If the fan is sitting on one corner of the desk or the other corner, you'd say the fan is ON the desk.
http://i.ebayimg.com/00/s/NDM2WDI2Nw==/z/EisAAMXQVERSsTH0/$_35.JPG?set_id=2
To be ABOVE the desk or OVER the desk, it would have to be mounted on the wall and above the surface of the desk or suspended from the ceiling.
1. The fan is above the desk.
(Can we use 'above' any time when the fan is on the ceiling, for example, in each corner. When the fan is directly over the desk, can we use "over" as in the sentence? Can we use 'above' when the fan is in each corner of the ceiling.)
2. The fan is over the desk.
(Do we use 'over' when the fan is directly above the desk verticallty? When the fan is in each corner, can't we use 'over? If the desk is in the midder of the room, the fan is not directly above the desk.)
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