Can a tetrahedron have all dihedral angles rational?

[Zafa Pi]

At each edge of a tetrahedron the 2 common faces form a dihedral angle. Can each of these 6 angles be rational multiples of pi?

 

[Buzz Bloom]

Hi Zafa:

I would assume that it would be useful to see the equations relating the six dihedral angles to the five degrees of freedom in establishing the geometry of a tetrahedron. Have you tried to develop these equations?

Regards,
Buzz

 

[Zafa Pi]

Hi Zafa:

I would assume that it would be useful to see the equations relating the six dihedral angles to the five degrees of freedom in establishing the geometry of a tetrahedron. Have you tried to develop these equations?

Regards,
Buzz

For the regular tetrahedron the dihedral angle is θ° = arccos⅓, which is irrational.
Equations good, give me some.

 

[Buzz Bloom]

Equations good, give me some.

Hi Zafa:

Developing these equations is not trivial. I estimate it would take me quite a few hours to do this, and I am not sufficiently interested in the problem to do it. I gather from your comment that at the present time you do not yet have the math skills to do it yourself. I think you will need some high school level algebra, some trigonometry, and perhaps also some solid geometry. So, sometime in the future you will likely be able to develop the equations part of your problem yourself. From these, you should then be able to also work out the rest of the problem as well.

I wish you good luck, and also the patience to wait if you cannot find someone to teach you the math skills you need.

Regards,
Buzz

 

[Zafa Pi]

Hi Zafa:

Developing these equations is not trivial. I estimate it would take me quite a few hours to do this, and I am not sufficiently interested in the problem to do it. I gather from your comment that at the present time you do not yet have the math skills to do it yourself. I think you will need some high school level algebra, some trigonometry, and perhaps also some solid geometry. So, sometime in the future you will likely be able to develop the equations part of your problem yourself. From these, you should then be able to also work out the rest of the problem as well.

I wish you good luck, and also the patience to wait if you cannot find someone to teach you the math skills you need.

Regards,
Buzz

LOL