The first computer I ever worked on had 8KB of core memory. It was an old Digital Equipment Corporation pdp-8/e. I loved that machine and its open face tale drives and teletype with paper tape punch and reader and card reader.
The touching of the cores isn't really important, what matter is which specific wires run through each core vs around/outside each core. That weave pattern defines the addressing scheme and the data stored in each address.
After digging some more, looks like core rope memory is different than the magnetic core memory in your link above (seems to describe magnetic core memory instead of rope memory). Core rope memory is used for ROM while magnetic core memory is used for RAM, and they have different working mechanism.
Another form of core memory called core rope memory provided read-only storage. In this case, the cores, which had more linear magnetic materials, were simply used as transformers; no information was actually stored magnetically within the individual cores. Each bit of the word had one core. Reading the contents of a given memory address generated a pulse of current in a wire corresponding to that address. Each address wire was threaded either through a core to signify a binary [1], or around the outside of that core, to signify a binary [0]. As expected, the cores were much larger physically than those of read-write core memory. This type of memory was exceptionally reliable. An example was the Apollo Guidance Computer used for the NASA Moon landings.
Basically, 1 if the sense wire going through the core, or 0 if the wire bypass the core. You basically woven 1 and 0 manually into the rope. And yeah, it doesn't matter if the cores touch each other as long as the wires are woven correctly. This article have some wiring diagram for the core rope memory: http://www.righto.com/2019/07/software-woven-into-wire-core-rope-and.html .
All this stuff is really cool! Looks like you can use one ring to store multiple bits by using multiple sense wire. So that's why there are a lot of wires in the images. Seems like there are 16 bits for each core?
I wonder how feasible it is for someone to built their own retro computers by soldering a bunch of nand gates and weaving their own ram and rom. The only problem seems to be getting the ferrite cores in huge quantities (are those still being sold these days?).
I totally want to try, I think ferrite toroids are everywhere, but the material that core/rope memory rings are made of has a specific magnetic response hysteresis which is important.