"Hyperbolic space has a lot more space in it than flat space. The circumference of a circle grows linearly with the radius in flat space, but exponentially in h

"Hyperbolic space has a lot more space in it than flat space. The circumference of a circle grows linearly with the radius in flat space, but exponentially in hyperbolic space. There's a huge amount of room even just a short distance away from some given point." So why can't the effect be simulated in Euclidean space by just using a bigger circle?