Rin is from the Japanese 凛 — cold, dignified, severe. Rin is one of the Japanese feminine names — central to traditional Japanese aesthetics emphasizing the concept of rin to shita (refined, dignified bearing).
Rin does not currently appear in the US Social Security Administration's top 1,000 girls' names, so we don't publish a US rank or birth count for it. That says nothing about the name's standing elsewhere in the world — only that it sits outside the ranked US data we rely on.
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In Pythagorean numerology the letters of Rin reduce to 5, The Seeker. This is a traditional interpretive system, not a factual claim about the name.