Is learning the opening 1. c4 e5 as white helpful in learning the opening 1. e4 c5? How about vice versa?
Are the ideas behind the Sicilian Defence as black very similar to the ideas behind the English Opening as white?
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Short answer: not too much
[disclaimer: The more openings you know the more basic understanding of chess you have -- in which case the answer would be "yes it helps you"]
Long answer: similar ideas but very different concrete variations
You would think that because of the similar pawn structures the English and Sicilian sometimes share, that knowledge in one would assist learning the other. In practice though the concrete lines are completely different.
Sure some ideas are the same like in a closed Sicilian Black wants to play b5-b4 and in the English White does b4-b5; that certainly helps. But open Sicilian is a sharp fight and the English equivalent structure results in very different play.
I would say the majority of answers to the English are nothing like the Sicilian defense so if you are thinking to save time by tackling learning these at the same time I don't think you will have much luck.
I'd say learning the Sicilian is graet for playing the English. Ideas from the Closed Sicilian, King's Indian attack and many others appear all the time in the English with reversed colors.
However, I don't think it works the other way around, as the main lines in the Sicilian come form the opened variation, where the extra tempo is often critical and therefore the ideas played change way too much.
This is quite a deep question that could be the subject of an interesting book. Without trying to go into depth, I think the main difference is that in the open Sicilian, White often feels encouraged by the non-developing nature of ..c5 to start a fierce attack with Nf3 and d4. In some lines (eg Najdorf) Black even provokes this, believing that it will misfire.
After the English sequence 1. c4 e5 if Black tries to mount an attack the lack of a tempo means that the attack is less likely to succeed, and more likely to misfire, so Black frequently avoids sharp play and plays more on the lines of the closed Sicilian with .. Nc6 and either ..Bb4 or ..Bg7. What this means is that knowledge from the English can be useful in the Sicilian and vice versa, but only if the lines correspond and the players motivations also correspond. This happens less often than you might expect.
Somewhat helpful but for the most part no. The fact that White gets an extra move in the English causes the types of positions to vary significantly. Mainly because Black is more hesitant to engage in the sharper systems being a tempo down.
Sir, preferably English opening may improve Sicilian Dragon and vice versa as both will have finachetto of Bishop with almost similar move orders from both colors.