Satisfying Maths

With thannks to Zirguezi

Homology cycles on a torus.

The eight solutions of Apollonius' problem.  The three given circles are shown in black. The eight solution circles are shown as four pairs (blue, magenta, yellow and grey).  In each pair, one solution circle encloses the given circles that are excluded by the other solution, and vice versa.  For example, the larger blue solution encloses the two larger given circles, but excludes the smallest; the smaller blue solution does the reverse.

Black hole bomb!

Tracking the modes

Dropping particles from a height above a rotating compact object with no event horizon. 

A geodesic of an almost swallowed particle near a rotating black hole

Stamping on top of the entire hole: This is actually a dispersive black hole bomb

A negative energy wave!

Modes freezing as they cross the cosmological horizon

Workflow 

Wavepacket of photons inside a material undergoing a non-adiabatic refractive index variation induced from a second, intense optical beam. 

A non-local initial condition spanning the entire ergo-region of a rotating black hole smoothly occupies a family of dispersion curves. 

Modification of the refractive index due to the non-linear Kerr effect inside a piece of glass as an intense laser pulse propagates through it. 

Epsilon Near Zero makes for interesting quantum predictions. Metamaterials can have near zero permittivity, making waves move extremely fast. The natural frequency of a time varying epsilon, near zero, gets smeared out, indicating a broad spectrum of quantum produced photons from the vacuum.