Visualising the order $2$ rotational symmetry of the cube, about the midpoints of opposite edges.

Question

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 :

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.

Answer 1

It looks like this: ${}{}{}{}{}{}{}{}{}$

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

Answer 2

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.