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From my understanding of our eyesight, we have red, blue, and green cones in our eyes that allow us to see color. If the wavelength of violet is smaller than blue (and green and red also) how can we see it? Shouldn't it be outside of our visible spectrum?

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The cones aren't receptive to just one color, but a spectrum. This spectrum also includes wavelength in the violet region. The colour is just defined by the maximum. See here

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