# Magnifying glass in hyperbolic space

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?

• They would move out of hyperbolic space into the ambient euclidean space where magnifying glasses scale things properly. – user21820 Mar 20 at 5:46

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.