I recently learned, through visual intuition, how the relative perception of time between two subjects changes as relative speed between them changes. It's because they are observing each other from an "angle" in the time dimension. And in that time dimension, angles do not trace circles, they trace paraboloids.

If I'm remembering correctly, this is also why the energy required to "reach" the speed of light for subjects with mass parabolically goes to infinity. I'm also guessing it can directly trace a proof down to why kinetic energy increases quadratically.