A light beam can use reflections to form a closed, N-sided polygon inside a solid, transparent cylinder

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A light beam can use reflections to form a closed, N-sided polygon inside a solid, transparent cylinder if N is sufficiently large. What is the minimum possible value of N for light inside a cylinder of

(a) Water,

(b) Polystyrene plastic,

(c) Cubic zirconia?

Assume the cylinder is surrounded by air.

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