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My grandmother used to read with a magnifying glass. What (an ideal) magnifying glass does, is basically a homothety: it scales the picture by some factor. Now, in a hyperbolic space there is no such thing as homothety. So, what a person living in a hyperbolic space would do to improve poor vision?

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  • $\begingroup$ They would move out of hyperbolic space into the ambient euclidean space where magnifying glasses scale things properly. $\endgroup$ – user21820 Mar 20 at 5:46
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What you say will still be true: a magnifying glass will still scale the picture by some factor. Let us say that the scale factor is $\ell > 1$.

The difference will be that the scaled picture will no longer be a picture of the old familiar hyperbolic space in which the sectional curvature is $-1$. Instead, it will be a picture of hyperbolic space with curvature $-\frac{1}{\ell^2}$; I'm using here that the units of curvature are basically $1/\text{(length)}^2$.

So, for example, a really powerful magnifying glass with scale factor $\ell >\!\!> 1$ will present a picture of a hyperbolic space whose curvature is nearly zero, being pretty much indistinguishable from Euclidean space.

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Even though a magnifying glass appears to scale the plane picture you're looking at uniformly, the actual image that forms on your retina lives on (the inside of) a sphere. So that actual image cannot actually be scaled uniformly.

Really, our immediate visual sensations lives not in our 3D space, but in the space of directions emanating from our eye. And that space is a sphere no matter whether the eye itself is embedded in hyperbolic or Euclidean space. More precisely, a sphere in hyperbolic space is always isometric to a sphere in Euclidean space; they just embed differently.

So a hyperbolic creature's retina might well have the same intrinsic geometry as ours does, and so it is completely conceivable that his magnifying glass might transform his visual sensation in the same (imperfect) way that our magnifying glasses do for our eyes.

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