>Also, I have always wondered if there's a way to translate these shapes to strings for quick "mental algorithm" matching. Something like, "5, left turn, 3, right turn..."

I was once measured to have abysmal spatial reasoning (which is probably backed up by poor hand-eye coordination and sense of direction) but I had no problem doing molecular geometry and group theory, I suspect it was because I applied logic rather than visualisation to compensate. (I am terrible at walking into an empty place an imagining how I might want to fill it)

Playing on mobile, it would be cool if the accelerometer rotated all the pieces at once.

I think that's basically what I did, just without elevating the forms to anything conscious. By the end, I was matching shapes too fast to have actually counted much of anything -- some sort of consolidated memory had started.

A small improvement: use a vowel to encode turns, and consonants to encode segment lengths. So each shape would be encoded by a "word", e.g. "bakitux". Brains have dedicated "word-processing units" and those are much faster than number-processing units.

Abstracting into words will lead to invention of words that are disconnected from the underlying geometry (i.e., shapes that are not possible due to collisions), but have strong verbal associations. Eventually, the word thinkers will redefine the system to allows them to play word games without dealing with the pesky geometric limitations.

> Brains have dedicated "word-processing units" and those are much faster than number-processing units.

*In most people.

> The first turn can be defined as always "right" or "left,"

No it can’t, not in 3D... it is just a turn.

You can however choose to rotate about the first segment until the turn is Right, then use this orientation for the rest.

In terms of normal form and restarts, who cares? Just generate the string both ways, it’s 2n computations, choose the lowest in lexicographic order, and life is good. If you get to billions of turns are humans really going to be doing the comparison? Surely we’d just stream the string into a hash function or something...

Right-handed rotation is different from left-handed rotation no matter what coordinate system you choose.

WLOG the shape starts at the origin (0,0,0), the first kink is at (0, 0, m), and the next point is (x, y, m), where x²+y²=n²

You can always rotate this to (n, 0, m) and view from the +y axis looking down, and it is a right turn.

The L shape is not complex enough to be chiral, but the post is (and you are!) is claiming that you need an extra bit of information to specify “which L, the right handed one or the left handed one?” ... That is a distinctively 2D question.

The keyword is "define"!

You can just define the first turn to be Right no matter how it is oriented.

If you are not happy with that then just rotate the first segment until it looks Right to you. Then label the other turns.

If you are still not happy with the word Right to denote the first term then just make up new notation. Define the first turn to be 1. Use numbers to denote turns if you object to using Left and Right.

> No it can’t, not in 3D... it is just a turn

Not if it is chiral. then whatever turn is defined as right is not left