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TWRoO said:
Isn't it actually 0.01, or perhaps 0.001 or something similar

Pure water at sea level can be solid/liquid or gaseous at 0.01 degrees Celcius/Centigrade... It is the triple point of water because it can be in all three of the main states.

Although I suppose if 0.01 is the tripple point then the freezing point would be 0 because if it is cooling down it would be water at 0.01, and ice at 0.
Wheras the melting point might be 0.02 or whatever.
(I may have made the last 3 lines up, I was just thinking it through)

You were almost right the triple point of water only happens at 0.01 degrees Celcius AND in an enviroment with pressure about 0.006 Atm. The triple point means that water exist in 3 forms at same time: solid, liquid and vapor.

Water can also pass from solid to vapor directly.