Here is the first of a set of animated gifs showing basic isometries of hyperbolic 3-space. This one displays a loxodromic transformation. I'll post one gif a day for a bit, but the shader rendering them all is ready for your tinkering! https://www.shadertoy.com/view/MstcWr pic.twitter.com/0JsK08BqjB
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There is a family of concentric beans (I like to call them bananas). If you added a second one in the pic, it would either be hidden inside this one or block it. You could add bananas for any other 2 ideal fixed points, but they'd be associated with a different transform.
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what about a lattice of bananas? or better yet, take a hyperbolic circle packing, and (fibred over a point?) have one banana per circle? Will sketch later. I think this is getting Hopf-fibrationy
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