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

# Tag: Mathematics

# 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

# 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

# Tinkering, Science, and (Not) Sharing It

I stumbled upon this research paper called PVC polyhedra: We describe how to construct a dodecahedron, tetrahedron, cube, and octahedron out of pvc pipes using standard fittings. ... In particular, if we take a connector that takes three pipes each at 120 degree angles from the others (this is called a “true wye”) and we … Continue reading Tinkering, Science, and (Not) Sharing It