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 Post subject: Menger Sponge Dissection
PostPosted: Wed Jan 08, 2014 12:51 pm 
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Joined: Sun Nov 23, 2008 2:18 am
The 1st Iteration of the process that produces the Menger sponge can be seen as a icosacube whose parts are arranged to form the skeleton of a cube. How many ways can this icosacube be devided into m n-cubes where m and n are integers and m*n = 20? For a given n, are ther any solutions consisting of all the same polycube or all different polycubes?

There is one way of dissecting the level 1 menger sponge into monocubes and could arguably be considered a trivial dissection.
I suspect that dissection into bicubes is impossible, but have no idea how to formulate a proof.
I have had no luck trying to find a dissection into 5 tetracubes, but suspect there is at least one such disection.
I have found a dissection using 4 copies of the same pentacube, but do not know if it is the only such dissection.
I have found 3 ways of bisecting the level 1 menger cube into a pair of identicle decacubes.

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 Post subject: Re: Menger Sponge Dissection
PostPosted: Wed Jan 08, 2014 3:46 pm 
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Bicubes are impossible because of checkerboard coloring.


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 Post subject: Re: Menger Sponge Dissection
PostPosted: Wed Jan 08, 2014 5:06 pm 
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I feel kind of silly for forgetting checkerboard coloring. Though that also means dissection into 5 l tetracubes is impossible since the l tetracube can itself be dissected into a pair of bicubes. Since the I, O, S, t, and chiral tetracubes can't fit in the level 1 Menger Sponge, and its fairly easy to show that inserting 3 or more tripod tetracubes either results in overlap or a isolated monocube, any disection into tetracubes must contain a combination of L and tripod tetracubes and no more than 2 tripod tetracubes. I think I have found a dissection with 3 l and 2 tripod tetracubes, but lack any good way of confirming the visual in my head.

P.S. If what I refer to as chiral and tripod tetracubes have accepted names, I do not know them, but I hope these names are unambiguous enough for what I have written to be understandable.

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