### Rotation but no reflection (C4) MAP rules

Posted:

**July 2nd, 2017, 4:59 am**These rules always preserve the rotational symmetry (i.e. a given pattern evolves in the same way when rotated by 90º) but not reflection symmetries (i.e. a pattern with a mirror plane does not preserve the mirror plane).

A reflection of a pattern evolves differently. C4 symmetry is preserved throughout.

Example: B36a*/S23 (clockwise symmetry)

Ironically, only C1-symmetric non-totalistic neighbourhoods can allow this kind of C4 MAP rules.

------Henceforth, a bit of a digression

Moore neighbourhood is D4 symmetry (because the corners are not equivalent to edges)

Each isotropic non-totalistic neighbourhood, due to the rotate4reflect, can be analysed with basic group theory.
NB. the (i) and (n) for σ are used because that is the way we tend to refer to the Hensel notation of neighbourhoods. I will consider them equivalent to v(vertical) and d(dihedral) respectively.

Assigning symmetry to each neighbourhood should be easy (I have somehow messed up though and I think I have the wrong approach). Considering the 8 neighbours, what I have done that probably is wrong is to check each transformation and count the number of unchanged cells within the neighbourhood.

E=8; 2C4=0; C2=0; 2σ(i)=2; 2σ(n)=2

which is a reducible representation equivalent to 2 A1 + B1 + B2 + 2 E.

How do I move on and get the symmetries of each neighbourhood within this representation? It's obvious that for this thread title you need "chiral" neighbourhoods to get any kind of rotate-but-not-reflect thing going on.

A reflection of a pattern evolves differently. C4 symmetry is preserved throughout.

Example: B36a*/S23 (clockwise symmetry)

Code: Select all

```
x = 21, y = 12, rule = MAPARYXfhZofugWaH7oaIDogBZofuhogOiAaIHogIAAgAAWaH7oaIHogGiA6ICAAIAAaIDogIAAgACAAIAAAkAAAA
3o5bo3b2o4b3o$2bo3bobo3bo5bobo$b2o3b3o3b3o3bo7$3o4b2o3bo5b3o$obo5bo3bo
bo3bo$2bo3b3o3b3o3b2o!
```

Ironically, only C1-symmetric non-totalistic neighbourhoods can allow this kind of C4 MAP rules.

------Henceforth, a bit of a digression

Moore neighbourhood is D4 symmetry (because the corners are not equivalent to edges)

Each isotropic non-totalistic neighbourhood, due to the rotate4reflect, can be analysed with basic group theory.

Code: Select all

```
.....E.....2C4.....C2.....2σ(i).....2σ(n)
A1...1.....1.....1.....1.....1
A2...1.....1.....1.....-1.....-1
B1...1.....-1....1.....1.....-1
B2...1.....-1....1.....-1.....1
E....2.....0.....-2.....0.....0
```

Assigning symmetry to each neighbourhood should be easy (I have somehow messed up though and I think I have the wrong approach). Considering the 8 neighbours, what I have done that probably is wrong is to check each transformation and count the number of unchanged cells within the neighbourhood.

E=8; 2C4=0; C2=0; 2σ(i)=2; 2σ(n)=2

which is a reducible representation equivalent to 2 A1 + B1 + B2 + 2 E.

How do I move on and get the symmetries of each neighbourhood within this representation? It's obvious that for this thread title you need "chiral" neighbourhoods to get any kind of rotate-but-not-reflect thing going on.