Can a tetrahedron have all dihedral angles rational?

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Discussion Overview

The discussion centers on whether all dihedral angles of a tetrahedron can be rational multiples of pi. It explores the geometric relationships and mathematical equations involved in defining the dihedral angles in relation to the tetrahedron's geometry.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested

Main Points Raised

  • One participant questions if all six dihedral angles of a tetrahedron can be rational multiples of pi.
  • Another participant suggests that understanding the equations relating the dihedral angles to the degrees of freedom in a tetrahedron's geometry would be beneficial.
  • A participant notes that for a regular tetrahedron, the dihedral angle is arccos⅓, which is irrational, implying that not all angles can be rational multiples of pi.
  • There is a suggestion that developing the necessary equations to explore this problem is complex and may require certain mathematical skills, which the original poster may not yet possess.

Areas of Agreement / Disagreement

Participants express differing views on the feasibility of having all dihedral angles as rational multiples of pi, with at least one example provided that suggests it may not be possible. The discussion remains unresolved regarding the generality of the claim.

Contextual Notes

The discussion highlights the potential need for specific mathematical knowledge, such as algebra and trigonometry, to fully engage with the problem. There are indications of missing assumptions regarding the conditions under which the angles might be rational.

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?
 
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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
 
Last edited:
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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
For the regular tetrahedron the dihedral angle is θ° = arccos⅓, which is irrational.
Equations good, give me some.
 
Zafa Pi said:
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
 
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Buzz 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
LOL:-p
 
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