Can you visualize this slightly more difficult 3D rotation?
The last one I gave involved a cube. This one uses a tetrahedron.
Imagine the tetrahedron with one triangular face at the bottom. We are going to connect the midpoints of two of the edges. Choose as one edge, one of the edges not in the bottom triangle. Choose as the other edge the only one that does not intersect the first one, which is located in the bottom triangle. The rotation axis is the line through the midpoints of the two edges.
Can you see that a 180 degree rotation leaves the tetrahedron occupying the same space? I can’t see it yet. I can see that the two edges attached to the rotation axis end up on top of themselves in the opposite direction. Where does the bottom face go? I wish there was an animation for this. I will have to see if I can construct a tetrahedron.
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I see it ending up in the same space.
As to the location of the bottom triangle, imagine that the bottom edge is facing you frontally, and the ascending edge behind it, looking like it intersects the bottom edge in the middle orthogonically. You see a triangle facing you. When you rotate it 180°, that triangle will move to the bottom, while the bottom triangle will move to be in the same place as you saw the triangle facing you.
@loli, I can almost see it. Darn, you are good at this.
@LostInParadise
My day job is in CG, so I have a little bit of an advantage.
No, you lost me at the rotation.
And, picking up a handy tetrahedron (which you didn’t mention you probably mean is an equilateral symmetrical tetrahedron, not any tetrahedron), and rotating it, I don’t think it is correct. Even the orientation does NOT end up the same.
You don’t specify direction of perspective, either, but after the rotation, the axis of rotation is now in the rear relative to my point of perspective.
That is, if you consider the two edges attached to the rotation axis, they have one point in common, which starts pointed roughly toward my perspective. After rotation, that point points away from me. Since the axis of rotation is NOT the axis of rotation, the tetrahedron does NOT occupy the same space. I think that more or less proves that the problem description’s “same space” claim in false.
At first I wasn’t sure that the orientation would also change, but it does.
I can visualize it better (and see it better when looking at the actual tetrahedron) if I choose a perspective that is looking directly at one face of the tetrahedron, so it appears as a triangle. Pick an axis of rotation between the midpoint of two sides. That axis divides the triangle into a smaller triangle, and a trapezoid. If you rotate that face around 180 degrees, the triangle and trapezoid will be on opposite sides, and the rest of the tetrahedron will now point TOWARD you, so it’s not the same orientation nor is it in the same space.
the line connecting the midpoint of the bottom left edge, and midpoint the top right edge.
Oh, I see. I seem to have mis-read the intended edges somehow.
That, of course, is an axis of symmetry, and if you look at the rotation from along that axis, the rotation is very easy to visualize, since you’ll see two triangles joined along one side, with the axis of rotation in the middle of it, and the rotation just trades places between the triangles from that perspective.
It helps to realize that if an edge turns 180 degrees and covers itself in the opposite direction then the two faces that share the edge are gong to exchange places.
@ragingloli Oh @ragingloli you are incredible!!! Thank you.
I could not wrap my head around it without your help!
I must admit to drinking 400 ml of sake with my sushi before attempting this.
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