The round trip, on P2₁/c
4 operations were generated from the standard generators of P2₁/c; the orbit of three points in general position was formed, the group was discarded, and 4 operations were rediscovered from the 12 points alone. The two sets are identical, which is what the figure asserts.
4 essays call
roundtrip-3d. The drawing above is what it returns with no arguments at all; every
call below passes it something, because a placement that passes nothing draws whichever member
of the family the generator happens to default to rather than the one its essay argues about.
Every one of this site's 393 essays names its parameters at the
call site, which the standard pass of 2026-08-09 established and param-floor
holds.
Where it is called
Changing this generator changes every one of these figures.
Into space
Forgetting a group in three dimensions
For three phases this site said its machinery was two-dimensional and decided nothing about a space group. That was true, and it was a limit rather than a principle — nothing in the decidability argument mentions the number two.
Into space
The operations nobody put in
A group is not a list of generators. Compose two of them and something arrives that neither contained — a screw where there were only mirrors, a glide where there was only a mirror and a centring vector — and in three dimensions most of a group's operations get there this way.
Symmetry at work
How many polytypes there are
One free choice per layer, repeated, gives a family of structures with the same composition, the same density and the same twelve neighbours — differing only in a sequence. Counting them up to rotation, reversal and relabelling turns "silicon carbide has hundreds of forms" into an enumeration.
Into space
The plan contains the group
A space-group diagram has always been treated here as a picture of the group. It is more than that: hand back the marks alone — no matrices, no operations, not even the centring — and the group comes out exactly, forty-five times out of forty-five.