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 Post subject: What is it with edge turning puzzles and shapeshifting?
PostPosted: Fri Sep 27, 2013 8:48 am 
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Joined: Fri Nov 05, 2010 2:20 am
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I've noticed that all solids where all edges turn will shapeshift.

Pyramorphinx, helicopter cubes, edge turning octahedra, dodecacopter...

But why? Anyone has an explanation?

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 Post subject: Re: What is it with edge turning puzzles and shapeshifting?
PostPosted: Fri Sep 27, 2013 2:24 pm 
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I'll try my best...
First of all, consider one edge on a polyhedron. That edge will have 2 endpoints. At either one of those endpoints, there will be at least 2 other edges that meet at that endpoint. There can't be one because that would mean there are also only 2 faces at that endpoint, or vertex, and there must be at least 3 faces around a vertex on a polyhedron. Therefore, there will be at least 2 edges (there can be more than 2, like on an octahedron or icosahedron). Both of those edges will create an angle which is symmetrical, so that means they also have the same vertical slope relative to the first edge. Keep in mind, though, that there are 2 endpoints for each edge. Because of this, there will always be 4 edges adjacent to any single edge with the same vertical slope(although there can be more than 4 total adjacent edges).
Now, certain sets of all of those edges that are adjacent will have the same vertical slope. If you imagine an edge turning puzzle, when you turn one edge, a part of one of the adjacent edges will be moving (unless the cut depth is infinitesimally small, but that doesn't really count). Because there are 3 other adjacent edges which have the same vertical slope, the cut of one can align with the cut of another and therefore you can turn another edge in mid-turn. I hope this makes sense.
The reason shapeshifting occurs is because edges only have a 180 degree rotational symmetry, so because you are turning it at an angle greater than or less than that, it causes shapeshifting and/or jumbling.

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Last edited by benpuzzles on Fri Sep 27, 2013 6:48 pm, edited 2 times in total.

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 Post subject: Re: What is it with edge turning puzzles and shapeshifting?
PostPosted: Fri Sep 27, 2013 2:57 pm 
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Joined: Thu Mar 07, 2013 2:38 pm
Location: U.K
Consider this; a megaminx can do 72* turns till you reach a end point to turn another face and a 3x3 takes 90* to do the same. Therefore the hexaminx would shapeshift and so would the mastermorphix. However, a rhombic dodecahedron takes 90* turns for jumbling and a 180* turn to stay in the rhombic dodecahdron shape. Reducing it to make a cube would cause it to jumble, not shape shift (e.g. helicopter cube). Not sure if I explained it well enough.

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