Rotating a regular tetrahedron so it looks like an egyptian pyramid?

Question

I have been able to easily create a mesh of a regular tetrahedron thanks to this answer:

However, as you can see, it looks like it's sitting on one of its edges, what I was looking for was for it to be orientated like an egyptian pyramid, i.e. sitting on one of its faces:

But as @YiFan cleverly pointed out, a pyramid is a pentahedron ! So my picture showing an egyptian pyramid is in fact wrong.

Back to my tetrahedron, I can correct somehow the orientation manually but it's imprecise, I rotate on X axis by ~35 degrees and on Z axis by 45 degrees.

Question:

How can I rotate a regular tetrahedron so it has the same orientation as an egyptian pyramid, so it looks like sitting on the floor ?

Answer 1

Currently I've found a way on how to solve my problem though I didn't figure out the math involved by myself, rather, a simple idea that ended up working:

A quaternion that rotates from/to and then transform all the points with it.

For the tetrahedron it is:

Quaternion.FromToRotation(Vector3.one, Vector3.up)

While it does work I would have loved to achieve it by myself!

I am posting this solely as a reference and not accepting my own answer, while indeed it solves the problem it doesn't really explain how :)

Thanks for anyone who took the time to help me out!

Answer 2

I don't know what software that is nor how to use it, but your request is impossible since the shape of the Egyptian pyramids are not regular tetrahedra. They have $5$ vertices! That is called a pentahedron, and a pentahedron one of whose faces lie on the $xy$-plane can be given by the coordinates for vertices $(-1,-1,0),(-1,1,0),(1,-1,0),(1,1,0)$ and $(0,0,1)$.

Answer 3

This is not an answer. It is just a demand of clarification with a scientific wording.

Here is an imaginary view of a tetrahedron (resting on one of its faces) and of an egyptian pyramid as seen from an observer at the ground level.

The question of how to rotate "in the best way" these two solids around their vertical axis in order that "they look the same" needs to be precised in the following way.

We have to transform this "best way" into an objective criteria. I propose for example to minimize the gap between the resp. apparent angles BAC and B'A'C', on one side and angles CAD and C'A'D' on the other. Is it a good criteria ? Are there other parameters ?