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 ... |โžš> = ฮฑ|โ†‘> + ฮฒ|โ†“> ... built from an up |โ†‘> state and a down state |โ†“>. ฮฑ and ฮฒ are complex numbers. The result |โžš> 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

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 Itโ€™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

My Elliptical Cone

I've still been thinking about this elliptical cone! It has been the main character in my geometric proof on stereographic projection mapping circles to circles. The idea has been to reduce a three-dimensional problem to a two-dimensional one, by noting that something has to be symmetric. A circle on a sphere is mapped to some … Continue reading My Elliptical Cone

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