  Breakthroughs in two-dimensional van der Waals heterostructures have revealed
that twisting creates a moir\'e pattern that quenches the kinetic energy of
electrons, allowing for exotic many-body states. We show that cold-atomic,
trapped ion, and metamaterial systems can emulate the effects of a twist in
many models from one to three dimensions. Further, we demonstrate at larger
angles (and argue at smaller angles) that by considering incommensurate
effects, the magic-angle effect becomes a single-particle quantum phase
transition (including in a model for twisted bilayer graphene in the chiral
limit). We call these models "magic-angle semimetals." Each contains nodes in
the band structure and an incommensurate modulation. At magic-angle
criticality, we report a nonanalytic density of states, flat bands,
multifractal wave functions that Anderson delocalize in momentum space, and an
essentially divergent effective interaction scale. As a particular example, we
discuss how to observe this effect in an ultracold Fermi gas.
