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Consider a cube that exactly fills a certain cubical box. As in Examples 8.7 and 8.10, the ways in which the cube can be placed into the box corresponds to a certain group of permutations of the vertices of the cube. This is the group of group of rigid motions (or rotations) of the cube. (It should not be confused with th e group of symmetries of the figure, which will be discussed in the exercises of Section 12.) How many elements does this group have? Argue geometrically that this group has at least three different subgroups of order 4 and at least four different subgroups of order 3.

Hard Solution Let the vertices of the cube be labeled by the lower case letters $a, b, ..., h$ and let the vertices (corners) of the box be labeled by the upper case letters $A,B,...,H.$ We will use the word vertex to talk about the verteces of the cube and the word corner to talk about the verteces of the box. The vertex $a$ can be placed at any of the eight corners of the box. $\color{red}{\text{For each such placement, there are 3 different ways in which the cube can be placed into the box.}}$ Hence there are 24 elements in the group of rigid motions (or rotations) of the cube.

Given a placement of the cube in the box, the rotations about a line joining the centers of opposite faces of the cube constitute a subgroup of group of rigid motions (or rotations) of the cube.. Such a subgroup has 4 elements. Since there are three pairs of opposite faces, there are three such subgroups.

Given a placement of the cube in the box, the rotations of the cube about a line joining diagonally opposite vertices constitute a subgroup of group of rigid motions (or rotations) of the cube. Such a subgroup has 3 elements. Since there are 4 such diagonally opposite vertices, there are 4 such subgroups.

enter image description here

Question 1. What's the most efficient way to visualize this? What program, software, or website can I use to visualize all these permutations? There has to be a better way than just plotting all these $24 \times 8 \times 8 $ points.

Question 2. Are there any animations? Is it easy to create one myself?

Question 3. I put this in red. I don't see why for each such placement, there are 3 different ways in which the cube can be placed into the box. Hypothetically assume I place vertex $a$ at vertex $A$ of the box. I choose $A$ for convenience. I can just call it something else. Because the box only has one opening or lid, there's only 1 way to place the cube into the box so that $a$ is at $A$?

This is from John B. Fraleigh page 86 exercise 8.45 A First Course in Abstract Algebra

EDIT @aPaulT - Could you please explain how you found this marvellous animation? I want to be able to do this in general for other shapes too.

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There's an interactive graphic here if you don't mind installing the plugin:

http://demonstrations.wolfram.com/RotatingCubesAboutAxesOfSymmetry3DRotationIsNonAbelian/

Select "3b" for one of the rotations and drag the slider to see the cube move through the three positions with a corner in a fixed position for your Q3.

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  • $\begingroup$ thanks. i upvoted for you. $\endgroup$ – Matthew Lau Jan 10 '14 at 8:14
  • $\begingroup$ I'm afraid I only found it by Googling "animation rotations of a cube" or something similar, so I don't have any idea where anything similar for other shapes might be found. $\endgroup$ – aPaulT Jan 10 '14 at 9:45

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