Gustaf Ullman

Exploring Nodal Lines in Random De Broglie Waves Exploring Nodal Lines in Random De Broglie Waves By Gustaf Ullman Introduction Nodal lines are a fascinating feature of random wave superpositions in quantum chaos. These lines, where either the real or imaginary parts of the wavefunction equal zero, reveal intricate patterns that merge randomness and structure. Using equation 1 from the paper "Nodal lines of random wavefunctions: perimeter corrections, statistics and scaling" by Saichev et al, we generate nodal lines for a superposition of De Broglie waves with random amplitudes, directions, and phases. Method The wavefunction is constructed as: ψ(r) = Σ a j exp(i(k j ·r + φ j )) Here, a j represents random amplitudes, k j is the wavevector with random directions, and φ j is the random phase. Nodal lines are identified where Re(ψ) = 0 or Im(ψ) = 0 . Results The generated nodal lines for R...

Creating Digital Art Using 2D Fourier Transforms Creating Digital Art Using 2D Fourier Transforms By Gustaf Ullman Introduction Digital art can explore concepts and techniques rooted in mathematics. One such method involves the use of two-dimensional Fourier transforms to create intricate and randomized patterns. This article outlines how to generate "lumpy" spatial frequency images in CMY (cyan, magenta, yellow) using a simple mathematical framework. Method The process begins with a radially symmetric function, such as: f(r) = A exp(-br) where r is the radial distance from the center of the image, A is the amplitude, and b controls the frequency distribution. This function is then assigned a random phase for each pixel, sampled uniformly between 0 and 2π. After applying a 2D Fourier transform to the resulting function, the output image exhibits a "lumpy" pattern, with spatial frequencies distri...

Lek, Genialitet och AI Att leka sig fram till genialitet Lek är något vi ofta förknippar med barndomen, men det kan vara minst lika betydelsefullt för oss vuxna. Detta har blivit tydligt för mig när jag studerat musikaliska underbarn som Wolfgang Amadeus Mozart och Emily Bear. Båda började i tidig ålder, inte för att de tvingades, utan för att musiken för dem blev en naturlig "leksak". Mozart kunde som barn sitta vid pianot i timmar, ungefär som barn ibland försjunker i att bygga med Lego, utan tvång eller krav. Emily Bear komponerade redan som mycket ung och fick som 10-åring en högskoleutbildning i filmmusik – hon sökte sig till musiken på eget initiativ. Dessa ovanliga och inspirerande exempel visar hur lekfullhet kan bestå ända in i vuxen ålder. Mozart, Emily Bear och även Felix Mendelssohn lekte fram sina kompositioner med kontrapunkt, harmoniska överraskningar och rytmiska finesser. Denna lekfullhet kan ibland tolk...

Norwood Russell Hanson and the Art of Hearing: Seeing Music as Inner Vision One of Norwood Russell Hanson’s central ideas is that observation is theory-laden—what we perceive depends on what we know. This insight extends beyond science and philosophy into the realm of music, where perception often requires training and understanding. Consider a fugue. To the untrained ear, it might sound like a complex but indistinct flow of notes. However, for someone with a trained ear, it is an intricate interplay of counterpoint, modulation, and thematic development. The same is true for recognizing syncope or modulation; these are not just elements of sound but concepts that must be "seen" with the mind. Music as a Visual Experience Interestingly, I choose the word "see" rather than "hear" when describing this experience, because I believe it mirrors the same inner vision we use in mathematics or other abstract disciplines. Just as a mathematician "sees...

Are Stars a Form of Life? A Metabolic Perspective One of the defining characteristics of life, as commonly understood in biological terms, is metabolism. Metabolism refers to the set of chemical processes that allow an entity to sustain itself, grow, and interact with its environment. In this context, the question arises: could stars, with their remarkable nuclear processes, be considered a form of life? The Metabolism of Stars Stars are born from vast clouds of gas and dust, primarily hydrogen. As they form, the immense pressure and temperature in their cores initiate nuclear fusion—a process where hydrogen atoms combine to form helium. This phase, known as the main sequence, dominates a star’s life and is analogous, in some sense, to an organism consuming a primary energy source to maintain itself. As stars exhaust their hydrogen, they begin to "metabolize" helium, producing heavier elements such as carbon, nitrogen, and oxygen. These elements are essential for the ...

Reflections on The Mission : A Review Some time ago, I wrote about the soundtrack to The Mission , expressing my admiration for its haunting beauty. When I watched the film, I found it to be deeply moving, a work that touched me on a profound emotional and spiritual level. At its core, the film carries strong influences from Christian anarchism, particularly its unwavering commitment to pacifism. However, like many works with a strong ideological bent, it is not without its flaws. The film is, fundamentally, a portrayal of Christian missionary work. In this respect, it mirrors the historical errors of many Christian missionaries by underestimating the wisdom and spirituality already present in indigenous cultures. The narrative frames the Christian mission in South America as a largely positive force, which, in the eyes of modern viewers, may feel deeply problematic. This critique is not directed at the spiritual core of Christianity, which emphasizes love and humility, but at hist...

Reviving the Blogosphere Reviving the Blogosphere I've written before about how much I miss the blogosphere. It more or less died out, replaced by Facebook. But this was a serious downgrade, both in terms of freedom of expression and content quality. One of the reasons Facebook succeeded is that it allows for short, hastily written posts. If you have a 5-minute break, you can dash off a quick post. Writing a well-crafted blog post, on the other hand, usually takes closer to an hour. That’s where ChatGPT comes in. I've revived my blog. It may not quite take just 5 minutes to write a blog post, but it might only take 15 minutes — short enough to make blogging compatible with managing family and work. I often repurpose posts I write here and cross-publish them on my blog. In a bit of meta-commentary and self-reference, I decided to write a blog post about t...