### Mirrors

Posted:

**January 18th, 2017, 3:37 pm**A rule based on biggiemac's idea in the CA Puzzles thread.

The behaviour of the states 2 and 3, complementary cells, is similar to that of special states in extendedlife, but not quite. Quite a few stable groupings of cells, as well as haplominos, are interesting catalysts or eaters.

- All still lifes in CGoL, when made with the states 2 and 3, are period 2 oscillators. They can therefore change in phase/parity (with the biblock potentially having an in phase or an out of phase combination).

I like to call them g for those with the live state in even generations and u for those with the live state in odd generations, from the group theory (applied to chemistry) nomenclature genau/ungenau.

There immediately is a marvellous application to this!! Pseudo still lifes, like the biblock, can be (g,u), i.e. each island in a different phase. However, for true still lifes to be period 2 oscillators, all the still life has to be in phase with itself - otherwise the cells that make it a true still life will be born, see here:

^ Proof that block on table is a true still life

Reactions can switch the parity of a SLP2:

^Glider or two cells change g-block to u-block

Proof of selective response:

A parity changing G to Century:

will function as an eater with same parity if the glider enters at a wrong phase:

Code: Select all

`@RULE Mirrors`

@TABLE

n_states:4

neighborhood:Moore

symmetries:permute

var a={0,1,2,3}

var a2=a

var a3=a

var a4=a

var a5=a

var a6=a

var a7=a

var a8=a

var l={1,3}

var l2=l

var l3=l

var l4=l

var l5=l

var l6=l

var l7=l

var l8=l

var d={0,2}

var d2=d

var d3=d

var d4=d

var d5=d

var d6=d

var d7=d

var d8=d

#BIRTH

0,l,l2,l3,d,d2,d3,d4,d5,1

2,a,a2,d,d2,d3,d4,d5,d6,3

2,l,l2,l3,l4,a,a2,a3,a4,3

#SURVIVE

1,l,l2,a,d,d2,d3,d4,d5,1

3,a,d,d2,d3,d4,d5,d6,d7,3

3,l,l2,l3,l4,a,a2,a3,a4,3

#CAT

1,a,a2,a3,a4,a5,a6,a7,a8,0

3,a,a2,a3,a4,a5,a6,a7,a8,2

@COLORS

1 255 255 255

2 35 35 35

3 225 255 255

The behaviour of the states 2 and 3, complementary cells, is similar to that of special states in extendedlife, but not quite. Quite a few stable groupings of cells, as well as haplominos, are interesting catalysts or eaters.

- All still lifes in CGoL, when made with the states 2 and 3, are period 2 oscillators. They can therefore change in phase/parity (with the biblock potentially having an in phase or an out of phase combination).

I like to call them g for those with the live state in even generations and u for those with the live state in odd generations, from the group theory (applied to chemistry) nomenclature genau/ungenau.

There immediately is a marvellous application to this!! Pseudo still lifes, like the biblock, can be (g,u), i.e. each island in a different phase. However, for true still lifes to be period 2 oscillators, all the still life has to be in phase with itself - otherwise the cells that make it a true still life will be born, see here:

Code: Select all

`x = 31, y = 19, rule = Mirrors`

2C24.2B$2C24.2B2$2C24.2C$2C24.2C11$4.2C23.2C$4.C24.C$.2C.C21.2B.C$.2C

.2C20.2B.2C!

^ Proof that block on table is a true still life

Reactions can switch the parity of a SLP2:

Code: Select all

`x = 6, y = 6, rule = Mirrors`

2.A$A.A$.2A2$4.2B$4.2B!

Code: Select all

`x = 3, y = 3, rule = Mirrors`

.2C$A2C$A!

^Glider or two cells change g-block to u-block

Proof of selective response:

A parity changing G to Century:

Code: Select all

`x = 11, y = 10, rule = Mirrors`

4.C$9.2B$9.2B2$6.C3$.A$.2A$A.A!

will function as an eater with same parity if the glider enters at a wrong phase:

Code: Select all

`x = 11, y = 10, rule = Mirrors`

4.C$9.2C$9.2C2$6.C3$.A$.2A$A.A!