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

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

complex-exp-1-z2

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