Both ice and TiO2 transmit light well in the visible region, well at least a few mm. What is also common is that they reflect light by having a refractive index different to air.
Having a range of particle sizes in varying orientations gives the diffuse reflectance we get to perceive a white surface.
Ice is transparent in large chunks but very small crystals are not. It is the same difference as between big crystals of sugar or transparent sugar glass mints and caster sugar. One is basically determined by the bulk properties of the material and the other by the refractive index difference at the copious air to material boundaries.
TiO2 works so well as a white pigment because it has one of the highest refractive indexes of relatively cheap materials (2.7/2.55).
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The same mechanism by which white flowers are white. And it is mostly because light is reflected by the refractive index discontinuity at the enormous surface area of the many small fractal crystals. That is why fresh powder snow is whiter than wet snow refrozen.
Put another way bulk ice in ice cubes is water clear but the frost in your freezer compartment is white. The difference is entirely due to the smaller size of crystals and their resulting boundary conditions.
It is to a pretty good approximation once you are under a foot or so. A TiO2 paint film can be made opaque within a few um.
Price largely determines why cables are made black. Carbon black is one of the cheapest pigments known since antiquity as lamp black (aka soot).
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