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Given a regular tetrahedron, make a uniformly scaled copy of it, such that both the original and the scaled copy (the scale factor is less than $1$) share the same centroid, and have the same orientation. Then rotate the scaled copy such that its four vertices lie on the four faces of the original tetrahedron.

Suppose the scale factor is specified, what would be involved in finding the rotation matrix that when applied to the scaled down tetrahedron gives the desired result of inscribing it in the original tetrahedron?

Has this problem been studied? Any citations, pointers, or hints are greatly appreciated.

EDIT: The range for the scale factor is $ \dfrac{1}{3} \le f \lt 1 $

Hosam Hajeer
  • 21,978
  • Isn't it a rotation about the $z$ axis by $\pi/3$, then a rotation about any horizontal line by $\pi$? – David G. Stork Feb 23 '22 at 04:59
  • This would be the case only if the scale factor is $\dfrac{1}{3}$ – Hosam Hajeer Feb 23 '22 at 05:26
  • No, here it is assumed that the scale factor is positive. – Hosam Hajeer Feb 23 '22 at 09:15
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    Rotate the inner tetrahedron around an axis that goes through the midpoints of opposing edges. By symmetry, its vertices will hit the outer tetrahedron faces simultaneously. This reduces the problem to a 2D rotation of a line segment inside a rectangle. You should get something like $\sin^{-1} \frac{1-f}{2f}$ for the angle I think. Note that there are 6 ways to do this (3 axes to choose from, two directions of rotation). I don't know if there are any other rotations that also work. – Jaap Scherphuis Feb 23 '22 at 09:23
  • Yes. This makes sense. I'll try it. – Hosam Hajeer Feb 23 '22 at 09:36
  • Aren't some scale factors ($<1$) "invalid," in the sense that the scaled tetrahedron simply cannot be placed as specified? For instance, surely a scale factor of $0.0001$ is invalid.... no? – David G. Stork Feb 23 '22 at 17:08
  • Yes. Of course. – Hosam Hajeer Feb 23 '22 at 22:45
  • @DavidG.Stork I've edited my question to explicitly specify the valid range for the scale factor. Thanks for your remark. – Hosam Hajeer Feb 23 '22 at 23:03
  • It would really help if you could plot even a single example of what you're seeking. I'm not convinced you can ever achieve what you seek. – David G. Stork Feb 23 '22 at 23:50
  • I think @JaapScherphuis has solved the problem in his comment above. I have verified that it does indeed result in the scaled/rotated tetrahedron having its four vertices on the four faces of the original tetrahedron. – Hosam Hajeer Feb 24 '22 at 01:26
  • @DavidG.Stork Check the .GIF animation in my solution below. – Hosam Hajeer Feb 24 '22 at 01:30
  • I posted an answer to a related question which may be of interest: https://math.stackexchange.com/questions/4391646/general-means-of-obtention-of-regular-tetrahedra-inscribed-into-a-regular-tetrah/4391802#4391802 – Intelligenti pauca Feb 27 '22 at 10:13
  • @Jaap Scherphuis See this follow up question https://math.stackexchange.com/q/4391646/305862 – Jean Marie Mar 03 '22 at 09:34

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This is not my solution. It is an implementation of the solution outlined by Jaan Scherphuis in his comment above. I've verified the formula he provided and it does work.

Below is an animation that depicts the solution.

enter image description here

Hosam Hajeer
  • 21,978