The perception of "cold" is due to your nerve cells. If an object is giving heat energy to your skin, you feel warmth. If it is taking away heat energy, you feel cold.

The answer in brief is:
the carpet has more insulation; the tile has greater heat conductance with your foot; and the tile has better conduction of heat away from your foot. Carpet is composed of fibres which trap air, creating I insulating layers. This is the same process whereby clothes (for example a woollen vest) trap your own body heat to warm you just as I told u early in of my facts.

So the carpet is sort of composed partly of fibres and partly of air. Air is much, much easier to heat than stone, tile, or metal. So the same amount of heat energy will cause a greater increase in the temperature of a carpet than of a tile. When you place your bare foot on the floor, the heat is drawn away from your body via thermal conductance, through the points of contact with the floor.
The smoother the surface, the more contact there is, and the greater the transfer of heat. So a smooth tile can draw away more heat than a ruffled carpet can. Also, the transfer of heat will depend on how good the substance is at conducting heat away.

Metal is very good, as it has all these free electrons in it, ready to carry currents and heat. Tile is probably reasonable good, while carpet would be quite bad at conducting heat away from you, because of those insulating layers of air.

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