# Non-product spacetime-group examples

For a spacetime symmetry group `Sigma`, let `K_s` be its pure spatial subgroup
and `K_t` its pure temporal subgroup.  Here “non-product” means that a mixed
generator of `Sigma` is not in `K_s K_t`; equivalently, its spatial and temporal
factors are not separately symmetries.  This is different from merely observing
that an ambient transformation can be written as a spatial map followed by a
temporal map.

Time below is lifted to the real line. `U:(x,t)->(x,t+T)` denotes the pure
temporal period and becomes the identity after passage to the periodic time
coordinate `S^1_T=R/TZ`.

## Mixed space-time glide

With full spatial periods `X,Y`, take

```text
G:(x,y,t) -> (-x, y+b/2, t+T/2).
G^2 = Y U.
```

Every odd power of `G` retains both the reflection/half-cell glide and a
half-integer time shift.  Thus the reflection, spatial half-translation, and
half-period temporal translation are not individual symmetries.  The animation
uses two vertically offset chiral kites and the color transposition `(0 1)`.

This is the mixed 2+1D time-glide operation explicitly described by
[Ke and Wu](https://arxiv.org/html/2604.05619v1).

## Glide time reversal

Let `X` be translation by one full spatial cell and define

```text
Q:(x,y,t) -> (x+a/2, y, -t).
Q^2 = X,
Q U Q^-1 = U^-1.
```

Neither the half-cell shift nor time-coordinate reversal is present separately. More
strongly, the time-orientation extension is non-split on the covering space:
every time-reversing lift `X^m U^n Q` squares to the odd spatial translation
`X^(2m+1)`, so no reversing lift has order two. The two motif orbits traverse
the same path with opposite time orientations and exchange colors.

This operation is called glide time-reversal by
[Xu and Wu](https://arxiv.org/abs/1703.03388) and by
[Ke and Wu](https://arxiv.org/html/2604.05619v1). The visualization represents
the coordinate map `t -> -t`; it does not encode a quantum antiunitary
operation.

## Rotary time reversal: magnetic point group 4-prime

On a square spatial lattice, take

```text
Q4:(x,y,t) -> (R_90(x,y), -t).
Q4^2 = R_180,
Q4^4 = identity.
```

Modulo the full time period, the mixed group is `C4`.  Its pure spatial kernel
is the `C2` generated by `R_180`, and its temporal image is coordinate-reversal
`C2`.  The extension

```text
1 -> C2 -> C4 -> C2 -> 1
```

does not split: `C4` has only one element of order two, whereas a complement
would require another.  Visually, opposite motifs share a pose at every frame;
adjacent motifs play the pose backward.  The alternating two-color action is
the quotient `C4 -> C2`.

The magnetic point group `4'` is one of the valid tetragonal 2+1D classes in
[Ke and Wu's classification](https://arxiv.org/html/2604.05619v1).

## Dihedral `D3` spacetime action

Let `S` be a threefold time screw and `M` a reflection combined with reversal
of the time coordinate:

```text
S = (R_120, t+T/3),
M = (mirror, -t),
S^3 = U,
M^2 = identity,
M S M = S^-1.
```

After quotienting by `U`, these operations generate a diagonal `D3`, not the
independent product of a spatial and temporal `D3`. The rotation, third-period
shift, reflection, and time-coordinate reversal are absent as separate generic
symmetries. The color actions are `rho(S)=(0 1 2)` and `rho(M)=(1 2)`. The
chosen dart motif gives a three-element orbit satisfying both relations.

This animation is a constructive inference combining the `3m'` class and time
screws in [Ke and Wu](https://arxiv.org/html/2604.05619v1) with the orbit method
used for [Choreographic Crystals](https://arxiv.org/abs/1407.5876).

## Loop constraints

All examples contain `U`, so they have a genuine pure temporal period.  Frames
are sampled at `t=k/N`, `k=0,...,N-1`; the duplicate endpoint is omitted.  The
default `N=60` is divisible by 2, 3, and 4, and the GIF writer sets `loop=0` for
infinite playback.  Tests verify the continuous group relations and `t=0=t=1`;
the decoded-GIF audit independently checks metadata, duration, and seam size.