Boosted

I have been playing with the geometry of special relativity again! The light cone signifies the invariance of the speed of light. There is a notion of length in four-dimensional spacetime, defined as c2t2 – x2 – y2 – z2. Surfaces of constant length are 4-dimensional hyperboloids. Light rays are null rays, as light travels…

Spins, Rotations, and the Beauty of Complex Numbers

This is a simple quantum state … |➚> = α|↑> + β|↓> … built from an up |↑> state and a down state |↓>. α and β are complex numbers. The result |➚> is in the middle, oblique. The oblique state is a superposition or the up and down base states. Making a measurement, you…

Lines and Circles

I poked at complex function 1/z, and its real and imaginary parts look like magical towers. When you look at these towers from above or below, you see sections of perfect circles. This is hinting at some underlying simplicity. Using the map 1/z, another complex number – w=1/z – is mapped to z. Four dimensions…