Tetrachromats also have a different concept of white. White is what we perceive when all our cones are equally stimulated. But you’d need a different blend of wavelengths to excite a bird’s quartet of cones than you would a human’s trio. Paper is treated with dyes that happen to absorb UV, so it wouldn’t look white to a bird. Many supposedly “white” bird feathers reflect UV and wouldn’t necessarily look white to birds, either.
