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LostInParadise's avatar

Can you visualize this simple rotation in 3 dimensions?

Asked by LostInParadise (28497points) 1 month ago

Imagine a line drawn through the middle of the top left edge of a cube and the middle of the bottom right edge. Now rotate the cube180 degrees around the line. Do you see that the cube ends up occupying the same space? I had read this and had the hardest time visualizing it. I used a Rubik’s cube to finally figure it out. Where do the top and bottom faces of the cube end up?

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6 Answers

ragingloli's avatar

If your view is frontal, I would imagine the bottom plane is now in the bottom right quadrant, the top plane in the upper left.

ragingloli's avatar

Oh, I misread it. I only rotated it 90°. If you rotate it 180, the bottom is now on the right, and the top on the left.

LostInParadise's avatar

Correct! Good visualizing.

Zaku's avatar

It assumes that the cube starts out facing us oriented in a “regular” way. The most unclear part for me was getting what you meant by “the middle of the top left edge”, which is only clear if you get that assumption.

After that, it was pretty easy to imagine, but what a line drawing of it would look like as it rotated wasn’t as clear to me.

It was clear that it would occupy the same space but I figured the top and bottom would have switched positions… it wasn’t until I looked at a physical cube (though I did not have to turn it physically) before I got that the top would end up on the left, and the bottom on the right, from my assumed perspective.

flutherother's avatar

The way I visualise it if the rotation is around this imaginary line the top left edge will still be the top left edge and the bottom right is still at the bottom right. However, the front and rear faces will change places. The line doesn’t move so the points of intersection with the edges won’t move either.

LostInParadise's avatar

What I found helpful was to imagine that the rotation axis is perpendicular to the ground. The left top edge is now at the very top with the top face to its right and the left face to its left.

Rotating the cube causes the left top edge to turn 180 degrees and end up where it started but in the reverse direction, causing the top and left faces switch positions.

The relationship between the right bottom edge, bottom face and right face is analogous. The right and bottom faces will swap positions,

Also, the front and back of the cube swap positions.

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