What is a good and easy way to fold a (50-100') rope for storage so that it resembles a caterpillar and has no loose ends getting undone? Close to the way it comes folded in a store like this at the Home Depot.
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As mentioned in comments, this option to coil a rope will get you some twists in it. So I do not recommend it for longer ropes (i.e. your climbing ropes), as twists are very inconvenient when belaying. This mode is mainly suited for cordelettes that you want to attach to your harness and therefore should be compact. To create such a coil follow these steps:
Wrap the rest tightly starting at one end making your way towards the other. I usually do not do this over the whole length, I just start in the middle as for my taste it still gets compact enough (I am lazy).
To secure the ending there are two options:
a) Leave only very little space at the end and then simply pull the cord through and clip this loop to a biner - fast and does not get loose as long as it is on the biner.
b) Use just enough rope to pull it through the end and then fold it back over the whole coil. This is neat even without a biner, but is somewhat more tedious to do and has no obvious attachment loop, so it is more suited for storage.
For long and/or big diameter ropes I recommend to use some variant of the butterfly coil (just google it, there are lots of instructions available). This is much faster to do and does give no twists, does not get loose and is great for transport.