Imagine you are an alien. You follow beams of diffracted light. A distant celestial object exuding white light. Thousands of Kelvins of blackbody radiation hitting thousands of fine wires. You see those pure spectral colors. You feel their mathematically exact intensities. You squint, and you tune in on – what we call – Red. We…

# Tag: Mathematics

## Stargate of Diffraction. Escaping the Labyrinth of Colorful Wires.

This blog has been many things. Experimental poetry playground, notes from the field by a small business owner, popular science and tech blog, compilation of dry research reports, self-referential musing of a web developer. The only intersection between all of this has been me being interested & invested in it. I had tech-blogged myself into…

## The Origin of Darkness

Where does it come from? Why and when did my physics/math/engineering art decide to only dwell on black backgrounds? I am sure there are proper theories in the history of art. I can come up with some armchair psychologist’s explanations – blaming popular culture, clichés, a collage of archetypical memories. I am hacker. We hackers…

## No Loose Ends

I don’t like loose ends. But my physics/math art curves have often been cut off, like fraying fabric. I have tried to let them taper off smoothly, letting them fade in the dark, but I would prefer closed lines. The real and imaginary towers have flat roofs. When using software, I tilt the structures, so…

## Imperfect Projections

___________________ _______ __ _ _ __ _______ ___________________ Imperfect Projections dissolve the inherent rectangularity dedifferentiating the materials used dependent on the subjective attitudes create interventions change the landscape of the city with a highlight of the spatial dimension a broad utility as a datum for the objective consideration engineers who are responsible resolve multiple objectives…

## They Shall Shine in the Dark

On February 24 I wrote: They shall shine in the dark. I use this sentence as a seed for Twitter search poetry – interlacing poetry with temperature waves again . ~ ~~ ~~~ ~~~~ They shall shine in the dark The interstellar void going down much faster determined to take over under renewed attack under…

## Loops Near the surface. Lumped Together in Space.

There was a time, when most articles here looked like lab reports or chapters of a thesis. Occasionally, there was a weird poem thrown in. Now is the time for art only, and the thesis-like postings provide for raw material. Temperature waves beneath the ground, driven by the oscillation of the temperature on the surface…

## True Colors

I have picked two colors. They shall shine in the dark. ~~~ Another version of my series Reality and Imagination. I used artistic license when tweaking the color values.

## Jellyfish of Diffraction

Diffraction patterns, again. But this time I tack them to an imaginary semi-circular screen. Screens grow bigger in radius with increasing wavelength – growing more reddish. If every wavelength would be diffracted in the same way, all peaks would lie on a radius of the circle. But as red is diffracted more, maxima move to…

## Creative Process. Evolution.

My creative process has been evolving gradually in the past year. ~ I am thinking about a little piece of physics, and how it is described with math. ~ Then I am creating a SageMath notebook (plus custom code) that outputs a set of functions as parametric curves in a three-dimensional space. Here, diffraction patterns…

## Newton’s Space Probes Investigate my Ribbons of Diffraction

I have been calculating diffraction patterns for visible light. Curves are displaced to turn the whole structure into a wavy ribbon built from colored wires or threads. I have turned these images into collages, adding Isaac Newton’s drawings from Opticks (1704). The more I moved Newton’s figures around, and the more I twisted the ribbons…

## Transforming the Celestial Sphere

A spherical spaceship swooshes by at 99% of the speed of light. What will it look like? Squashed because of Lorentz contraction – like an ellipsoid? No. The outline of a moving sphere will remain spherical. Roger Penrose explained this first in 1958 – 50 years after Einstein’s formulation of the theory of special relativity….

## 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…

## 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,…

## 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…

## 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…

## 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…

## 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…

## 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…

## 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…

## 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…

## Reality and Imagination

Grey and colorful. Cutting through each other. Chasing each other. Meeting in the center, leaning on each other, forming an infinite line. ~ Reality and Imagination: Real and imaginary part of complex function 1/z: ~ The real part of 1/z is painted in shades of grey, the imaginary part in rainbow colors. Plots are created…

## 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…

## Super Motivational Function

I’ve presented a Motivational Function, a while back. It is infinitely flat at the zero point: all its derivatives are zero there. Yet, it manages to lift its head – as it is not analytic at zero! If you think of it as a function of a complex argument, its weirdness becomes more obvious. Turn…

## Dirac’s Belt Trick

Is classical physics boring? In his preface to Volume 1 of The Feynman Lectures on Physics, Richard Feynman worries about students’ enthusiasm: … They have heard a lot about how interesting and exciting physics is—the theory of relativity, quantum mechanics, and other modern ideas. By the end of two years of our previous course, many…

## Motivational Function

Deadly mutants are after us. What can give us hope? This innocuous-looking function is a sublime light in the dark. It proves you can always recover. If your perseverance is infinite. As x tends to zero, the exponent tends to minus infinity. The function’s value at zero tends to zero. It is a zero value…

## Gödel’s Proof

Gödel’s proof is the (meta-)mathematical counterpart of the paradoxical statement This sentence is false. In his epic 1979 debut book Gödel, Escher, Bach Douglas Hofstadter intertwines computer science, math, art, biology with a simplified version of the proof. In 2007 he revisits these ideas in I Am a Strange Loop. Hofstadter writes: … at age…

## Integrating The Delta Function (Again and Again) – Penrose Version

I quoted Nobel prize winner Paul Dirac’s book, now I will quote this year’s physics Nobel prize winner Roger Penrose. In his book The Road to Reality Penrose discusses not-so-well-behaved functions like the Delta Function: They belong in the category of Hyperfunctions. A Hyperfunction is the difference of two complex functions: Each of the complex…

## The RSA Algorithm

You want this: Encrypt a message to somebody else – using information that is publicly available. Somebody else should then be able to decrypt the message, using only information they have; nobody else should be able to read this information. The public key cryptography algorithm RSA does achieve this. This article is my way of…

## Integrating the Delta Function (Again) – Dirac Version

The Delta Function is, roughly speaking, shaped like an infinitely tall and infinitely thin needle. It’s discovery – or invention – is commonly attributed to Paul Dirac[*]. Dirac needed a function like this to work with integrals that are common on quantum mechanics, a generalization of a matrix that has 1’s in the diagonal and…