In formal writing, when more than two things are being discussed, ‘one another’ is often preferred. But, in fact, according to Fowler's Modern English Usage, either phrase can be used. They are interchangeable and the choice becomes one of personal taste.
‘Penny and Prudence were always arguing, getting at each other by any means possible so that their friendship was constantly in doubt.’
‘Fred and Josh were the best of friends and would lend one another any gadget or item of clothing without hesitation.’
The picture is nothing to do with the words. But my widget failed to identify anything relevant, so I thought |I'd give you this picture of Yorkshire instead.