Then, I had the following observation: if we turn pieces one-sixteenth of a full turn, then also interesting things happen. In particular, a queen becomes a knightrider! Now, this observation may shatter your hope of the end of the Turning Chess story, perhaps?
Ouch. Have you ever known anybody so cruel and heartless? I thought I could just write about Turning once and be done with it, and instead look what he does!
In fact, Hans is right, but he's also wrong. If you turn an AD one sixteenth of a turn, it becomes a Knight; if you turn a WF one-sixteenth of a turn, it becomes -- what?
This is actually an important question, because the Queen's first step is a King move, a WF move; if you turn a Queen one sixteenth of a turn, her second, fourth, sixth, etc., steps are on the squares a Knightrider would touch, but what of the odd-numbered steps?
There are several possibilities.
First, perhaps the odd-numbered steps simply land on the nearest square to their proper geometrical location; now the 16th-turned-Queen becomes a horrid piece able to move, for example, in a "straight line" from d1 to d2 to c3, c4, b5, b6, a7, a8; or perhaps also from d1 to c2 to c3, b4, b5, a6, a7 -- ouch!
Secondly, perhaps the 16th turn makes the Q go to a place between two squares so she can not move at all! This is not very interesting, is it?
Third, the 1/16th turn might make the Q go to a place between two squares, where she can stop on the line between the two squares, thus expanding the board. This possibility would only be useful in a contest to devise a game on a small board, and who ever heard of such a contest as that?
Finally, perhaps the Q can simply skip over the nonexistent squares, and thus make, after all, the same moves as a KnightRider! This is actually the solution I like the best, even though the 1/16th Queen is no stronger than the 0.0625 AADD. This is the solution I will use to describe what happens when you turn the Queen one twenty-fourth of a turn:
You will notice that the AD rotated by a sixteenth of a turn becomes a Knight, and I'm sure you've also noticed that the AD and the Knight between them fill the outside edges of a 5 by 5 square of squares, and of course the edges of 5x5 number 16, which is 5 squared minus 3 squared. Obviously, 7 squared minus 5 squared is 24, and so the Long Knight (the 1,3 leaper) has 24 possible rotations, and rotating a Queen 1/24th of a turn makes her become first a (1,3)-rider, then a (2,3)-rider, then a Queen again.
81 - 49 = 32, of course, but please don't mention it. This could go on forever.
I believe the game is more interesting if the player chooses which direction to rotate the other Knight (or Rook or Bishop or HFD).
I mention Synchronized Turning just to show I had my own followup ideas to Turning Chess.
Since Hans is Dutch and since the Rose is named after a flower, I'd like to call the q16[NAD] the "Tulip", if that name for a piece is not already taken. But I think it is already taken, so I'll name this piece the Hans, in honor of the clever fellow who suggested the idea of 1/16 turns.
The Rose is a Knightrider that turns 1/8th of a turn at each step it takes, always turning in the same direction; its funny notation was qN, but we see it should be q8[N]. (In my page on circular riders, some of the square brackets have been turned into capital As, which makes the notation hard to understand. This has inspired me with yet another thing that I might get around to writing...)
The Hans is quite simply a Knightrider that makes one-sixteenth of a turn at every step. For example, it could move from h1 to h3, g5, e7, c8, a8, and oops it already ran out of room. In fact, this piece cannot express its full potential movement on anything smaller than a 25x25 chessboard, and even then it needs to be in the center square. Even on an 8x8 board, it is quite powerful, as it moves in 16 different directions, and moves in a curve of leaps, something like the circular grand jetees in Swan Lake, a path that may allow it to avoid obstacles that would block a Queen; in fact, it can attack through a Pawn chain -- a horrifying thought.
If, instead, we postulate a Crooked Sixteenth Knightrider, we see that its path from b1 would go to c3 and then c5 and then d7, or from b1 to c3-e5-f7. Such things cannot be allowed to exist.
And as for 1/16th corkscrew pieces in 3D Chess, I think normal corkscrew pieces are already confusing enough.
Now I can turn to other things.