I've noticed that some surprising properties of negatively curved (hyperbolic) space also hold in flat Hilbert space, and higher dimensional flat spaces in general.
Q: Are are these similarities just coincidence, or is there more of a connection between these spaces?
I'll refer to Hilbert (and high dimensional) flat space as HILL, and negatively curved low dimensional (mainly 2D) space as HYP.
As the amount of curvature below 0 is equivalent to the size of a figure drawn in that space, I'll phrase statements as being about sizes/distances instead of curvature.
Examples:
Take a regular tree in HILL with 4 equidistant angles between branches and equal branch lengths. If the branching cycles between turning 90 degrees in different coordinates for successive branching points (X, Y, Z, W, …), then the tree should fit correctly and maintain the same distance between all branches the same number of connections away from any vertex.
Consider if this tree is instead situated in HYP, with N (at least 3) branches per vertex and branch lengths of M. Provided M is large enough given N then the branches end up spaced out like in the HILL example. In this case at least it's so much that it's possible to move (translate) in the space away from the entire tree to arbiary large areas of open space that have a rapidly vanishing range of angles leading back to the tree. There is even space to house a copy of the whole tree in every one of these inter-branch spaces, and to do this recursively.
Endless random walks in HILL are likely to never return to near the origin.
Such walks in HYP share this property.
There are infinite independent straight lines in HILL.
Similarly straight lines in HYP become increasingly independent with distance.