Complex Alien Eclipse

My colorful complex function lived in a universe of white light. I turned off the light. Turned it into its negative. Expected it to look bleak. Like thin white bones on black canvas, cartoon skeletons of imaginary alien creatures. But it is more like the total solar eclipse I watched in 1999. There is interference, … Continue reading Complex Alien Eclipse

Joys of Geometry

Creating figures with math software does not feel like fabricating illustrations for science posts. It is more of a meditation on geometry. I want to literally draw every line. I am not using grid lines or rendered surfaces. I craft a parametric curve for every line. A curve is set of equations. Yet, playing with … Continue reading Joys of Geometry

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 … Continue reading Spins, Rotations, and the Beauty of Complex Numbers

Galaxies of Diffraction

These - the arrangement of points in the image below - are covectors, sort of. I wrote about them, some time ago. They are entities dual to vectors. Eating vectors, spitting out numbers. Vectors are again 'co' to vectors; they will eat covectors. If vectors live in a space with axes all perpendicular to each … Continue reading Galaxies of Diffraction

Elliptical Poetry

look at these towers Using the map creating a distorted image projected up to the sphere All connecting rays follow this rule Imaginary number i makes an appearance that borders on the poetic It’s nothing more than a whisper construct the proof for yourself something of a dying art avoid thinking about anything there has … Continue reading Elliptical Poetry

Circles to Circles

Using stereographic projection, you create a distorted image of the surface of a sphere, stretched out to cover an infinite plane. Each point on the sphere is mapped to a point in the equatorial plane by a projection ray starting at a pole of the sphere. Draw a circle on the sphere, e.g. by intersecting … Continue reading Circles to Circles

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 … Continue reading Lines and Circles

Vintage Covectors

Covectors in the Dual Space. This sounds like an alien tribe living in a parallel universe hitherto unknown to humans. In this lectures on General Relativity, Prof. Frederic Schuller says: Now comes a much-feared topic: Dual vector space. And it's totally unclear why this is such a feared topic! A vector feels familiar: three numbers … Continue reading Vintage Covectors