We studied electromagnetic wave propagation in a system that is periodic in both space and time, namely a discrete 2D transmission line (TL) with capacitors modulated in tandem externally. Kirchhoff's laws lead to an eigenvalue equation whose solutions yield a band structure (BS) for the circular frequency ω as function of the phase advances kxa and kya in the plane of the TL. The surfaces ω(kxa, kya) display exotic behavior like forbidden ω bands, forbidden k bands, both, or neither. Certain critical combinations of the modulation strength mc and the modulation frequency Ω mark transitions from ω stopbands to forbidden k bands, corresponding to phase transitions from no propagation to propagation of waves. Such behavior is found invariably at the high symmetry X and M points of the spatial Brillouin zone (BZ) and at the boundary ω = (1/2)Ω of the temporal BZ. At such boundaries the ω(kxa, kya) surfaces in neighboring BZs assume conical forms that just touch, resembling a South American toy ``diábolo''; the point of contact is thus called a ``diabolic point''. Our investigation reveals interesting interplay among geometry, critical points, and phase transitions.
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