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Given a cube am facing great difficulty in visualising rotation symmetries along the axis formed by the midpoints of diagonally opposite edges.

Given two diagonally opposite edges, need consider the rest of coplanar edges, and the two long diagonals formed through them.

Need to swap the vertices of these long diagonals, for order $2$ rotation symmetry, apart from swap of the two vertices of each diagonally opposite edge.

Say, cube is given as : enter image description here

And need to find the rotational symmetry being given by the axis formed by the midpoints of edges determined by the vertices $A,F$ and $C,H$ respectively. Then, the correct answer is given by the rotational symmetry: $(AF)(CH)(DG)(BE).$

Say, here have great difficulty in visualising how the vertices $D,G$ get swapped.

Have found a link on desmos at: cube

If could modify parameters here, to visualise the rotation symmetry given above, then would be able to visualise easily. Or, if could provide some even better visualization tool.

jiten
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2 Answers2

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It looks like this: ${}{}{}{}{}{}{}{}{}$ Cube edge rotation

This animation was made in the Asymptote vector graphics language. You can find the source code here.

Kevin Carlson
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  • Better change the labeling to the one here. – jiten Sep 27 '22 at 05:46
  • Would be highly useful for all - if the way it was built be specified, or some hint as how to acquire the skill and what is name of the tool. Your rotation axis is about the midpoints of the edges determined by $2,6,$ and $4, 8,$ respectively. The rotational symmetry is:$(26)(48)(35)(17).$ – jiten Sep 27 '22 at 05:57
  • Continuing with my last comment, this tool seems better than geogebra and desmos. So, request some details. – jiten Sep 27 '22 at 06:02
  • Request to provide another gif, that takes the reflection axis as passing through the midpoints of edges determined by $2, 3,$ and $5, 8,$ respectively. Reason : faced difficulty in cases where reflection axis is determined by edges parallel to the base of the cube. Hence, OP specified such a case. – jiten Sep 27 '22 at 06:08
  • @jiten If you turn your head $90$ degrees to the left, the edge on the far right of the cube will look like an edge on the bottom of the cube. – David K Sep 27 '22 at 12:34
  • @DavidK A better alternative could be the knowledge of the tool, so could make new gif accordingly. – jiten Sep 27 '22 at 12:51
  • I confess, though, that I also obscured my meaning by writing tongue-in-cheek. You don't actually have to turn your head. You could instead realize that if a rotation exchanges certain vertices in a certain way when the rotation axis passes through vertical edges, it will still exchange the same vertices in the same way if we first reorient the entire cube so that one edge the axis passes through is on the base of the cube. This is a mental visualization tool. – David K Sep 27 '22 at 12:55
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    @jiten I’ve added some info about how I made this. – Kevin Carlson Sep 27 '22 at 15:41
  • Kindly help with the symmetries shown in terms of the four long diagonals as at: https://math.stackexchange.com/q/4582458/424260. – jiten Nov 22 '22 at 09:41
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Visualization works better if you draw the most important things you are trying to visualize -- in this case, the axis through the midpoints of edges $AF$ and $CH.$

The following links to an interactive example where the midpoints of those two edges have been labeled $I$ and $J$ and there is an axis $IJ$ passing through those points. (The labeling of the vertices of the cube is the same as yours although the initial orientation is not.)

https://www.geogebra.org/3d/dg73wzzw

The remaining four vertices form the vertices of the rectangle $DEGB,$ also shown in that example. The axis $IJ$ is perpendicular to the rectangle $DEGB$ and passes through its center. Therefore a $180$-degree rotation around that axis rotates the rectangle $180$ degrees. You should be able to figure out what that does.

You could try replicating this construction in the style of the cube you already drew, but you should choose a perspective that does not project all the vertices $D,$ $E,$ $G,$ and $B$ onto the same line on the two-dimensional drawing. That makes it very hard to draw the rectangle or the diagonals between those points.

David K
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