∗632: build the three mirrors from two generators

∗632 is usually given by three mirrors P, Q, R, the sides of its kaleidoscope triangle. Two moves make all three: α, a sixth-turn about the 6-centre O, and the mirror R. Click them until your copy (left) sits where the target puts it (right).

Starting point: ∗632 (p6m) in Two Generators, where α = PQ. Drawn like Describing a wallpaper group by its generators.

Yours
Target

6-centre up-centre down-centre 2-centre where your copy starts the copy what the motion is

Each click moves

Steps

    Keys: a α · s α−1 · r R · ⌫ undo · h hint

    How two moves make three mirrors

    R is one of the two moves

    R

    One click. R runs through the 3-centre ▲ and the 2-centre ◆ of the kaleidoscope triangle.

    Q is R, moved

    Q = R α−1 R α R = g R g−1,  g = R α−1

    The motion g carries R's line onto Q's. Both lines meet at the 3-centre ▲, where a third-turn swaps them: Q and R are one kind of mirror.

    P is Q, turned

    P = α Q = α R α−1 R α R

    A sixth-turn after Q is the mirror 30° away. P is the other kind: it swaps ▲ and ▽, so every word for it has an odd number of α's.

    Two parities every word must pass

    Each R flips your copy. Each α, either way round, moves its 3-corner between ▲ and ▽, and R never does. So the motion fixes how many R's and α's its word has, mod 2. The chips under each pane track both.

    BuildFlipped3-cornerCount of RCount of αFewest

    With α−1 taken away, Q takes 9 clicks and P 10, since α−1 = α5.

    Spoiler: six clicks that pass R, then Q, then end on P

    With each click moving the plane: R, α, R, α−1, R, α. The word grows on the left, because each new move acts after the ones before it.

    Clicks that move the plane, or the copy

    The plane (the default): α always turns about O and R always flips in its own line. A click acts after everything before it, so its letter goes on the left of the word. Clicking R, α, R, α−1, R writes R α−1 R α R = Q.

    The copy: α turns your copy about its own 6-corner and R flips it across its own R side, so a flipped copy turns the other way. Each letter now goes on the right, and you click a word in the order it is written: R, α−1, R, α, R makes Q again. Both make every motion; only the click order flips.