Plural Form

Take a figure of a historical patent, meticulously drawn before there was software. It's a Plural Grating Spectrograph. Wonder about the appeal of antique technical drawings. So modest, so sublime. Copy it, multiply it. Slowly rotate the color wheel, until the purples meet the reds again. This is an impossible spectrum. There are no purples … Continue reading Plural Form

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 … Continue reading Loops Near the surface. Lumped Together in Space.

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 … Continue reading Jellyfish of Diffraction

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 … Continue reading Creative Process. Evolution.

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 … Continue reading Newton’s Space Probes Investigate my Ribbons of Diffraction

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. … Continue reading Transforming the Celestial Sphere

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 … Continue reading Boosted

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

Trapped inside the Light Cone

Trapped inside the light cone on a path of eerie fire. Causally connected to your past . . Onward. Celestial spheres embrace the future cone gather round your burning self.

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

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

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 … Continue reading Reality and Imagination

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

Super Motivational Function

I've presented a Motivational Function, a while back. $latex f(z) = e^{\left(-\frac{1}{z^{2}}\right)}&s=3$ 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 … Continue reading Super Motivational Function

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 … Continue reading Dirac’s Belt Trick

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. $latex e^{\left(-\frac{1}{x^{2}}\right)}&s=3$ As x tends to zero, the exponent tends to minus infinity. The function's value at zero tends to zero. It is … Continue reading Motivational Function

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 … Continue reading Gödel’s Proof

Enthalpy

When you move from fundamental principles (in physics)  to calculating something 'useful' (in engineering), you seem to move from energy to enthalpy. Enthalpy is measured in Joule, as well as energy. It is assigned to a 'system', a part of the physical world separated from other parts by interfaces. The canonical example is a vessel … Continue reading Enthalpy

Statistical Independence and Logarithms

In classical mechanics you want to understand the motion of all constituents of a system in detail. The trajectory of each 'particle' can be calculated from the forces between them and initial positions and velocities. In statistical mechanics you try to work out what can still be said about a system even though - or … Continue reading Statistical Independence and Logarithms

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 … Continue reading Integrating The Delta Function (Again and Again) – Penrose Version

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 … Continue reading The RSA Algorithm

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 … Continue reading Integrating the Delta Function (Again) – Dirac Version

Delta Function Haiku

I have proved that a Lorentzian bell curve becomes the Dirac Delta Function in the limit. Now I want to look at another representation of the Delta Function. As this is a shorter proof, a haiku will do. ~ Infinite numbers of oscillations added. Need to damp them down Symmetrically attach an exponential for each … Continue reading Delta Function Haiku

The Improper Function and the Poetry of Proofs

Later the Delta Function was named after their founder. Dirac himself called it an improper function. This time, the poem is not from repurposed snippets of his prose. These are just my own words to describe a proof: ~ In the limit the Lorentzian becomes the improper function. In the limit of tiny epsilons it … Continue reading The Improper Function and the Poetry of Proofs

Heat Conduction Cheat Sheet

I am dumping some equations here I need now and then! The sections about 3-dimensional temperature waves summarize what is described at length in the second part of this post. Temperature waves are interesting for simulating yearly and daily oscillations in the temperature below the surface of the earth or near wall/floor of our ice/water … Continue reading Heat Conduction Cheat Sheet

Can the Efficiency Be Greater Than One?

This is one of the perennial top search terms for this blog. Anticlimactic answer: Yes, because input and output are determined also by economics, not only by physics. Often readers search for the efficiency of a refrigerator. Its efficiency, the ratio of output and input energies, is greater than 1 because the ambient energy is … Continue reading Can the Efficiency Be Greater Than One?