A learning journey for symmetry

how to refer to the "laws of nature"? Rules to enable reliable reasoning primary maths as resource : deals with lines of symmetry, mirror leading to rotation symmetry, as a kind of pattern recognition need translational symmetry, so need to work out a learning journey restricted to dealing with symmetries associated with physical space only.

{>> PID to flip though different constraint relationships. radio button to change terms <<}

Identifying what does not change

Drop a ball. Move 3 m to your left and repeat. Drop it again. You’d expect the same behaviour. Physics does not change if you move around. Nor does it change if you turn around. Or if you drop the ball again later.

If Alice watches you, she’d agree:

Physics is invariant under translation in the distance, under rotation, under translation in time

That’s not to say that Alice will see precisely what you do — she might be a long way off and have to compensate what she records to account for that — but in these three cases, she would easily be able to figure out what you saw. You're the local witness, and after making sensible adjustments the remote watcher(in this case Alice) would agree on the physics.

Dropping the ball whilst moving at constant velocity past Alice requires different adjustments to recover what you record, but again it’s possible.

That physics stays the same under these circumstances is an excellent economical principle: allowing for parsimonious descriptions of the world. More strongly than that, it agrees with our common sense, developed as a result of successfully interacting with the lived-in world. If the world were more capricious, surviving and flourishing would be hard.

Alice could probably figure out what you’d record if you dropped the ball yet again, but now whilst accelerating.

That Alice, or you, can figure out what other points of view record is essential to physics being universal. It’s a good principle and seems to work out, although it’s hard to see how you could prove it.

In particular, physics is assumed to be the same:

You can change your point of view in any of the ways above and still rely on the same physics. In particular, you(the watcher from afar) can figure out what someone who is close to the process will record(the local witness): someone who can say here, now, co-moving, co-rotating, co-accelerating with the process.

Relativity is about figuring out what other points of view notice and record.