- #1
Zafa Pi
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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?
For the regular tetrahedron the dihedral angle is θ° = arccos⅓, which is irrational.Buzz Bloom said: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
Hi Zafa:Zafa Pi said:Equations good, give me some.
LOLBuzz Bloom said: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
Yes, a tetrahedron can have all dihedral angles rational. In fact, there are infinitely many tetrahedra that have all rational dihedral angles.
A rational dihedral angle is one that can be expressed as a ratio of two integers. For example, an angle of 90 degrees is rational because it can be written as 90/1.
No, there are no restrictions on the size of a tetrahedron with all rational dihedral angles. It can be any size, as long as all the angles are rational.
This is proven through a theorem called the Pythagorean-hodograph (PH) theorem. It states that for any given set of rational numbers, there exists a tetrahedron with those numbers as its dihedral angles. Since there are infinitely many rational numbers, there are also infinitely many tetrahedra with all rational dihedral angles.
Yes, a tetrahedron can have all dihedral angles irrational. In fact, the majority of tetrahedra have at least one irrational dihedral angle. It is much more rare for a tetrahedron to have all rational dihedral angles.