Electrodes and number of pins in vacuum tubes
"If a diode has a cathode, an anode, and a supplementary anode (the control grid or plate) 
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No it doesn't. A diode has a cathode and an anode, so it has two (active) connections. (Additionally, it will probably have one (direct heating) or two (indirect heating) filament connections).

Add a control grid and you have 3 connections, and that thingy is called a triode.