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Oscillator glide symmetry classes

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Oscillator glide symmetry classes

Postby Tropylium » July 24th, 2011, 5:41 pm

I've suggested on the wiki classifying patterns by their symmetry. This seems to be now well underway.

However, a lot of oscillators cannot be placed in the usual planar symmetry classes, since they have glide symmetry some sort (along the time axis). How should these symmetry classes be denoted? I think this list covers all the possibilities:

Glide rotations
—any single phase is asymmetric, sum of all phases has S2 symmetry (blocker)
—any single phase is asymmetric, sum of all phases has S4 symmetry (dinner table)
—any single phase has S2 symmetry, sum of all phases has S4 symmetry (windmill)

Glide reflections
—any single phase is asymmetric, sum of all phases has D1 symmetry (orthogonally: griddle; diagonally: mold) (glide reflectiv spaceships also fall here!)
—any single phase has S2 symmetry, sum of all phases has D2 symmetry (orthogonally: Achim's p144, diagonally: bipole)
— any single phase has S4 symmetry, sum of all phases has D4 symmetry (phoenix, Achim's p16)

Glide rotations+reflections
—any single phase has D1 symmetry, sum of all phases has D2 symmetry (orthogonally: smiley, tumbler; diagonally: mazing)
—any single phase has D2 symmetry, sum of all phases has D4 symmetry (orthogonally: blinker, monogram; diagonally: washing machine)

These are of course a subset of the three-dimensional line groups, but the Sn/Dn notation apparently does not extend to them.

There's also a question of what to do with billiard tables that require induction coils. A tubber stabilized by two bookends is D2; stabilized by four snakes it can be S4; stabilized by four houses, D4. A similar case is emulators, which can be either "cis" (falling then in the same class as smiley), or "trans" (falling then in the same class as Achim's p144).
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Re: Oscillator glide symmetry classes

Postby Jason Summers » July 25th, 2011, 5:51 pm

Here is Hickerson's classification of the symmetries of finite oscillators, from his "stamp collection":

There are 43 types of symmetry that an oscillator can have, taking into
account both the symmetry of a single generation and the change of
orientation (if any) M generations later.  There are 16 types of
symmetry that a pattern can have in a single generation.  Each of these
is given a one or two character name, as follows:

   n   no symmetry

   -c  mirror symmetry across a horizontal axis through cell centers
   -e  mirror symmetry across a horizontal axis through cell edges

   /   mirror symmetry across one diagonal

   .c  180 degree rotational symmetry about a cell center
   .e  180 degree rotational symmetry about a cell edge
   .k  180 degree rotational symmetry about a cell corner

   +c  mirror symmetry across horizontal and vertical axes meeting
       at a cell center
   +e  mirror symmetry across horizontal and vertical axes meeting
       at a cell edge
   +k  mirror symmetry across horizontal and vertical axes meeting
       at a cell corner

   xc  mirror symmetry across 2 diagonals meeting at a cell center
   xk  mirror symmetry across 2 diagonals meeting at a cell corner

   rc  90 degree rotational symmetry about a cell center
   rk  90 degree rotational symmetry about a cell corner

   *c  8-fold symmetry about a cell center
   *k  8-fold symmetry about a cell corner

For a period P/1 object, specifying the symmetry of generation 0 tells
us all there is to know about the oscillator's symmetry.  For a period
P/2 or P/4 object, we also need to know how gen M is related to gen 0.
For the P/2 case, gen M can either be a mirror image of gen 0, or a 180
degree rotation of it.  For the P/4 case gen M must be a 90 degree
rotation of gen 0.  In any case, if we merge all gens which are multiples
of M, the resulting pattern will have more symmetry than the original
oscillator.  We describe the complete symmetry class of the oscillator
by appending the one or two character description of the union's symmetry
to that of gen 0's symmetry.  For example, if gen 0 has 180 degree
rotational symmetry about a cell center, and gen M is obtained by
reflecting gen 0 across a diagonal, then the union of gens 0 and M is
symmetric across both diagonals, so its symmetry class is denoted ".cxc".

The 43 possible symmetry types are:

   period/mod = 1:  nn    -c-c  -e-e  //    .c.c  .e.e  .k.k  +c+c
                    +e+e  +k+k  xcxc  xkxk  rcrc  rkrk  *c*c  *k*k

   period/mod = 2:  n-c   n-e   n/    n.c   n.e   n.k
                    -c+c  -c+e  -e+e  -e+k
                    /xc   /xk
                    .c+c  .cxc  .crc  .e+e  .k+k  .kxk  .krk
                    +c*c  +k*k  xc*c  xk*k  rc*c  rk*k

   period/mod = 4:  nrc   nrk

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