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 Post subject: Transforming Polyforms
PostPosted: Sun Oct 13, 2013 1:40 pm 
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Joined: Sun Nov 23, 2008 2:18 am
This is a puzzle concept inspired by the recent thread on Hexahexes, and was mentioned in a reply I made to that thread, but was overlooked there.

1. Start with a polyform.
2. Any vertex where two or mor monoforms meet a space can serve as a hinge.
3. A move consists of choosing a hinge, an edge shared by two monoforms that meet at the hinge, and twisting the two parts of the polyform either side of this edge about the hinge to form another polyform.
4. A move is only valid if their is enough space to complete the twist without monoforms overlapping and the post-move shape being a valid polyform.

To give examples using Tetrominoes:
Example 1:
Start with an I tetromino.
There are six hinges, three along each long edge.
Either end monomino can be rotated about one of the two hinges attaching it to the rest of the tetromino to form either an L or J tetromino.
Either of the middle hinges can be used to turn the I into an O tetromino.
Example 2:
Start with an O tetromino.
There are four hinges, 1 at each midpoint of the polyform's edges.
Any of these hinges can be used to turn the O into an I, J, or L Tetromino depending on how its split.
Example 3:
Start with an L or J tetromino.
There are five hinges, two along the long side, 1 long the base, one at the inner corner, and on on the inside of the upright.
Depending on the chosen hinge, you can tranform this tetromino into an S/Z, O, T, or I.

I think this concept would work with polyamonds, polyominoes, polyhexes, polycubes, polyrhombs, and any other polyform where the monoform is:
1. N-dimensional and capable of tiling n-space.
2. All of its n-1 faces are congruent.
3. it is rotational symmetric under any possible move.

I am pretty confident that the Transforming Tetromino can take the sape of any Tetromino, but I wonder if this would be true of the transforming polyomino for all N. How about for other transforming polyforms?

Considering that the parts of a polyform transformed as described rotate relative to each other, is it generally possible for the monoforms to be in any order with any set of orientations, or does this type of movement put restrictions on permuation and orientation?

Are there any other polyforms with which this concept should work?

Any ideas on how to make a physical model of a transforming polyomino?

I know some of what I said above would probably make more sense with visual aids, and I apoligize that I am incapable of creating illustrations using computer graphics and can't think of a good way to illustrate the idea with ASCII art.

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