Atul Tripathi

🚨 BREAKING: Two metal plates… in a perfect vacuum… should feel nothing. But they move toward each other. No magnets. No charge. No particles pushing them. So what’s pulling them together? This is called the Casimir Effect and it proves something most people miss: “Empty space” isn’t empty. It’s structured. Between the plates, only certain wavelengths can exist. Outside, more possibilities exist. So the system becomes imbalanced… And the result? A force. Not from matter. Not from energy being added. But from structure resolving under constraint. Think about that. The vacuum isn’t nothing it’s a field of possible states. And when you limit those possibilities… you change reality. This is the key: Force doesn’t always come from something pushing. Sometimes it comes from what’s allowed to exist. That’s not just physics… that’s structure defining outcome. If this makes you rethink what “empty space” really is… follow me I’m breaking down how structure shapes reality across physics.

√1 = 1 √121 = 11 √12321 = 111 √1234321 = 1111 √123454321 = 11111 √12345654321 = 111111 √1234567654321 = 1111111 √123456787654321 = 11111111 √12345678987654321 = 111111111

"I think the universe is pure geometry - basically, a beautiful shape twisting around and dancing over space-time." - Antony Garrett Lisi

When φ awakens geometry. A lone figure forges order from the void; unraveling golden spirals, taming icosahedra, collapsing hypercubes into fractal harmony.

Thomas young's double slit experiment proved light behaves like a wave by creating an interference pattern that only waves can produce. He shown light through two narrow slits cut into a barrier and recorded the pattern on a screen behind it.

“If you want to find the secrets of the universe, think in terms of energy, frequency and vibration.” - Nikola Tesla

Yuri Kovalenok, @jurijkovalenok1, is a physics teacher who represents physics and mathconcepts (and formulas) in an artistic fashion. Here is an animation from his seemingly infinite notebook. Used with permission.

Linear dynamics x' = Ax in ℝ³, ruled by the eigenvalues of A. Real λ → flows along eigendirections. Complex pair → spirals. Triple repeated λ (defective) → it tries to spiral but can’t quite.

Schrödinger, 1926: iℏ ∂ψ/∂t = Ĥψ. Classical trajectories discarded for the time-evolution of the complex wavefunction ψ(x,t). Watch this 3D Gaussian packet propagate, disperse, and interfere; |ψ|² yields probability densities, phase encodes momentum.

Two rods rotate at different constant speeds to show that π is irrational. Even close approximations like 22/7 or 355/113 almost match—but never exactly. This shows π cannot be written as a simple fraction.

Most engineers have seen this formula. P(A|B) = P(B|A) × P(A) / P(B) Almost none can explain what it actually does. Here's Bayes' Theorem in plain English, and where it's hiding inside systems you use every day. The core idea in one sentence: Bayes' Theorem updates your belief about something after seeing new evidence. That's it. Four terms: Prior → what you believed before the evidence Likelihood → how probable the evidence is, given your hypothesis Evidence → how common the evidence is overall Posterior → your updated belief after seeing the evidence A concrete example: Say 40% of all emails are spam (your prior). You see a new email containing the word "lottery." 10% of spam emails contain "lottery." Only 1% of legitimate emails do. Plug into Bayes: P(spam | "lottery") = (0.10 × 0.40) / P("lottery") ≈ 87% The word "lottery" updated your belief from 40% → 87%. That's Bayes in action. Prior belief + new evidence = updated belief. Where it lives in AI: 1/ Spam filters The Naive Bayes classifier, the algorithm behind most spam filters - applies this exact calculation word by word across an entire email. Each word shifts the probability up or down. It's called "naive" because it assumes each word is independent of the others, which isn't realistic, but works remarkably well in practice. 2/ Medical diagnosis AI A patient has symptom X. What's the probability of disease Y? Bayes updates the base rate (how common the disease is) with the likelihood of seeing that symptom in patients who have it. Same formula, different domain. 3/ Your LLM's uncertainty Modern language models don't just predict the next token, they assign a probability to every possible token. The sampling process (temperature, top-p) is directly working with those probability distributions. Bayesian reasoning is embedded in every response your model generates. The insight most engineers miss: Bayes doesn't give you certainty. It gives you a rational way to update uncertainty. That's exactly why it's foundational to AI - real-world systems are never certain. They're always working with incomplete, noisy, probabilistic information. Every model that learns from data is, at its core, doing some version of this: Start with a belief. See evidence. Update the belief. That's Bayes. That's machine learning.

Hamiltonian Monte Carlo: probability as physics. Endow particles with momentum, then let Hamilton’s equations (dq/dt = ∂H/∂p, dp/dt = −∂H/∂q) carve reversible, volume-preserving trajectories through phase space.

Graph Tricks

Stokes’ Theorem relates the circulation around a boundary curve C to the curl over the surface. When the surface is split into small pieces, inner edges cancel out—leaving only the outer boundary C.

The Galton quincunx: each bead traces a discrete random walk through pegs, governed by Bernoulli trials. The ensemble converges to the Gaussian; Pascal’s binomial coefficients etched perfectly in the histogram. Central Limit Theorem made visible.

Physics notes

For over 160 years, mathematicians have struggled with a simple-looking but unsolved problem called the Riemann Hypothesis. It is considered one of the greatest mysteries in mathematics, with a $1 million prize for a correct proof. The problem is about prime numbers—the basic building blocks of all numbers. Primes seem to appear randomly, but over large scales, they show signs of a hidden pattern. In 1859, Bernhard Riemann suggested that this pattern follows a deep mathematical rule. Since then, trillions of cases have been tested, and all agree with his idea—but no one has proven it for all numbers. If proven true, it could improve our understanding of numbers and have real-world impact, especially in areas like encryption.

Visualization of Fourier Series

The true beauty of mathematics lies in its patterns and symmetries.