At the atomic level, nature tends to rely on predictable geometric patterns when printing materials with rich functional and electronic behaviours.  In the last 100 years, we have become exceedingly good at determining electronic, optical, and other emergent phenomena of materials from their internal geometries — so good that we can tweak these geometries to tailor those phenomena and even induce novel states of matter, such as topological insulating states. What if we were to disturb these geometries in a much more pronounced way, starting from a completely different geometric playbook, such as hyperbolic geometries instead of the usual flat ones?  Does our predictive power regarding the behaviour of these materials survive?  Can we even find or make such materials?  And if we can, which new technologies might they inspire?  Can they help with quantum computation or error correction?  We will answer some of these questions together.