Physics & Chemistry

Ukrainian mathematician Maryna Viazovska solved a problem that had puzzled mathematicians for over 400 years. Even Johannes Kepler and Isaac Newton couldn’t crack it.

We live in a three-dimensional world, but Maryna solved a puzzle in an eight-dimensional space—something that’s very hard even to imagine.

She was born in Kyiv, studied at Taras Shevchenko University, worked in Bonn and Berlin, and at just 33 became a professor in Lausanne.

So what was the problem? It’s about how to pack identical spheres as tightly as possible in space. This question was first asked by Kepler back in 1611. Over time, scientists found answers for two and three dimensions—but not for eight.

Maryna proved that in eight dimensions, the densest packing is formed by a special mathematical structure called a lattice. What’s even more amazing is that she did it in just 23 pages, while earlier attempts took hundreds.

In 2022, she was awarded the Fields Medal, the most prestigious prize in mathematics. She became only the second woman in history to receive it.

Today, Maryna Viazovska works in Lausanne, supports Ukrainian mathematicians, and brings pride to Ukraine with her achievements.

Mathematically ideal structure

cot^{10}x integral.

Planck’s Law ✍️

It explains how hot objects glow by revealing how energy is emitted in tiny, discrete packets rather than a smooth, continuous flow. Imagine heat inside an object as countless microscopic vibrations. These vibrations don’t release energy in just any amount—they can only emit specific “chunks” called quanta. When an object heats up, these quanta are released as light of different colors.

At lower temperatures, the object mostly emits longer wavelengths like infrared, which we can’t see. As the temperature rises, the energy of these quanta increases, shifting the glow toward visible light—first red, then orange, and eventually white or even blue at very high temperatures. This is why heated metal changes color as it gets hotter.

Planck’s Law precisely describes how much light is emitted at each wavelength for a given temperature. By studying this pattern, scientists can determine an object’s temperature and understand the behavior of energy at the atomic scale.

D’Alembert’s Principle ✍️

It explains motion by turning a moving system into something that behaves as if it were perfectly balanced. Imagine observing an object in motion—forces are acting on it, causing acceleration. Now, instead of chasing that motion directly, we introduce an imaginary “inertial force” that opposes the acceleration. This added effect doesn’t exist physically, but it cancels out the motion in our calculations.

With this clever adjustment, the system can be treated as if it’s in equilibrium, where all forces balance out. The real forces and these imagined inertial forces together create a condition where no net work is done for any tiny, possible movement (called a virtual displacement).

This transforms a complex dynamics problem into a simpler statics-like one. Scientists and engineers use this principle to analyze motion with greater ease, especially in systems with many interconnected parts.

Peano Axioms ✍️

They explain how the entire structure of counting numbers can be built from a few simple, crystal-clear rules. Imagine starting with a single starting point—called zero (or one, depending on convention)—and a rule that lets you move forward step by step. Each move gives you the “next” number, like climbing an endless ladder where every rung is uniquely defined.

These axioms ensure that no two numbers share the same successor, so every step forward is distinct and unambiguous. They also guarantee that there’s no looping back—once you move ahead, you never return to a previous number. This creates a clean, infinite progression: 0, 1, 2, 3, and so on.

The most powerful idea comes from induction. It works like a logical domino effect: if something is true at the starting point, and you can prove it stays true when moving from one number to the next, then it must be true for all numbers. This principle allows mathematicians to establish truths across the entire number system with confidence.

Together, these simple rules act like the blueprint of arithmetic. From them, all familiar properties of numbers—addition, multiplication, and beyond—can be carefully constructed and understood.

The Periodic Table is a systematic arrangement of chemical elements based on their atomic number, electron configuration, and recurring chemical properties. The modern periodic table is based on the law proposed by Henry Moseley, which states that the properties of elements are periodic functions of their atomic numbers.

Earlier, Dmitri Mendeleev arranged elements according to atomic mass, but Moseley’s work corrected inconsistencies and led to the modern form. The table is organized into periods (rows) and groups (columns). Elements in the same group have similar valence electrons and therefore similar chemical behavior.
The periodic table is divided into blocks such as s-block, p-block, d-block, and f-block, depending on the orbital in which the last electron enters. It also shows important trends like atomic size, ionization energy, electronegativity, and metallic character.

Overall, the periodic table helps in predicting the properties, reactivity, and bonding behavior of elements, making it a fundamental tool in chemistry.

Standard Model of Particle Physics ✍️

It explains how the universe is built from a small set of fundamental particles and the forces that govern their behavior, much like a cosmic blueprint. Instead of everyday objects, it focuses on tiny building blocks like quarks and leptons, which combine to form everything we see around us. These particles interact by exchanging force carriers—special particles that act like messengers, transmitting forces such as electromagnetism and the strong and weak nuclear forces. Most of the time, these interactions are invisible and subtle, but under high-energy conditions, like those created in particle accelerators, their effects become measurable. Scientists observe the traces and patterns left behind after particles collide, using them to confirm predictions and uncover new details. The result is a powerful framework that allows us to understand matter, energy, and their interactions at the deepest known level.

He could not see world, yet he showed us the universe: Leonhard Euler
Leonhard Euler’s life is one of the most extraordinary stories in science—a saga of brilliance, blindness, and boundless perseverance. Born in Basel in 1707, Euler was mentored by Johann Bernoulli, who quickly recognized his unmatched mathematical talent. By his twenties, Euler was already reshaping mechanics and analysis and soon joined the St. Petersburg Academy in Russia. But his journey was marked by immense struggle. In 1735, after exhausting himself on astronomical calculations, Euler suffered a severe fever that left him blind in one eye. Later, cataracts robbed him of sight completely. Yet this darkness became the dawn of his greatest achievements.
Blindness never slowed him. He dictated his ideas to assistants and family members, producing over 800 papers and books, more than any mathematician in history. His productivity was so immense that the St. Petersburg Academy continued publishing his manuscripts for decades after his death. He averaged nearly one paper per week—a rhythm unmatched in science.
Euler’s contributions were vast and enduring. He gave us the notations e, i, and π, unified trigonometry with complex numbers, and discovered the identity:

e^(iπ) + 1 = 0

—uniting five fundamental constants in a single expression: e (the base of natural logarithms), i (the imaginary unit), π (the circle constant), 1 (the multiplicative identity), and 0 (the additive identity). This equation is often called the most beautiful in mathematics, a perfect harmony of nature’s deepest truths.

But Euler’s genius extended far beyond this elegance. He formulated Euler’s Theorem, a cornerstone of number theory that connects modular arithmetic with the Euler Totient Function. It states that if a and n are coprime integers, then:

a^φ(n) ≡ 1 (mod n)

This theorem generalized Fermat’s Little Theorem and became the foundation of modern cryptography—used today in RSA encryption and secure digital communication. Euler’s insight into modular arithmetic showed how pure mathematics could one day protect human privacy and information.
He also solved the Königsberg bridges problem, founding graph theory, and laid the groundwork for number theory, fluid dynamics, optics, astronomy, and engineering. His Euler–Bernoulli beam theory still underpins modern structural engineering. He worked on practical problems for the Russian Empire, from artillery design to navigation, proving that mathematics was not only abstract beauty but also a tool for nations.
Behind these achievements was a man of resilience. Euler endured the deaths of several children, the loss of his first wife, and constant ill health. Yet he remained cheerful, deeply religious, and devoted to teaching. His students recalled his warmth and generosity, even as his mind raced ahead of everyone else. When blindness confined him, Euler joked that he had fewer distractions and could concentrate better on mathematics.
On 18 September 1783, while discussing the orbit of Uranus with colleagues, Euler collapsed from a brain hemorrhage. The Russian Academy recorded his passing with words that captured his essence: “Euler ceased to live and to calculate.”
His blindness became not a weakness but a symbol of his indomitable spirit. Euler proved that true vision lies not in the eyes but in the mind. His perseverance in darkness gave humanity the mathematical language to describe nature itself. He remains the mathematician who wrote more than anyone in history, and his life is a beacon to this era: obstacles are only shadows before the light of determination.
Euler’s Theorem is not just a relic of 18th-century mathematics—it powers modern cryptography, digital signatures, and secure data transmission. Every time you send an encrypted message or make an online transaction, Euler’s logic silently protects our information.

Stirling's approximation is a useful formula that helps estimate very large factorials.

Factorials grow extremely fast, so calculating them exactly becomes difficult for large numbers.

Stirling’s approximation gives a simple way to estimate them using π and exponential functions.

The formula was developed by the Scottish mathematician James Stirling.

It is widely used in probability, statistics, and physics when dealing with very large numbers.