When Albert Einstein was about four or five years old, his father showed him a magnetic compass. The young Einstein was fascinated by one simple fact: however he turned the compass, the needle kept returning to a definite direction. There was no visible push, no string, no obvious contact. Decades later, Einstein still remembered the experience as something that made a deep and lasting impression on him. It suggested that behind ordinary appearances there could be hidden physical laws waiting to be understood. A small compass did not teach Einstein relativity. But it gave him something just as important: the suspicion that nature hides structure beneath what we can see. Source: Albert Einstein, Autobiographical Notes, first published in Paul Arthur Schilpp (ed.), Albert Einstein: Philosopher-Scientist (1949).

A room of just 35 randomly selected people is enough to make intuition stumble. The probability that at least two share the same birthday is about 81.44% despite there being 365 possible dates. The calculation is easier from the opposite direction: find the probability that all 35 birthdays are different, then subtract it from 1. Probability often becomes most interesting exactly where intuition becomes unreliable.

The debate over AI proofs is moving beyond one question: “Is the proof correct?” At a mathematics seminar yesterday, Mathias Stout raised two deeper unresolved questions about AI-generated formal proofs: Why should we trust LLM-generated Lean certificates? And how do we verify that the theorem encoded in Lean actually corresponds faithfully to the original mathematical problem? Machine verification may check a proof inside a formal system. But humans still have to verify that we formalized the right statement. That distinction could become crucial as AI enters serious mathematical research.

The zeros of sine are encoded directly into an infinite product. Analysis and number patterns meet in one elegant formula. This is Euler’s Infinite Product for the Sine Function. This remarkable identity says that the sine function can be built from an infinite product over the positive integers. Its zeros are immediately visible: whenever x = ±1, ±2, ±3, …, one factor vanishes, exactly matching the zeros of sin(πx).

An equilateral triangle has six symmetries: three reflections, two non-trivial rotations, and one surprisingly easy-to-forget symmetry, doing nothing at all. That “do nothing” move is the identity symmetry, and it is essential in group theory. Even more interesting, for the triangle, a reflection followed by a rotation is generally not the same as doing them in the opposite order: MR ≠ RM A simple shape already contains the basic idea of a non-commutative group.

Florence Nightingale showed that statistics can do more than summarize numbers. They can change decisions. During the Crimean War, she organized mortality data so that readers could immediately see a striking pattern: far more soldiers were dying from disease than from battle wounds. Her charts separated deaths by cause and month, turning a dense table of figures into a visual argument for better sanitation in military hospitals. This is an early lesson in data visualization: a well-designed graph does not merely display data. It helps people understand what the data is saying.

One of the deepest features of mathematics is that an idea can be developed with no application in sight, then decades later become exactly the language needed to describe nature. Mathematics often advances first by following its own internal logic; physics discovers the relevance later.

Count the lattice points inside and on the boundary, and the area appears automatically.

On this day in 1905, Albert Einstein’s paper On the Electrodynamics of Moving Bodies was published. It introduced the framework we now call special relativity, replacing absolute space and time with a theory built around the relativity principle and the constant speed of light. This one paper permanently changed how physics understands space and time.

Goldbach is famous for his conjecture, but he also had a beautiful argument showing that there are infinitely many primes. In a 1730 letter to Euler, Goldbach used the Fermat numbers: Fₙ = 2^(2ⁿ) + 1. These numbers are pairwise relatively prime. Therefore, each Fermat number contributes a prime factor different from those of all the others, forcing infinitely many distinct primes. A surprisingly elegant side of Goldbach that is rarely discussed.

Babbage corrects Tennyson Charles Babbage once objected to a line in Tennyson’s poem The Vision of Sin, where the poet wrote that every moment one person dies and one is born. Babbage pointed out that this would imply a perfectly constant world population, whereas births slightly exceeded deaths. He even suggested replacing the line with a more “accurate” numerical rate, before admitting that the true decimal was far too long for poetry. A perfect meeting of mathematics and literal-minded humour.

This curious identity is known as the “Sophomore’s Dream.

Oxford mathematician Jon Keating says mathematics needs a “sober reckoning” with AI. As AI becomes increasingly capable of solving research problems, he raises three fundamental questions: Who deserves credit? Is mathematics primarily about solving problems, or creating new concepts and theories? And will AI-generated proofs be written so humans can actually understand and learn from them? Keating’s conclusion is measured: “radical change is coming,” but mathematics should adapt and work with AI rather than abandon what makes mathematical understanding valuable.

Sine and cosine from the unit circle

Prime numbers can look completely chaotic. Marcus du Sautoy provided a striking example: among the 100 numbers immediately below 10,000,000 there are nine primes, while among the 100 numbers immediately above it there are only two.

As a boy, Alan Turing once watched wild bees flying across the Scottish heather, traced their flight paths, and used the intersection to locate their nest. Long before computers and cryptography, his instinct was already clear: observe a pattern, model it, and solve the problem.

Fields Medallist June Huh has a nuanced warning about AI in mathematics. He says the danger is not that AI can solve hard problems, but that mathematics could become a race simply to obtain answers. “Obtaining the answer is not the entirety of research.” Huh says he uses AI in his research every day and calls this an exciting era for mathematicians. But he argues that the human role remains deeper: asking new questions, preserving intellectual curiosity, and building new mathematical horizons. Perhaps the real Math–AI question is no longer “Can AI solve it?” but “What do humans understand after it does?”

A cube becomes surprisingly crowded in higher dimensions. An n-dimensional cube has exactly n × 2ⁿ⁻¹ edges. So a 7-dimensional cube has 128 vertices but 448 edges.

A little linear algebra humor: “That math professor’s marriage is falling apart!” “No wonder. He’s into scientific computing, and she is incalculable.”