I saw a few minutes of a documentary that mentioned the domed roof of the largest mosque in Istanbul is supported by the strength of sets of three arches assembled so they share a common pillar. I'd like to learn more about that building technique but I'm not sure what the name is for that kind of structure. Also, can four stone arches share a central column or would the shared column likely collapse?
The best answer I can think of is a form of “Flying Buttress” (FB).
A flying buttress is one half of a catenary arch. Therefore, the dome you reference would consist of six FB’s aligned in pairs. A FB receives upper end reaction supports from one or more support/supported structure(s). In the case of the Mosque, vertical reactions for all six FB’s are resisted by the column. It can be shown by simple vector analysis that horizontal reactions due to gravity cancel at the peak of the dome. A special connection detail is required at the top of the arches to facilitate horizontal force interaction & transfer. If our FB’s will also support the infill, the central column must support about the same gravity load regardless of the number of FB’s. Pictures of variations of Flying Buttresses are easy to find on the internet.