The answer is very simple. As described in the accepted answer to the related question about alpha-subunits vs alpha-helices the alpha- and beta- are arbitrary names. It could easily have been 1 and 2 or A and B based on ordering of letters or numbers; indeed, there are "type I" and "type II" turns.
The history, from the wikipedia page on alpha-helices says:
In the early 1930s, ... William Astbury initially proposed a kinked-chain structure for the fibers. He later joined other researchers (notably the American chemist Maurice Huggins) in proposing that:
- The unstretched protein molecules formed a helix (which he called the α-form)
- The stretching caused the helix to uncoil, forming an extended state (which he called the β-form).
(Emphasis added). In other words, he named the "α-form" first, and considered the "β-form" to be an uncoiled form of the first one. These names later became "α-helix" and "β-sheet". It could easily have been the other way around.
Indeed there are other types of helix; the 310 helix and the π-helix. If either of these had been discovered first, perhaps they would be the alpha-helix. Of them all, the 310 helix has the most 'logical' name as:
the helix has three residues per turn, and ... has 10 atoms in the ring formed by making the hydrogen bond
so if we followed a similar naming scheme, the alpha helix would be a 3.613 helix. Admittedly this is a bit of a cumbersome name.
As well there are the alpha sheet - a sheet made of helical strands - and the beta helix - a helix made of strands. For both of these, the "alpha" and "beta" part of the name has been chosen because of the naming of the original structures they are related to.