I think the main reason is that mirrors are smooth so reflect light at nice angles. Towels are bumpy and rough and when they reflect light, it gets diffuse and goes in all sorts of directions.
Great question, like Dan said, its because the towel is bumpy so the light is reflected in all directions from all parts of the towel whereas the mirror reflects it straight back.
It’s true that light rays bounce off both white objects and mirrors. However the way they do this is different. Mirrors tend to have very smooth surfaces and reflect light in a certain way such that the angle of incidence is the same as the angle of reflection. The light rays bounce off in a structured way. A white towel for instance doesn’t absorb light rays either but there is no structure to how the light rays are bounced off: they scatter in any which way.
It’s the smoothness that is important – you can put a thin layer of glass on top of black surface and it will be reflective of some of the light. Even white paper is bumpy at a microscopic level so you can’t see your face in it.
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