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Nov 2, 2001, 2:13:55 PM11/2/01

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Hi,

I'm new to this list but I've been playing around with Life for a while

using Life32 and recently MCell. I've looked around various Life / CA

web pages to see if anybody else has discovered this and couldn't find

anything.

I'm new to this list but I've been playing around with Life for a while

using Life32 and recently MCell. I've looked around various Life / CA

web pages to see if anybody else has discovered this and couldn't find

anything.

In exploring the 2x2 rule of Life, I came across this oscillator. It has

a period of 4194302 (verified with MCell 4.20) or 2^22-2. This

oscillator is not constructed with ships of any sort. There are some

other interesting (I think anyway) properties that are described in the

file included below.

I have a theory about this oscillator that would need proof and a

question about this oscillator (see file below for more info):

Theory: Solid (rectangular) blocks of cells form only at generations

2^N-2 where 1<=N<22 for the oscillation period.

Question: Why are 3 block combinations missing?

Eric Weddington

MCell file below:

#Life 1.05

#R 125/36

#D The large 4x94 block is an interesting oscillator:

#D 1. period 4194302 or 2^22-2

#D 2. solid blocks, such as the 4x94 block at generation 0, seem to only

form at generations

#D 2^N-2, where 1<= N < 22.

#D 3. The height and width values of the solid blocks are even numbers

that range from 2 to 96

#D and always sum to 98.

#D 4. There are 3 width / height combinations of solid blocks that are

missing: 84x14, 28x70, 56x42.

#D 5. If all the areas of the solid blocks are factored, all the primes

from 2 to 51 are present, except

#D for 7. The missing blocks above all contain 7 in the

factorization of their areas.

#D Eric B. Weddington, 2001.10.31

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Nov 16, 2001, 1:19:39 PM11/16/01

to

Eric Weddington (umg...@attglobal.net) wrote:

> I'm new to this list but I've been playing around with Life for a while

> using Life32 and recently MCell. I've looked around various Life / CA

> web pages to see if anybody else has discovered this and couldn't find

> anything.

It's been known for several years at least, but I don't know of any

published descriptions of it.

> In exploring the 2x2 rule of Life,

That name is unclear, since there are many Moore-neighborhood, totalistic

rules in which patterns made of 2x2 blocks in gen 0 are made of 2x2 blocks

in every generation. The specific rule that you're using is B36/S125 in

standard notation; i.e. birth occurs if there are 3 or 6 neighbors,

survival if there are 1, 2, or 5. (But it works the same in all rules

between B3/S5 to B3678/S012567.)

> I came across this oscillator. It has

> a period of 4194302 (verified with MCell 4.20) or 2^22-2. This

> oscillator is not constructed with ships of any sort. There are some

> other interesting (I think anyway) properties that are described in the

> file included below.

In RLE notation your pattern is:

x = 4, y = 94, rule = B36/S125

4o$4o$4o$4o$4o$4o$4o$4o$4o$4o$4o$4o$4o$4o$4o$4o$4o$4o$4o$4o$4o$4o$4o$

4o$4o$4o$4o$4o$4o$4o$4o$4o$4o$4o$4o$4o$4o$4o$4o$4o$4o$4o$4o$4o$4o$4o$

4o$4o$4o$4o$4o$4o$4o$4o$4o$4o$4o$4o$4o$4o$4o$4o$4o$4o$4o$4o$4o$4o$4o$

4o$4o$4o$4o$4o$4o$4o$4o$4o$4o$4o$4o$4o$4o$4o$4o$4o$4o$4o$4o$4o$4o$4o$

4o$4o!

This pattern emulates one in the 1-dimensional XOR CA, also known as Rule 90

in Wolfram's notation. In each phase, your pattern is an exclusive-or of a

bunch of concentric rectangles of size N x (98-N) for even numbers N.

Here are the first few phases:

gen 0: 94x4

gen 1: 96x2 + 92x6

gen 2: 90x8

gen 3: 92x6 + 88x10

gen 4: 94x4 + 86x12

gen 5: 96x2 + 92x6 + 88x10 + 84x14

gen 6: 82x16

Using a 1 to indicate the presence of a rectangle and a 0 to indicate its

absence, this becomes:

gen 0: 01000000000...000

gen 1: 10100000000...000

gen 2: 00010000000...000

gen 3: 00101000000...000

gen 4: 01000100000...000

gen 5: 10101010000...000

gen 6: 00000001000...000

Here the first position represents a 96x2 rectangle, the next represents

a 94x4 rectangle, and so on; the 48'th position represents a 2x96 rectangle.

Each 0 or 1 above is the XOR of its two immediate neighbors in the preceding

generation, except for the end cells, where we must imagine that there are

permanent 0's to the left of the leftmost cell and to the right of the

rightmost cell.

We can eliminate the inelegance of those permanent 0's by reflecting the

entire pattern about each of them, infinitely often:

gen 0: ... 0 000...00000000010 0 01000000000...000 0 000...00000000010 0 ...

gen 1: ... 0 000...00000000101 0 10100000000...000 0 000...00000000101 0 ...

gen 2: ... 0 000...00000001000 0 00010000000...000 0 000...00000001000 0 ...

gen 3: ... 0 000...00000010100 0 00101000000...000 0 000...00000010100 0 ...

gen 4: ... 0 000...00000100010 0 01000100000...000 0 000...00000100010 0 ...

gen 5: ... 0 000...00001010101 0 10101010000...000 0 000...00001010101 0 ...

gen 6: ... 0 000...00010000000 0 00000001000...000 0 000...00010000000 0 ...

Now we're dealing with an infinite, spatially periodic pattern in Rule 90.

This rule has been studied quite a bit, and can be understood by algebraic

techniques related to binary polynomials. See, for example, section 4 of

the article "Algebraic Properties of Cellular Automata" by Stephen Wolfram,

at:

www.stephenwolfram.com/publications/articles/ca/84-properties/index.html

> I have a theory about this oscillator that would need proof and a

> question about this oscillator (see file below for more info):

>

> Theory: Solid (rectangular) blocks of cells form only at generations

> 2^N-2 where 1<=N<22 for the oscillation period.

I'm sure this can be explained using the algebraic method, but I haven't

tried to work through the details.

Incidentally, rule 90 can be emulated more directly in the rule B7/S23457.

For example, the pattern below also has period 4194302; each cell in its

central row (except the ends) corresponds to one rectangle in your pattern:

.oooooooooooooooooooooooooooooooooooooooooooooooooo.

o..o...............................................o

.oooooooooooooooooooooooooooooooooooooooooooooooooo.

In RLE notation that's:

x = 52, y = 3, rule = B7/S23457

b50o$o2bo47bo$b50o!

Dean Hickerson

de...@math.ucdavis.edu

Nov 19, 2001, 2:14:38 PM11/19/01

to de...@math.ucdavis.edu

Dean Hickerson wrote:

> Eric Weddington (umg...@attglobal.net) wrote:

>

> > I'm new to this list but I've been playing around with Life for a while

> > using Life32 and recently MCell. I've looked around various Life / CA

> > web pages to see if anybody else has discovered this and couldn't find

> > anything.

>

> It's been known for several years at least, but I don't know of any

> published descriptions of it.

>

Thanks for letting me know that it's already been known.

>

> > In exploring the 2x2 rule of Life,

>

> That name is unclear, since there are many Moore-neighborhood, totalistic

> rules in which patterns made of 2x2 blocks in gen 0 are made of 2x2 blocks

> in every generation. The specific rule that you're using is B36/S125 in

> standard notation; i.e. birth occurs if there are 3 or 6 neighbors,

> survival if there are 1, 2, or 5. (But it works the same in all rules

> between B3/S5 to B3678/S012567.)

>

The name "2x2 rule variant of Life" is what I've seen using the programs MCell

and Life32. I agree, that it's not a very descriptive or mnemonic name.

>

> > I came across this oscillator. It has

> > a period of 4194302 (verified with MCell 4.20) or 2^22-2. This

> > oscillator is not constructed with ships of any sort. There are some

> > other interesting (I think anyway) properties that are described in the

> > file included below.

>

> In RLE notation your pattern is:

>

> x = 4, y = 94, rule = B36/S125

> 4o$4o$4o$4o$4o$4o$4o$4o$4o$4o$4o$4o$4o$4o$4o$4o$4o$4o$4o$4o$4o$4o$4o$

> 4o$4o$4o$4o$4o$4o$4o$4o$4o$4o$4o$4o$4o$4o$4o$4o$4o$4o$4o$4o$4o$4o$4o$

> 4o$4o$4o$4o$4o$4o$4o$4o$4o$4o$4o$4o$4o$4o$4o$4o$4o$4o$4o$4o$4o$4o$4o$

> 4o$4o$4o$4o$4o$4o$4o$4o$4o$4o$4o$4o$4o$4o$4o$4o$4o$4o$4o$4o$4o$4o$4o$

> 4o$4o!

>

I should've put the rle notation in the message, as it's shorter. Instead I just

cut and pasted directly from MCell.

But now this is what I find odd.... If the theory holds that blocks *only* form

at generations 2^N-2 then there are 3 blocks missing in the (N x (98-N) for even

N) pattern.

The blocks missing are (width x height): 84x14, 28x70, and 56x42. It's

interesting to note that each of those dimensions are some multiple of 7.

The reasoning (accurate or not) behind listing the dimensions above in that

manner is that for all blocks that are formed at g 2^N-2, the width mod 4 = 0 and

the height mod 4 = 2. The above missing blocks would fit this pattern.

How does this mesh with what you described above about Wolfram rule 90?

>

> Incidentally, rule 90 can be emulated more directly in the rule B7/S23457.

> For example, the pattern below also has period 4194302; each cell in its

> central row (except the ends) corresponds to one rectangle in your pattern:

>

> .oooooooooooooooooooooooooooooooooooooooooooooooooo.

> o..o...............................................o

> .oooooooooooooooooooooooooooooooooooooooooooooooooo.

>

> In RLE notation that's:

>

> x = 52, y = 3, rule = B7/S23457

> b50o$o2bo47bo$b50o!

>

Thanks for the pattern!

Eric

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