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### rectangles

Posted: **June 11th, 2011, 3:02 pm**

by **wintersolstice**

I have considered looking at the fate of rectangles

a 162x162

x = 162, y = 162, rule = B3/S23

162o$162o$162o$162o$162o$162o$162o$162o$162o$162o$162o$162o$162o$162o$

162o$162o$162o$162o$162o$162o$162o$162o$162o$162o$162o$162o$162o$162o$

162o$162o$162o$162o$162o$162o$162o$162o$162o$162o$162o$162o$162o$162o$

162o$162o$162o$162o$162o$162o$162o$162o$162o$162o$162o$162o$162o$162o$

162o$162o$162o$162o$162o$162o$162o$162o$162o$162o$162o$162o$162o$162o$

162o$162o$162o$162o$162o$162o$162o$162o$162o$162o$162o$162o$162o$162o$

162o$162o$162o$162o$162o$162o$162o$162o$162o$162o$162o$162o$162o$162o$

162o$162o$162o$162o$162o$162o$162o$162o$162o$162o$162o$162o$162o$162o$

162o$162o$162o$162o$162o$162o$162o$162o$162o$162o$162o$162o$162o$162o$

162o$162o$162o$162o$162o$162o$162o$162o$162o$162o$162o$162o$162o$162o$

162o$162o$162o$162o$162o$162o$162o$162o$162o$162o$162o$162o$162o$162o$

162o$162o$162o$162o$162o$162o$162o$162o!

### Re: rectangles

Posted: **June 11th, 2011, 4:05 pm**

by **Lewis**

a 83x83 produces a P6 Unix on each of its digaonals (I may have posted this before in a different topic, I'm not sure) :

Code: Select all

```
x = 83, y = 83, rule = 23/3
83o$83o$83o$83o$83o$83o$83o$83o$83o$83o$83o$83o$83o$83o$83o$83o
$83o$83o$83o$83o$83o$83o$83o$83o$83o$83o$83o$83o$83o$83o$83o$
83o$83o$83o$83o$83o$83o$83o$83o$83o$83o$83o$83o$83o$83o$83o$83o
$83o$83o$83o$83o$83o$83o$83o$83o$83o$83o$83o$83o$83o$83o$83o$
83o$83o$83o$83o$83o$83o$83o$83o$83o$83o$83o$83o$83o$83o$83o$83o
$83o$83o$83o$83o$83o!
```

**EDIT**: I might do a bit of an investigation into

*n*x

*n* recatngles, the 41x41 box turned up an interesting still life at it's centre:

Code: Select all

```
x = 41, y = 41, rule = 23/3
41o$41o$41o$41o$41o$41o$41o$41o$41o$41o$41o$41o$41o$41o$41o$41o
$41o$41o$41o$41o$41o$41o$41o$41o$41o$41o$41o$41o$41o$41o$41o$
41o$41o$41o$41o$41o$41o$41o$41o$41o$41o!
```

**Edit 2**:

I checked all filled nxn boxes from 1x1 to 100x100. Here are some of the more interesting results:

- Longest lifespan : 80x80 box (n=80) (1167 generations)
- Pattern dies out completely for n=0, 4, 8, 12, 15 and 20.
- The interesting 20-cell still life is the centre for n=41, 45, 61, 77 and 93.
- n=58 is the smallest rectangle to emit gliders.
- The aforementioned
**Unix** when n=83
- When n=84, 4
**Pentadecathlons** are formed.
- When n=87, 4 16-cell still lifes (Cis-Mirrored R-Mango I think) are formed.
- When n=99, 8 Hats are formed.

Still not found:

- A rectangle with infinite growth.
- A rectangle that emits Spaceships other than gliders.
- A rectangle that leaves a Pulsar or (even better) an Octagon II at it's centre.

I have full census results in table form for these first 100 in an Excel file; I'll upload them if anyone's interested.

I'll probably look more into this, or into

*n*x

*m* rectangles some time.

**Edit 3**:

n=102 produces some 14-cell still patterns made from 2 loaves joined in an unusual way.

n=110 procudes 4x P2

**Spark Coils**.

### Re: rectangles

Posted: **June 12th, 2011, 12:00 pm**

by **ssaamm**

Simple way of doing this:

Set random fill density to 100%

Create selection rectangle, note its size

fill it "randomly"

### Re: rectangles

Posted: **June 12th, 2011, 12:22 pm**

by **Lewis**

ssaamm wrote:Simple way of doing this:

Set random fill density to 100%

Create selection rectangle, note its size

fill it "randomly"

I've been doing something similar to this, because in WinLife32 (I need for the census feature) the 100% density fill leaves empty cells occasionally...

I'm suprised there hasn't been more research into this (like with the Life Digits). Rectangles and squares seem a logical thing to first draw into a Life program.

It'd be nice to find the smallest area rectangle with infinite growth; so far none of the prefect squares I've looked at have.

### Re: rectangles

Posted: **June 12th, 2011, 12:44 pm**

by **calcyman**

It'd be nice to find the smallest area rectangle with infinite growth; so far none of the prefect squares I've looked at have.

Or, even better, an impossibility proof.

### Re: rectangles

Posted: **June 12th, 2011, 1:22 pm**

by **ssaamm**

Well, the proof for perfect squares is that in very large patterns, it repeats itself

Go ahead, make a 5000x5000 and look at the corners

I'm not sure how valid this is.

### Re: rectangles

Posted: **June 12th, 2011, 1:27 pm**

by **137ben**

Well, the proof for perfect squares is that in very large patterns, it repeats itself

Go ahead, make a 5000x5000 and look at the corners

I'm not sure how valid this is.

But when the "fuses" at the corners finally reach each other at the center of each side, there's know guarantee that they won't form a symmetrical backrake on each side (or gun, for that matter).

### Re: rectangles

Posted: **June 12th, 2011, 7:24 pm**

by **ssaamm**

I see what you are saying now.

The inside is chaotic even though it forms fractal patterns; the things making up those patterns are unpredictable. Unfortunately it is unlikely that the inside of the pattern will create something that emerges on the outside, making infinite growth very unlikely, but not very disprovable.