bo2bo$16b2o!

Glider, loaf, beehive and blinker => block-layer, the switch-engine

evolving in the same way as above:

x = 17, y = 16, rule = B3/S23

2bo$obo$b2o5$6bo$5bobo$5bo2bo$6b2o6b3o3$12b2o$11bo2bo$12b2o!

Glider, two loaves and a block => block-layer:

x = 22, y = 17, rule = B3/S23

18b2o$18b2o6$bo$2bo$3o3$8b2o$7bo2bo8b2o$8bobo7bo2bo$9bo9bobo$20bo!

Glider, beehive, loaf and block => glider-streamer; the switch-engine evolves in the same way as for the block-layer immediately above:

x = 32, y = 22, rule = B3/S23

18bo$19b2o$18b2o3$22bo$21bobo$21bobo$22bo2$29b2o$28bo2bo$29bobo$30bo7$

2o$2o!

The same, with only the block moved:

x = 14, y = 18, rule = B3/S23

12b2o$12b2o3$o$b2o$2o3$4bo$3bobo$3bobo$4bo2$11b2o$10bo2bo$11bobo$12bo!

Glider, beehive, loaf and block => glider-streamer again, but the glider, beehive and loaf are differently placed relative to each other. However, the switch-engine evolves in the same way:

x = 25, y = 18, rule = B3/S23

obo$b2o$bo5$6b2o$5bo2bo$6b2o4$7bo$6bobo$5bo2bo$6b2o15b2o$23b2o!

Glider, beehive, loaf and block => glider-streamer once more, with yet another placement of glider, beehive and loaf giving the same switch-engine evolution.

x = 25, y = 18, rule = B3/S23

obo$b2o$bo5$6b2o$5bo2bo$6b2o4$7bo$6bobo$5bo2bo$6b2o15b2o$23b2o!

Here's a 9-glider slow salvo creating a block-layer out of a blinker, using the same basic switch-engine evolution as the "glider, two loaves and a block" pattern above:

x = 417, y = 457, rule = B3/S23

bo$2bo$3o44$35bo$36b2o$35b2o38$71bo$72b2o$71b2o62$111bo$112bo$110b3o

39$160bo$161b2o$160b2o38$194bo$192bobo$193b2o38$240bo$238bobo$239b2o

163$403bo$404bo$402b3o12$415bo$416bo$414b3o4$413b3o!

Finally, here's as 8-glider slow salvo creating a glider-streamer out of a blinker - I'd forgotten this one until I prepared this post:

x = 670, y = 572, rule = B3/S23

bo$2bo$3o19$21bo$22bo$20b3o24$24bo$25bo$23b3o8$41bo$42bo$40b3o18$90bo$

91bo$89b3o405$565bo$566bo$564b3o62$646bo$647b2o$646b2o15$658bo$659bo$

657b3o4$667b3o!

And a slight variant:

x = 670, y = 572, rule = B3/S23

bo$2bo$3o19$21bo$22bo$20b3o24$24bo$25bo$23b3o8$41bo$42bo$40b3o6$90bo$

91bo$89b3o417$565bo$566bo$564b3o62$646bo$647b2o$646b2o15$658bo$659bo$

657b3o4$667b3o!

Eight cells can get a switch-engine going, but its exhaust catches up and destroys it. Here are the best-known pattern, and two others, which take either one or two more steps to reach the same phase. There are quite a few others, but all quite close to one of these.

x = 4, y = 6, rule = B3/S23

o$2o$o2bo2$bobo$2bo!

x = 5, y = 7, rule = B3/S23

o$bo$2bo$bo$o$2b3o!

x = 4, y = 7, rule = B3/S23

o$o$b2o$3bo$3bo$3bo$3bo!

There are various ways of adding a blinker or preblock to any of these to produce an 11-cell pattern with infinite growth (either a block-laying or glider-stream switch-engine can be produced), but as noted above, Paul found some 10-cell examples, reproduced below:

Block-layer:

x = 8, y = 6, rule = B3/S23

5bobo$5bo$3bo$bobo$2obo$bo!

Another block-layer:

x = 15, y = 58, rule = B3/S23

13bo$12b2o$13b2o52$bo$o$b2o$3bo!

And a glider-streamer:

x = 18, y = 35, rule = B3/S23

b2o$2o$bo29$16bo$15bobo$15bo$14bo!

This one (another block-layer) was, I think, found by Tom Rokicki:

x = 59, y = 29, rule = B3/S23

57bo$56b2o$57b2o24$o$2o$obo!

The last 3 of these 4 could be altered in trivial ways by substituting different r-pentomino predecessors (there are 5 5-cell 1-step predecessors of the classic r-pentomino, and 3 5-cell 2-step predecessors, and of course all 4 could be rotated and reflected, but aside from those variants, does anyone know any more?

Paul also looked for infinite-growth patterns with small bounding boxes, and I think found one in a 5-by-5 box, which I don't have, and over 40 in 8-by-8 boxes (all switch-engines), which I won't post unless anyone particularly wants them. I think he also produced various interactions of 3 synchronised gliders from the same direction

colliding with a block to produce a surviving switch-engine, but again, I don't have these. I do have the following pattern creating a block-layer, also starting with three gliders and a block, but with one glider coming from the NW and the other two from the SE, and in which there can be an arbitrarily long gap between the first and second gliders and the remaining pair:

x = 337, y = 264, rule = B3/S23

bo$2bo$3o24$22b2o$22b2o214$271b2o$271bobo$271bo18$334b3o$334bo$335bo!

A blinker can be substituted for the block:

x = 337, y = 267, rule = B3/S23

bo$2bo$3o24$20b3o218$271b2o$271bobo$271bo18$334b3o$334bo$335bo!

Also, this can easily be converted to a 4-glider construction of a

block-layer, like this:

x = 343, y = 270, rule = B3/S23

bo$2bo$3o7$5b2o$5bobo$5bo236$277b2o$277bobo$277bo18$340b3o$340bo$341bo!

or this:

x = 344, y = 262, rule = B3/S23

9b3o$9bo$10bo2$3o$2bo$bo233$278b2o$278bobo$278bo18$341b3o$341bo$342bo!

or this:

x = 346, y = 276, rule = B3/S23

9bobo$9b2o$10bo$bo$2bo$3o248$280b2o$280bobo$280bo18$343b3o$343bo$344bo!

Moving on, I have found various interactions between a single glider and three small still lifes or oscillators (many derived from Paul Callahan's 8-by-8 starting points), all of which start by creating some successor to the r-pentomino, and in all of which a blinker or block simply tames the exhaust.

Glider, two beehives and a blinker => block-layer:

x = 20, y = 8, rule = B3/S23

2bo$obo16bo$b2o16bo$6b2o11bo$5bo2bo$6b2o8b2o$15bo2bo$16b2o!

Glider, loaf, beehive and blinker => block-layer, the switch-engine

evolving in the same way as above:

x = 17, y = 16, rule = B3/S23

2bo$obo$b2o5$6bo$5bobo$5bo2bo$6b2o6b3o3$12b2o$11bo2bo$12b2o!

Glider, two loaves and a block => block-layer:

x = 22, y = 17, rule = B3/S23

18b2o$18b2o6$bo$2bo$3o3$8b2o$7bo2bo8b2o$8bobo7bo2bo$9bo9bobo$20bo!

Glider, beehive, loaf and block => glider-streamer; the switch-engine evolves in the same way as for the block-layer immediately above:

x = 32, y = 22, rule = B3/S23

18bo$19b2o$18b2o3$22bo$21bobo$21bobo$22bo2$29b2o$28bo2bo$29bobo$30bo7$

2o$2o!

The same, with only the block moved:

x = 14, y = 18, rule = B3/S23

12b2o$12b2o3$o$b2o$2o3$4bo$3bobo$3bobo$4bo2$11b2o$10bo2bo$11bobo$12bo!

Glider, beehive, loaf and block => glider-streamer again, but the glider, beehive and loaf are differently placed relative to each other. However, the switch-engine evolves in the same way:

x = 25, y = 18, rule = B3/S23

obo$b2o$bo5$6b2o$5bo2bo$6b2o4$7bo$6bobo$5bo2bo$6b2o15b2o$23b2o!

Glider, beehive, loaf and block => glider-streamer once more, with yet another placement of glider, beehive and loaf giving the same switch-engine evolution.

x = 25, y = 18, rule = B3/S23

obo$b2o$bo5$6b2o$5bo2bo$6b2o4$7bo$6bobo$5bo2bo$6b2o15b2o$23b2o!

Here's a 9-glider slow salvo creating a block-layer out of a blinker, using the same basic switch-engine evolution as the "glider, two loaves and a block" pattern above:

x = 417, y = 457, rule = B3/S23

bo$2bo$3o44$35bo$36b2o$35b2o38$71bo$72b2o$71b2o62$111bo$112bo$110b3o

39$160bo$161b2o$160b2o38$194bo$192bobo$193b2o38$240bo$238bobo$239b2o

163$403bo$404bo$402b3o12$415bo$416bo$414b3o4$413b3o!

Finally, here's as 8-glider slow salvo creating a glider-streamer out of a blinker - I'd forgotten this one until I prepared this post:

x = 670, y = 572, rule = B3/S23

bo$2bo$3o19$21bo$22bo$20b3o24$24bo$25bo$23b3o8$41bo$42bo$40b3o18$90bo$

91bo$89b3o405$565bo$566bo$564b3o62$646bo$647b2o$646b2o15$658bo$659bo$

657b3o4$667b3o!

And a slight variant:

x = 670, y = 572, rule = B3/S23

bo$2bo$3o19$21bo$22bo$20b3o24$24bo$25bo$23b3o8$41bo$42bo$40b3o6$90bo$

91bo$89b3o417$565bo$566bo$564b3o62$646bo$647b2o$646b2o15$658bo$659bo$

657b3o4$667b3o!