How many times does the light reflect?

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SUMMARY

The discussion centers around calculating the number of reflections a light beam makes when projected from a corner of an isosceles triangle before reaching the base. The key insight is that the angle of incidence equals the angle of reflection, which allows for the development of a mathematical model to predict the beam's behavior. The inquiry references a specific geometry problem aimed at a younger audience, emphasizing the relationship between the angle of the beam and the number of reflections. A general formula for calculating the height of the beam based on its angle and the number of reflections is sought.

PREREQUISITES
  • Understanding of basic geometry concepts, particularly isosceles triangles.
  • Familiarity with the law of reflection (angle of incidence equals angle of reflection).
  • Basic algebra for setting up equations and solving for variables.
  • Knowledge of trigonometric functions to relate angles and distances.
NEXT STEPS
  • Research the derivation of the law of reflection in geometric optics.
  • Explore trigonometric identities relevant to angles in triangles.
  • Learn about geometric series and their applications in calculating reflections.
  • Investigate existing mathematical models for light behavior in triangular geometries.
USEFUL FOR

Students studying geometry, educators teaching reflection principles, and anyone interested in mathematical modeling of light behavior in triangular shapes.

Papatom
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My question is inspired by this thread:
https://www.physicsforums.com/threads/can-you-solve-this-geometry-problem-for-nine-year-olds.890794/

If you an isosceles triangle and you put in a corner (or somewhere else) a light beam (laser) the beam will reflect one or more times before it comes down to the base. See picture. Is there a general equation for how many times the light is reflected by the legs? Can you calculate from the angle and how many reflections there are how high the beam comes. It is evident that all beams must come down to the base again. Is there a proof for?

Triangle number 1 is the 9-years old question. If the angle of the laser with the base gets larger, the beam gets more and more reflected until it is reflected downwards again. Points A and C are the special cases were the beam is reflected back to the same corner.

Does anyone knows if there math available to calculate this?
 

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At every reflection, the incoming angle (relative to the surface) is equal to the outgoing angle. That allows to set up a short equation how the angle (relative to the base, for example) changes with every reflection. Based on that it is possible to find a general formula.
 

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