Still photography and motion pictures have, over history, used aspect ratios such as 4:3, 3:2, or for printing 4×5, 8×10 and what not.
These choices were arbitrary – based on practical design and implementation considerations of the time.
The popular 8×10 paper size came from how fine paper was originally manufactured and sliced down to size by hand, in Dutch paper mills and corresponded to the equipment size readily handled by the length of the arms of the mill workers. These cut 8×10 sheets were later cut to create 8×5 sheets, which in turn were sliced to 4×5 sheets. (I could not verify these claims independently but could not dispute them either. Of interest, the 8 1/2 by 11 inch sheet of paper we take for granted also seems to have come out of similar issues and stuck with us because of practical issues regarding manual typewriters, issues that no longer exist today.)
The 35mm standard came from early still photo film which happened to be 70mm wide, but was split down the middle by Thomas Edison to save money for making a movie film. After adding holes along side the film for pulling the film through their movie camera, the image area became 24mm wide measured across the film. Each image was limited to 18mm in the length direction – becoming a 24 x 18mm or 4:3 aspect ratio image.
This film was then adopted for new still cameras (Leica) which chose to double the 18mm to 36mm, hence 24mm by 36mm (the well known 35mm format) in a 2:3 (or 3:2) aspect ratio. The 1:1 ratio photo came from waist-level viewfinder cameras – since it was not easy to turn the camera sideways, they chose a 1:1 ratio.
The result is that today’s modern digital camera and print aspect ratios are arbitrary and based on design choices that occurred out of practical considerations in the 19th century and the early 20th century.
And then there is the 16:9 aspect ratio of HD, which is the compromise that came out of a committee that wanted to create a new TV standard to deal well with older 4:3 content and wide screen content which is wider than 16:9. Basically, an arbitrary compromise value.
There is also similar information on how did we end up with audio reel-to-reel tape recording at 7 1/2 inches per second? I was told it was because this was the speed at which 16mm film, with an optical soundtrack on the film, operated. I could not quickly verify if this was true though and could only work out that 16mm film seemed to go through at 7.1″ inches per second at 24 fps.