Is There a Geometric Interpretation for the Multiplication or Power of Angles?

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Discussion Overview

The discussion revolves around the geometric interpretation of the multiplication and powers of angles, exploring whether such interpretations exist and how they might relate to concepts like solid angles and areas on a sphere.

Discussion Character

  • Exploratory
  • Debate/contested
  • Conceptual clarification

Main Points Raised

  • Some participants question the geometric meaning of multiplying two angles, suggesting that it may be akin to scaling since angles lack dimensions.
  • One participant notes that while the sum of angles can be visually represented, the product of angles lacks a clear geometric illustration.
  • Another participant states they have not encountered a scenario where angles are multiplied, implying a lack of geometric interpretation for such operations.
  • Some participants propose that solid angles might relate to the product of plane angles, suggesting an analogy to multiplying lengths to find an area.
  • A specific example is raised regarding the area of the surface of a unit sphere defined by angles θ and φ, which could be approximated by the product θ × φ, but there is a desire for a more exact correlation.
  • One participant encourages further exploration of the exact mathematical formulation for the area of a portion of a sphere in relation to angle products.

Areas of Agreement / Disagreement

Participants express differing views on whether a geometric interpretation exists for the multiplication of angles, with some asserting it does not and others suggesting potential relationships to solid angles and areas. The discussion remains unresolved regarding a definitive geometric interpretation.

Contextual Notes

Participants note the lack of dimensionality in angles and the ambiguity surrounding the multiplication of angles, highlighting the need for precise definitions and mathematical formulations.

Jhenrique
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Exist an geometric interpretation for the multiplication of 2 angles? Or exist an geometric interpretation for the square/cube of an angle?
 
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Your question is rather vague.Please Explain it more.
 
As in: ##\theta_1+\theta_2## would mean that an object underwent two rotations ... so what would ##\theta_1\times\theta_2## mean?

Angles don't have any dimensions - so multiplying angles is the same as scaling them.
 
adjacent said:
Your question is rather vague.Please Explain it more.

If I had asked what geometrically means the sum of 2 angles, you, probably, show me a draw like:

imagem.jpg


But if I ask what means α×β, what draw you show me for illustrate such product?

Simon Bridge said:
Angles don't have any dimensions - so multiplying angles is the same as scaling them.

But exist the solid angles...
 
I have never come across a situation where angles are multiplied together. To the best of my knowledge there is no reason to do this, so there is no geometric interpretation.
 
I was thinking if the solid angle could have some relationship with the product between plane angles...
 
Jhenrique said:
I was thinking if the solid angle could have some relationship with the product between plane angles...
We-ell, by analogy to multiplying two lengths you could argue for an angle-equivalent to an area but I don't know what that would mean.

The area of the surface of a unit sphere inside angles ##\theta## and ##\phi## would be (approximately) ##\theta\times\phi## ... that the sort of thing you are thinking of?

(Here the angles have to be specially defined.)
 
Last edited:
Simon Bridge said:
The area of the surface of a unit sphere inside angles ##\theta## and ##\phi## would be (approximately) ##\theta\times\phi## ... that the sort of thing you are thinking of?

Yeah! But, I was looking for a exact correlation not approximate...
 
Well you could work out the exact version if you like - that's just algebra.
How do you find the area of a bit of a sphere?
 

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