n-Cell Thick Patterns (1)

Couldn’t hold my curiosity any longer: “What is the minimum length of an n-cell thick pattern that exhibits infinite growth?”

From November 1997 to October 1998 Paul Callahan found several compact patterns that exhibit infinite growth. Two of these patterns give useful upper and a lower bounds, respectively.

1-cell thick

Minimum-length pattern that exhibits infinite growth:  1 x 39

5-cell thick

Minimum-length 5-cell thick pattern that exhibits infinite growth: 5 x 5

What’s in between 1-cell thick and 5-cell thick patterns? How about 2-cell thick, 3-cell thick, and 4-cell thick patterns? I have modified my script in Perl for Golly to explore this question.

See details below for 2-cell thick and 3-cell thick patterns…

2-cell thick

There is only one minimum-length 2-cell thick pattern that exhibits infinite growth and fits inside a 2 x 12 rectangle. Initially, there are 17 alive cells.

Pattern (RLE)

#CXRLE Pos=0,0
x = 12, y = 2, rule = B3/S23
o2b2ob4obo$6ob2o2bo!

It produces a block-laying switch engine, and achieves stability at generation 2638:

The final population includes five gliders.

3-cell thick

There are twenty minimum-length 3-cell thick patterns that exhibits infinite growth and fit inside a 3 x 9 rectangle.

• Pattern #1 (initially, there are 12 alive cells)

Pattern (RLE)

#CXRLE Pos=0,0
x = 9, y = 3, rule = B3/S23
7b2o$5bobo$3ob5o!

It produces a block-laying switch engine, and achieves stability at generation 2338:

The final population includes nine gliders.

• Pattern #2 (initially, there are 14 alive cells)

Pattern (RLE)

#CXRLE Pos=0,0
x = 9, y = 3, rule = B3/S23
6b3o$bob2o2bo$4obob2o!

It produces a glider-making switch engine, and achieves stability at generation 8084:

• Pattern #3 (initially, there are 15 alive cells)

Pattern (RLE)

#CXRLE Pos=0,0
x = 9, y = 3, rule = B3/S23
3bo4bo$5ob3o$o2b2o2b2o!

It produces a block-laying switch engine, and achieves stability at generation 995:

The final population includes five gliders.

• Pattern #4 #5 (initially, there are 18 alive cells)

Pattern (RLE)

#CXRLE Pos=0,0
x = 9, y = 3, rule = B3/S23
3b3o2bo$o2bob3o$9o!

Pattern (RLE)

#CXRLE Pos=0,0
x = 9, y = 3, rule = B3/S23
3b3obo$o2bob3o$9o!

Both produce the same glider-making switch engine, and achieve stability at generation 3454:

• Pattern #6 (initially, there are 16 alive cells)

Pattern (RLE)

#CXRLE Pos=0,0
x = 9, y = 3, rule = B3/S23
2bo4b2o$3o2b2obo$obob5o!

It produces a block-laying switch engine, and achieves stability at generation 4589:

The final population includes nineteen gliders.

• Pattern #7 (initially, there are 17 alive cells)

Pattern (RLE)

#CXRLE Pos=0,0
x = 9, y = 3, rule = B3/S23
b2o4b2o$3ob3obo$obo2b4o!

It produces a glider-making switch engine, and achieves stability at generation 1480:

• Pattern #8 (initially, there are 16 alive cells)

Pattern (RLE)

#CXRLE Pos=0,0
x = 9, y = 3, rule = B3/S23
b2o2b2obo$o2b2o2b2o$3obo2b2o!

It produces a block-laying switch engine, and achieves stability at generation 657:

The final population includes two gliders.

• Pattern #9 (initially, there are 17 alive cells)

Pattern (RLE)

#CXRLE Pos=0,0
x = 9, y = 3, rule = B3/S23
b2o2b4o$3ob3obo$obo4b2o!

It produces a glider-making switch engine, and achieves stability at generation 1231:

• Pattern #10 #16 #17 (initially, there are 14, 15, and 16 alive cells)

Pattern (RLE)

#CXRLE Pos=0,0
x = 9, y = 3, rule = B3/S23
b2ob2o2bo$2ob2ob2o$o2bo4bo!

Pattern (RLE)

#CXRLE Pos=0,0
x = 9, y = 3, rule = B3/S23
o4bo2bo$b2ob2ob2o$o2b2ob3o!

Pattern (RLE)

#CXRLE Pos=0,0
x = 9, y = 3, rule = B3/S23
o4bob2o$b4o2b2o$o2b2ob3o!

They produce a glider-making switch engine, and achieve stability at generation 2147:

• Pattern #11 #12 #13 #14 (initially, there are 16, 17, 18 and 18 alive cells, respectively)

Pattern (RLE)

#CXRLE Pos=0,0
x = 9, y = 3, rule = B3/S23
b3ob3o$2obobob2o$o2bo3b2o!

Pattern (RLE)

#CXRLE Pos=0,0
x = 9, y = 3, rule = B3/S23
b3ob4o$2obobob2o$2o3bo2bo!

Pattern (RLE)

#CXRLE Pos=0,0
x = 9, y = 3, rule = B3/S23
b3ob4o$2obobob2o$2o2bo2b2o!

Pattern (RLE)

#CXRLE Pos=0,0
x = 9, y = 3, rule = B3/S23
b3ob4o$2ob3ob2o$2o5b2o!

They produce the same glider-making switch engine that achieves stability at generation 4883:

• Pattern #15 (initially, there are 16 alive cells)

Pattern (RLE)

#CXRLE Pos=0,0
x = 9, y = 3, rule = B3/S23
b4o2b2o$b2obobo$obobob3o!

It produces a block-laying switch engine, and achieves stability at generation 1883:

The final population includes three gliders and a lightweight spaceship.

• Pattern #18 #19 (initially, there are 16 and 16 alive cells, respectively)

Pattern (RLE)

#CXRLE Pos=0,0
x = 9, y = 3, rule = B3/S23
o3b3obo$2b2o2b3o$3ob2o2bo!

Pattern (RLE)

#CXRLE Pos=0,0
x = 9, y = 3, rule = B3/S23
o3b3obo$2b2obob2o$3obobobo!

They produce a block-laying switch engine, and achieve stability at generation 2873:

The final population includes five gliders.

• Pattern #20 (initially, there are 16 alive cells)

Pattern (RLE)

#CXRLE Pos=0,0
x = 9, y = 3, rule = B3/S23
o2b2o2b2o$ob2ob2obo$2o2bobobo!

It produces a glider-making switch engine, and achieves stability at generation 2918: