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Jeffery Mewtamer

Post subject: Magics, polyominos, and squashed polycubes. Posted: Tue Mar 05, 2013 12:43 am 

Joined: Sun Nov 23, 2008 2:18 am

My thoughts randomly float to a series of 4tiled magics based on the tetrominos as they appear in Tetris and I realized that any unfolded magic(or at least one that follow conventions regarding how tiles connect to each other) is a foldable polyomino. Thinking about the different shapes magics can take on, I realized that fullyfolded positions(that is all tiles lie in a series of parallel planes, none are lying in an orthogonal plane) can be thought of as a polycube that has been squashed along one of its axes. Now, it is obvious that, under normal constructions, an ntile magic cannot assume all nominos.
For example, the clasic rubik's magic can be unfolded into the 2*4 octomino and the 3*3 minus a corner octomino, but to my knowledge, it cannot assume any other octomino shape, and the 8 tile magic that forms a ring octomino is a completely different puzzle within this family of folding puzzles.
Expanding into what I have dubbed squashed polycubes, the classic magic can assume all three of the squashed 1*2*4 octcubes, the squash 2*2*2 octocube, plus a few noncuboid squashed polycubes.
Considering these observations, some questions that come to mind: 1. Given the set of ntiled magics, which one can achieve the greatest subset of the squashed ncubes. 2. For the magic that answer's 1, is there a shape that is particularly hard to fold the magic into. In other words, would it make an interesting puzzle based on solving for shape alone. 3. Are there any magic that do not satisfy 1, yet still make for an interesting puzzle solving for shape? 4. Are there any squashed ncubes achieveable by 2 or more distinct ntiled magics. 5. In general, can a magic folded into one polycube of a chiral pair be refolded into the other of the chiral pair? 6. In general, can a magic folded into a asymmetric squashed polycube be refolded to be squash along the other axes? 7. Are there any ncubes not reachable by any ntiled magic? 8. Is their a polyform related to magics in states that have tiles lying in orthogonal planes?
_________________ Just so you know, I am blind.
I pledge allegiance to the whole of humanity, and to the world in which we live: one people under the heavens, indivisible, with Liberty and Equality for all.
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kastellorizo

Post subject: Re: Magics, polyominos, and squashed polycubes. Posted: Sun Mar 17, 2013 7:20 am 

Joined: Fri Nov 04, 2005 12:31 am Location: Greece, Australia, Thailand, India, Singapore.


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Jeffery Mewtamer

Post subject: Re: Magics, polyominos, and squashed polycubes. Posted: Mon Mar 18, 2013 4:35 am 

Joined: Sun Nov 23, 2008 2:18 am

I will admit that squash was simply the first adjective that came to mind when comparing to regular ploycubes, and I have no idea if their is a name for such in proper polyform nomencleture.
To try and better explain what I mean by a squashed polycube: Start with a halfcube(that is a cuboid with dimentions of 1, 1, and .5) as your monoform. you hav two allowed joints: 1. two half cubes joined by their square faces(forming a diform that is cubic in shape). 2. Joined by rectangular faces, forming a diform that is a 2*1*.05 cuboid. Joining a rectangular face to a square face is invalid. Joining two face with only partial overlap is invalid. joining along an edge or at a single vertex is invalid.
What I described above produces a set of polycubes that have been squashed by 50% along one axis, but you can set the unequal dimension of the monoform to any thickness(and could even create elongated polycubes by setting it greater than the equal dimensions). Naturally, if we view the tiles of a magic as being cuboids with a square base, we end up replace the .5 in the above polyform description with a rather small number(i.e. the thickness of a magic tile relevant to their length and width).
In other words, if we take a magic tile as our monoform, the set of polyforms I am interested in is that formed by allowing the following joints: 1. two tiles stacked on top of each other, all edges lined up. 2. two tiles joined edgetoedge with nontocuhing eges and faces flush.
And I am interested in the subsets of these polyforms that are reachable with the movements allowed by a magic's folding and unfolding.
_________________ Just so you know, I am blind.
I pledge allegiance to the whole of humanity, and to the world in which we live: one people under the heavens, indivisible, with Liberty and Equality for all.
My Shapeways Shop


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kastellorizo

Post subject: Re: Magics, polyominos, and squashed polycubes. Posted: Mon Mar 18, 2013 7:21 pm 

Joined: Fri Nov 04, 2005 12:31 am Location: Greece, Australia, Thailand, India, Singapore.

If it makes sense, by using tiles to create cubes, i.e. going from two to three dimensions, we are able to create (from a given open magic) ANY polycube (squashed or not), and in more than one ways. To be more specific, I will give an example: A 1x1x18 open magic (presented as a continuous segment) can create any squashed 4polycube. A 1x1x24 open magic (presented as a continuous segment) can create any nonsquashed 4polycube. In both cases we will need to stabilise the parts, probably with rubber bands. Now, if we use open magics presented as noncontinuous segments, I suspect that they can also create all cases (or at least MOST of them), but I've got no proof of this on hand. Pantazis PS. In any case, I hope we are on the same page, or else some images would be extremely useful!
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