The 19 ways to split a 2x2x2 cube into 2 equal tetracubes.
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The 19 ways to split a 2x2x2 cube into 2 equal tetracubes.
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The 19 ways to split a 2x2x2 cube into 2 equal tetracubes.
@ruuddotorg Could it be that some of them are the same, but the overall results is rotated?
It looks like the top left one, the one at the rightest right and the one in the center of the bottom line are the same.
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@ruuddotorg Could it be that some of them are the same, but the overall results is rotated?
It looks like the top left one, the one at the rightest right and the one in the center of the bottom line are the same.
If you fully account for mirroring and rotational symmetry, only 3 ways remain I think
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The 19 ways to split a 2x2x2 cube into 2 equal tetracubes.
@ruuddotorg I looked for symmetries in the layout and then realised there’s no privileged element to put in the center. Now I’m trying to figure out why 19, symmetry groups etc. Lovely stuff.
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The 19 ways to split a 2x2x2 cube into 2 equal tetracubes.
@ruuddotorg Nice work, this is great!
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@ruuddotorg I looked for symmetries in the layout and then realised there’s no privileged element to put in the center. Now I’m trying to figure out why 19, symmetry groups etc. Lovely stuff.
@tikitu There are 3 variants, one with 3 orientations (the “pancakes”), one with 4 orientations (the “y”) and one with 12 orientations (the “snakes”, actually 6 for each mirror image), for a total of 19.
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@tikitu There are 3 variants, one with 3 orientations (the “pancakes”), one with 4 orientations (the “y”) and one with 12 orientations (the “snakes”, actually 6 for each mirror image), for a total of 19.
@ruuddotorg @tikitu It's neat that in this case we always get two equivalent pieces. For example in the decomposition of 4x4x4 cubes into 32-cube chunks I don't think this would be the case. We could start with two pancakes and then swap two cubes not related by a symmetry. (And of course for 3x3x3 same volume is impossible!)
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The 19 ways to split a 2x2x2 cube into 2 equal tetracubes.
@ruuddotorg
In any case the graphic design is ace -
The 19 ways to split a 2x2x2 cube into 2 equal tetracubes.
@ruuddotorg if you allow disconnected tetracubes you get 28 more possibilities, I think, made by three disconnected-tetracube-within-2x2-cube shapes:
(1) a 2 cube line and another parallel 2 cube line diagonally opposite it. 3 orientations (one per axis).
(2) three cubes in an L shape plus a disconnected cube diagonally opposite the corner of the L. 8 (choices of corner) * 3 (orientations of L) = 24 orientations, I think.
(3) four disconnected cubes. 1 orientation, I think.
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@ruuddotorg if you allow disconnected tetracubes you get 28 more possibilities, I think, made by three disconnected-tetracube-within-2x2-cube shapes:
(1) a 2 cube line and another parallel 2 cube line diagonally opposite it. 3 orientations (one per axis).
(2) three cubes in an L shape plus a disconnected cube diagonally opposite the corner of the L. 8 (choices of corner) * 3 (orientations of L) = 24 orientations, I think.
(3) four disconnected cubes. 1 orientation, I think.
@ruuddotorg argh, I think you have to divide the 24 orientations for (2) by some symmetry factor. I'm guessing in half but need to double-check...
EDIT: yeah, I think it's 12. you can generate them uniquely by starting with 2 pancakes (3 options) and then choosing one of the four corners in in the plane of their orientation to swap between the two pancakes. 3 * 4 = 12.
if this is right, that's 12+3+1 = 16 disconnected tetracube decompositions.
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