I think that you can get an arbitrarily large disk. The proof is by crochet. Since there's a pattern for crocheting constant negative curvature disks where you increase the radius as you go and since we live in 3-space, it follows that you can get arbitrarily large disks.

See this TED talk for some cool applications of hyperbolic crochet to biology, or this article for a more rigorous explanation.

I think that you can get an arbitrarily large disk. The proof is by crochet. Since there's a pattern for crocheting constant negative curvature disks where you increase the radius as you go and since we live in 3-space, it follows that you can get arbitrarily large disks.

See this TED talk for some cool applications of hyperbolic crochet to biology, or this article for a more rigorous explanation.