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Is there a systematic way to find out what all the symmetries of the following cube are?

Naturally, rotations and reflections along a diagonal or a plane are taken into account.

enter image description here

Of course, by inspection one may be able to find all the symmetries, but what I really mean is;

Given an $n\times n \times n$ cube. If we know that $k$ unit cubes are black and the rest are white.

Is it possible to answer the same question in the general case?

There might be no symmetry at all, but what is the best criterion that can handle the general case?

What is the upper bound of symmetries? and what arrangements of black and white unit cubes give us the upper bound?

P.S. Sorry for asking too many questions in one post.

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2 Answers 2

It is not clear what you mean by a good criterion to handle the general case. In order to test whether a certain symmetry preserves colours, you need to inspect all cubes. For instance the cube in the illustration suggests that rotation of $2\pi/3$ around the diagonal through the visible white corner cube preserves colours, but one cannot be sure without seeing the rear view (and indeed the $8$ invisible interior cubes). Reminds of the joke about a mathematician stating that in Scotland there is at least one sheep which is black on at least one side.

As for the upper bound of symmetries: all $2^dd!$ symmetries of a $d$-dimensional (hyper-)cube could arise; that's $48$ symmetries in dimension $3$. For instance if all corners are white and all other squares blck, but there are many other possibilities.

By the way, the checkerboard colouring suggested by the illustration admits half that amount of symmetries: the other half interchanges black and white.

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There are 32 crystallographic point groups. Since you want a 3d cube the linear and planar groups wouldn't work. So there are 14 possibilities left. See:

http://en.wikipedia.org/wiki/Crystallographic_point_group

and

http://en.wikipedia.org/wiki/Crystal_classes

The Wikipedia article has a link at the bottom has a link to a flow chart to determine which of the 32 point groups a particular arrangement characterizes a particular configuration of blocks. See: http://webhost.bridgew.edu/shaefner/symmetry/pointgroup/tutorial.html#flowchart

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Dear MaxW, in fact, the crystallographic point of view was one of my major motivations to ask this question. –  Ehsan M. Kermani Feb 11 '12 at 20:53

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