Leiden Declaration on Artificial Intelligence and Mathematics

This declaration calls for action to address the challenges posed by the use of artificial intelligence within mathematics research. It is the result of a community initiative and is endorsed by the International Mathematical Union (IMU).

Technological developments have repeatedly transformed the practice of mathematics. Recent artificial intelligence technologies, including symbolic and neural methods for the generation and formalization of mathematics, may already have initiated a significant chapter in this long history. Among researchers, artificial intelligence has produced a wide range of reactions: enthusiasm for its potential to yield new discoveries; intimidation by the pace of developments; indifference to these rapid changes; and concern for the implications, both for mathematics and in wider society.

Mathematicians have a choice about whether and how to adopt artificial intelligence in the conduct of their research. They also have a responsibility to ensure the continued flourishing of the discipline. This Declaration calls upon mathematicians to exercise this responsibility, and provides recommendations for individuals, institutions, government, and industry.

Although we adopt the perspective of mathematical research, much of what we write applies equally to other aspects of mathematics. This includes work in the broader mathematical sciences, education, mentoring, publishing, funding, science policy, and use of mathematics in the wider world.

...

https://leidendeclaration.ai/

The declaration warns that reliance on AI-generated proofs threatens the accuracy, reliability and independent verifiability of mathematical research, and that the growing involvement of AI companies risks skewing research priorities toward problems amenable to AI methods while disadvantaging researchers without access to them. It calls on mathematicians, professional organizations and policymakers to disclose the use of AI, uphold peer review, and protect the openness and values of the discipline.

The declaration was written by 16 mathematicians, including Jim Portegies (Eindhoven), who led the project, Mateja Jamnik (Cambridge), Ursula Martin (Oxford), Rodrigo Ochigame (Leiden), and Michael Harris (Columbia).

The declaration expresses concerns about risks to the accuracy and reliability of mathematical papers raised by reliance on AI-generated proofs. It says: "Technology companies' involvement in research raises the risk that research questions are prioritized and incentivized because of their amenability to AI methods and models, rather than their deeper significance to understanding." Researchers without access to AI methods are put at a disadvantage.

The declaration says that "models trained on published works frequently return outputs that do not properly cite the human works they synthesize" and that these models' training data has been gathered by "exploiting licenses and access arrangements" or "simply violating copyright protections."

Corporate press releases about mathematical discoveries by AI are described as being on "market timelines before the accepted processes of community evaluation in mathematics can take place."

The declaration recommends disclosing the use of AI in research papers, ensuring that AI-assisted papers are peer-reviewed, and providing legal resources and public funding so that academia and for-profit companies can compete on equal terms.

Contributors describe the purpose of the declaration as to stimulate debate and help mathematicians articulate the values of their profession. Michael Harris said: "The purpose, from my perspective, is to recover control of the narrative about the values and the goals of mathematics from the A.I. industry." He describes mathematicians' values as being "openness, honesty, giving credit where credit is due, sharing, transparency about methodologies, and access for independent verification of results." Writing in the Communications of the ACM, co-author Mateja Jamnik described the declaration as a call on the "mathematical community to make its values explicit at a time when AI systems are increasingly becoming part of mathematical research".

https://en.wikipedia.org/wiki/Leiden_Declaration_on_Artificial_Intelligence_and_Mathematics

Thoughts about the Leiden Declaration

(By Timothy Gowers)

[...]

One of the concrete outcomes of the workshop was the Leiden Declaration, which has now been signed by over 3000 people. Given that I was part of the workshop, it might seem a bit strange that I am not one of the signatories of the resulting declaration. The reason is not so much that I disagree with it in any concrete way, but more that in several places it makes confident assertions and recommendations that I feel somewhat uncertain about. So instead I prefer to try to articulate my views about the issues raised by the declaration and put them in this blog post. Before I do that, I would like to make clear that I am very glad that the Leiden Declaration exists and I think that it has done a lot of good in focusing people’s minds on the issues that AI is forcing the mathematical community to grapple with, which are more acute now than they were last September.

...

https://gowers.wordpress.com/2026/07/26/thoughts-about-the-leiden-declaration/

listening to this article is available on the site

In the mid-noughties, when music by the Killers and Franz Ferdinand blared out of every pub and nightclub I passed, I spent my days and nights struggling through a Ph.D. in applied mathematics. My research focused on simulating how special light waves interact in liquid crystals and using simple equations to approximate and understand those interactions. When I look back at my thesis now, liquid crystal technology is old hat, and I imagine my work could be completed with AI assistance in a matter of days—maybe hours.

But the same cannot be said for the work of the pure mathematics Ph.D. students with whom I shared a cramped office at the University of Edinburgh. At the time, I felt sorry for these colleagues, who day after day sat at their desks, seemingly tearing their hair out and making no progress. (Though I was struggling too, I was at least always making some headway.) When we finished and went our separate ways, some hadn’t even published a paper.

Now, in hindsight, I finally understand why they toiled for years on abstract mathematical problems that only a handful of people in the world care about. It wasn’t arrogance, as I thought at the time; they weren’t trying to prove their superior intelligence by being the first to solve a seemingly intractable mathematical problem. It wasn’t even a form of masochism (which was my second guess)—penance for some imagined inadequacy. I realized they derived joy, satisfaction, and meaning from the long journey toward understanding.

“Sometimes, understanding just strikes you as being very beautiful. Sometimes it’s a feeling of accomplishment, like completing a marathon,” muses Carnegie Mellon University mathematician Jeremy Avigad. “But it’s not quite either of those: It’s just a wonderful feeling when you’ve been thinking long and hard about something complex, difficult, and then—all of a sudden—it just comes together.”

This feeling has driven mathematicians throughout history. Likewise, the way mathematicians pursue that feeling has changed little over the centuries. They notice or imagine links, patterns, or properties in numbers, shapes, or logical structures. From this, they write conjectures—unproven statements of their speculation. They or other mathematicians then use logical reasoning and the tools of mathematics in often creative ways to prove or disprove those conjectures. Finally, yet other mathematicians verify (or challenge) the proofs.

Invariably, this process requires a whole heap of thinking time. “I went to a pure maths camp with classes where we would sit with hard maths problems for half an hour and no one would say anything—everyone was just thinking,” says Krystal Maughan, a mathematician and computer scientist about to get her Ph.D. at the University of Vermont. “But then we would work together and kind of tease out the problem.”

This is the age-old joy of math in action. But today’s AI systems are starting to make inroads into bypassing this slow, deliberative process. Taking this trend to its logical conclusion, what happens if AI makes the mathematician’s struggle completely unnecessary? Might AI even sideline humanity completely?

AI’s Growing Role in Mathematics

For decades, computation has accelerated mathematical progress. This began 50 years ago, when mathematicians used a computer to prove the four-color theorem, which asks whether any map can be colored using no more than four colors, with no adjacent regions sharing the same color. The answer is yes, and the computer proved it, controversially, by checking 1,936 cases in a way no human could realistically verify.

Yet throughout this computational era, even in proofs relying on massive computational resources, the role of the human mathematician has remained central. Humans propose conjectures, guided by intuition. They devise strategies to prove them, guided by creativity and experience. And humans verify whether those proofs are correct.

Now AI is challenging the status quo. In just a few years, large language models (LLMs) have evolved from “stochastic parrots,” capable of little more than regurgitating basic mathematics scraped from the internet, into advanced mathematical reasoning machines.

Last summer, systems from Google DeepMind and OpenAI reached a level equivalent to the world’s most mathematically gifted high school students, achieving gold-medal status at the International Mathematical Olympiad. In this annual competition, contestants must solve six notoriously difficult problems from various areas of mathematics.

Earlier this year, Google DeepMind’s experimental AI system Aletheia achieved an even more significant milestone when it autonomously produced publishable Ph.D.-level research results. While the work itself is obscure mathematically—calculating structure constants in arithmetic geometry—the significance lies in the complex reasoning it displayed in tackling an unsolved mathematical problem. And more recently, a new general-purpose AI system from OpenAI disproved an important conjecture in combinatorial geometry. This result would have been worthy of publication in a major mathematics journal if humans had been the authors, and top mathematicians hailed the feat as a milestone for AI in mathematics, demonstrating independent, original, and sophisticated thinking.

Another shift has come from combining LLMs with mathematical tools known as proof assistants, which have been around for more than a decade. These systems—such as Isabelle, Lean, and Rocq—are specialized programming languages that check mathematical proofs step-by-step, verifying their logical correctness. Traditionally, mathematicians have had to translate their theorems and proofs into this machine-readable format by hand, a laborious process known as formalization. Now, LLMs are starting to remove this bottleneck, automating the translation of informal proofs into formal code that proof assistants can verify.

"From Human Proof to Formal Proof

Euclid’s famous proof that there are infinitely many prime numbers appears very different when formalized in Lean, a proof assistant. Human mathematicians routinely skip steps and rely on shared understanding; formalization makes every assumption and inference explicit so a computer can verify the proof."

Versions of such systems, sometimes called reasoning agents, are becoming highly sophisticated. In February, for example, the AI company Math, Inc. used its aspirationally named reasoning agent Gauss to formalize a proof that had earned the mathematician Maryna Viazovska, of EPFL, in Switzerland, a Fields Medal in 2022. Gauss first helped human mathematicians complete the formalization of Viazovska’s solution to the 8-dimensional sphere-packing problem in a matter of days, and then autonomously formalized the more complicated 24-dimensional case in just two weeks.

Such achievements suggest that AI is already capable of handling some mathematical tasks long considered uniquely human. As the technology advances, more of the day-to-day work of human mathematicians is likely to become fair game for AI.

Mathematicians Debate AI’s Role in Discovery

In September 2025, I attended the 12th Heidelberg Laureate Forum—an annual conference that brings hundreds of young mathematicians and computer scientists together with their intellectual idols. AI dominated the conversation and, from the get-go, tension was in the air.

Speakers described a future in which superhuman AI mathematicians transcend human knowledge and capabilities: forming conjectures, searching solution spaces, proving conjectures, and finally verifying the proofs and generalizing the results, all without human involvement. If this future comes to pass, Yang-Hui He of the London Institute for Mathematical Sciences memorably declared, human mathematicians could become “priests to oracles.”

While such startling predictions were being voiced on stage, my gaze was drawn to the audience. Frowning, fidgeting, and exchanging furtive glances—the crowd’s unease was palpable. Trill White, a student at Australia’s Deakin University, later recalled sitting in that hall and thinking: “ ‘That’s devastating. What will people have to contribute to mathematics? Will it become something that no one understands?’ I did get a sense that this is going to change everything.”

Jessica Randall, a South African mathematician for Google Developer Groups, says she sensed a collective existential dread rising among the young mathematicians. “I could feel everyone was worried, because they hadn’t thought that far ahead,” she says. “It was like a big bombshell that hit us, and we certainly started realizing AI has the potential to replace us.”

Some established mathematicians, including He, seem comfortable with AI taking on tasks that are currently the preserve of human mathematicians. That’s because they just want to know the answers to the biggest questions in mathematics—such as the six remaining Millennium Prize Problems—even if AI does it all. “A lot of mathematicians are pragmatic and just want to understand. They would sell their soul for the solution to a problem,” jokes Avigad. “Whatever it takes, right?”

But this “just want to know” camp is by no means the only faction: Most mathematicians do not hope or expect AI to replace them entirely. Instead, two broad alternatives are emerging. The first is a human-centric aspiration that prioritizes human understanding of mathematics and treats AI as a tool, much like a calculator. The second is a collaborative “teamwork makes the dream work” vision, where humans and AI work together to tackle problems neither could solve alone.

The Human Role in Mathematics

Fields Medalist and Princeton mathematician Akshay Venkatesh has been thinking about this topic from the human-centric viewpoint for years. In 2022, he used his Fields Medal Symposium to implore the mathematics community to deeply consider what AI might mean for the practice of mathematics. At the time, the idea that AI could replace mathematicians seemed far-fetched. Now, he says, “we’re reaching the point where, for at least some tasks with abstract mathematical reasoning, computers are becoming competitive with humans.”

For Venkatesh, the question is not just what computers can do, but what mathematics is for. “Sometimes I think when we use numbers, it’s not so much that we are describing phenomena that are intrinsically numerical, but that we can all agree exactly what the numbers mean,” he says. “It’s a way of bringing us to agreement.”

Mathematician and machine learning expert Maia Fraser, of the University of Ottawa, shares this sentiment. She says the joy she derives from mathematics is something distinctly human that integrates the subconscious and conscious mind. She describes starting with an intuitive sense that a certain thing should be true and gradually bringing out something that she can express in a rigorous proof. Communicating and sharing these deep-born thoughts is “a form of collective intelligence that is something beautiful about the human spirit,” she says.

By these arguments, an AI proof of a mathematical conjecture that has stubbornly resisted human efforts would be useful only if comprehensible to humans. “That the statement can be proved by AI is already useful information,” concedes Fraser. “But then it’s still an open problem to come up with an elegant, beautiful human proof.” Even if no such proof exists, she says, searching for it “is still a valuable endeavor.”

AI and the Future of Mathematical Collaboration

A more collaborative approach to AI in mathematics comes from Terence Tao, who first competed in the math Olympiad at the age of 10. In 1986, 1987, and 1988, he won bronze, silver, and gold medals, respectively, making him the youngest winner of each of the three medals in Olympiad history. Now a Fields Medalist and professor at the University of California, Los Angeles, he has earned a reputation as one of the most gifted mathematicians alive.

Unlike some of his peers, Tao is neither dismissive of AI nor fearful. Instead, he sees it as the catalyst for a fundamental shift in the discipline—a transition toward what he calls “big mathematics.” He envisions a future of large-scale, decentralized collaborations between humans and machines, where complex mathematical tasks can be diced and sliced, with humans claiming the creative parts and AI doing the lion’s share of the technical grunt work.

Already, Tao is experimenting with this concept, working on problems alongside scores of online collaborators, some using AI tools. “A hundred years ago, almost every mathematics paper was single author,” he says. “But now I collaborate with people I’ve never met—and maybe in the future, I won’t even know if they are AI or real people.”

The key to Tao’s vision is uniquely mathematical: formalization. When a proof is translated into code and checked step-by-step by proof assistants, it removes any chance of human error or dishonesty. This approach changes how collaboration works, because trust is established through verification rather than reputation or rapport. An idea from an unknown researcher or even an amateur can be taken seriously if it has a formal proof.

“If it wasn’t for this formal verification layer, opening projects up without any safeguards would just be a disaster,” adds Tao. “But in math, we can completely check and verify outputs, and this really filters out a lot of the rubbish.”

The Risks of AI in Mathematics

From the young researchers at the Heidelberg Laureate Forum to some of the biggest names in the field, mathematicians all seem to agree on one point: AI has the potential to transform their discipline. But there’s far less consensus on what that transformation will mean in practice.

Some worry about the accessibility of AI tools. Traditionally, mathematicians have required little more than intuition, training, and a pen and paper to advance their field. If this slow, deliberative process is no longer valued by society, and particularly by research funders, then mathematics could become an elitist activity, only practiced by select organizations that can afford to work with proprietary AI models.

Another concern is motivation. As AI systems take on more of the work, the incentive to engage deeply with difficult problems may weaken. Princeton’s Venkatesh says that the long human process of formulating and understanding a proof may be hard to justify, not just to funders, but even to mathematicians themselves. “There have been times where I’ve spent years thinking about something, and I’ve slowly struggled to understand it,” he says. “If your computer can do large chunks of that for you, will you have the motivation to spend that time?”

That concern extends to the next generation. If students can use AI to jump straight to answers, they most likely will. But every time they skip the struggle, they miss an opportunity to build the foundations of their own unique intuition. Over time, some worry, the next generation of mathematicians may suffer from a form of intellectual atrophy, unable to think outside the AI box that trained them.

In response to such fears, the mathematics community is taking action. Individuals are writing essays, organizing workshops, and debating in journals, while institutions and community groups are developing guidelines for how AI should be used in research and publication. Indeed, mathematicians are applying the same rigor and curiosity that they use every day to reckon with the challenges of AI. Taken together, these efforts reflect a broad effort to try to retain control over the direction of mathematics in the era of AI.

So, is AI sucking the soul out of math? In one way, it is doing the opposite. It is forcing mathematicians to confront deep questions about what mathematics is, why they have devoted their lives to it, and the purpose math serves in society. At the same time, though, it is reshaping the practice of mathematics in a way that may be difficult to reverse.

“Mathematics makes me a better problem solver at normal problems, because it frames my mind to think in a very logical, rational way,” says Randall, who noted the existential dread at the Heidelberg Forum. “It helps with every aspect of my life.” As AI transforms mathematics, many researchers wonder whether future mathematicians will be able to say the same.

Hinged dissections are dissections with a chain-like property: we can disassemble an original figure in such a way that all the pieces are linked at least at one point, and a new rearrangement is possible by continuously moving of the pieces, without destroying the chain.

According to Frederickson, who published in 2002 the most remarkable book on this subject, the idea of hinged dissections originated in early twentieth century with Henry Ernest Dudeney. Famously, Dudeney also gave a hingeable dissection of the equilateral triangle to a square:

https://pi.math.cornell.edu/~mec/GeometricDissections/2.2%20Hinged%20Dissections.html

Make regular triangle ABC , then we would find the lengh d of the square

s.t. the regular triangle & the square have same superficial measure

make D the midpoint of BC, P on ray AD s.t. 2 PD=BC

find Q on ray BC s.t. the angle AQP be a right angle, then DQ=d

Since we have d, dissect the regular triangle into 2 polygon.

Find a line segment perpendicular with DE & with lengh d/2

by 3, we dissect the upper polygon into 2 polygon.

make M the midpoint of DE, symmetry L1 by M , then we dissect the lower polygon into 2 polygon.

then we have the 4 polygon can decomposing a regular triangle into a square.

https://oz.nthu.edu.tw/~u9521202/make.html

About a decade ago Tonan Kamata, now a mathematician at the Japan Advanced Institute of Science and Technology (JAIST), stood mesmerized in front of a math museum’s origamilike exhibit. It featured a triangular tile cut into four pieces that were connected by tiny hinges. With a simple swivel, the pieces spun around to transform the triangle into a square.

The exhibit traces its origin to a mathematical puzzle published in a 1902 newspaper. Henry Dudeney, a self-taught English mathematician and puzzle columnist, asked his readers to dissect an equilateral triangle into the smallest number of pieces that could be rearranged into a square. In his next column two weeks later, he noted that a “Mr. C. W. McElroy of Manchester”—Charles William McElroy, a clerk who frequently wrote to Dudeney with puzzle solutions—had a four-piece solution. After two more weeks, Dudeney reported that none of the newspaper’s other readers had bested the solution, and since then, the record has stood. It remained unproven, however, whether a solution with fewer pieces existed.

The puzzle became known as “Dudeney’s dissection” or the “haberdasher’s problem,” [...] .

Now, more than 122 years after it was first proposed, Kamata and two other mathematicians have finally proved that a solution with fewer pieces is impossible. Their result was posted to the server arXiv.org in a December 2024 preprint entitled “Dudeney’s Dissection Is Optimal.”

“I believe many who appreciate mathematics would agree that the simpler an unsolved problem appears, the more profoundly captivating it becomes to those who love mathematics,” Kamata says.

Together with Massachusetts Institute of Technology mathematician Erik Demaine and JAIST mathematician Ryuhei Uehara, Kamata had been developing a new approach to tackle origami-folding problems using graph theory. In graph theory, a graph is essentially a collection of lines, or edges, and vertices, the points where edges meet. The edges and vertices of one graph can be compared with those of another to explore deeper relationships between the two structures—an approach that Kamata thought might help to resolve Dudeney’s dissection.

One part of the problem is fairly simple: a two-piece solution can be ruled out by thinking about the problem’s constraints. For starters, the triangle and square must have equal areas because the pieces are the same. For a square, its longest possible cut is across the diagonal. A bit of pen-and-paper math shows that, unfortunately, the diagonal’s length is too short for the edge of its equal-area triangle, which rules out a two-piece solution.

Proving that there are no three-piece solutions is much trickier, however, and that’s the reason for the century-long delay. Although it is a simple three-piece puzzle, there are an infinite number of ways to cut up the triangle, Demaine says. “Each of those pieces could have arbitrarily many edges to it, and the coordinates of those cuts start at arbitrary points,” he says. “You have these continuous parameters where there’s lots and lots of infinities of possible choices that makes it so annoyingly hard. You can’t just brute-force it with a computer.”

To tackle the problem, the group categorized possible dissections of an equilateral triangle based on how the cuts intersect the triangle’s edges. First, the researchers sorted the infinity of ways to cut the triangle into five unique classifications. They then repeated the exercise for a square and found 38 distinct classifications.

Next, the researchers tried to match a triangular graph to a square one by tracing all the possible paths in each shape and comparing the resulting collections of edge lengths and angles. If one of the squares’ paths had matched that of a triangle’s, it would have meant that the researchers had discovered a three-piece solution.

The approach transformed the continuous problem into a discrete one—almost. Within each classification, there are “still infinitely many places all these vertices could go,” Demaine says. In the end, the group derived a collection of complex lemmas, or intermediate steps in a theorem, that, together with the categories, used proof by contradiction to find no matching paths.

Smith College computer scientist Joseph O’Rourke, who has wrestled with the puzzle on and off for decades and was not involved with the current work, thinks the group’s proof can probably be simplified. He notes that it took a messy collection of highly specific lemmas to eliminate all the possibilities, “Many other researchers would have given up.”.

Crucially, if the authors can simplify their proof, the matching-diagrams technique could unravel a slew of related origamilike open questions. “These problems remind us how much there is yet to discover,” Kamata says. “Anyone can become a pioneer in this frontier.”

By Lyndie Chiou

https://www.scientificamerican.com/article/mathematicians-find-proof-to-122-year-old-triangle-to-square-puzzle/

Deux des dix disques de la figure ont été noircis. Noircissez d’autres disques de façon que l’on ait toujours 1 ou 3 disques noirs sur chacun des 8 alignements tracés.

Two of the ten discs in the figure have been blackened. Blacken additional discs so that there are always either 1 or 3 black discs on each of the 8 lines drawn.

https://www.ffjm.org/fr/

https://youtu.be/n5MAsOkAENU

L’instabilité de Rayleigh-Taylor affecte une couche de fluide plus lourd, disposée initialement au-dessus d’une couche de fluide plus léger. Sous l’effet de la gravité, les fluides tendent vers une configuration plus stable, avec la position des fluides inversée. L’instabilité concerne le mouvement de l’interface entre les deux fluides, initialement en équilibre instable.

Dans cette simulation, l’effet est illustré à l’aide de particules interagissant via un potentiel de Lennard-Jones. Les particules bleues sont 10 fois plus lourdes que les particules rouges. La température affichée est une mesure de l’énergie cinétique moyenne, alors que l’entropie est donnée par

-(pln(p) + (1-p)ln(1-p)) / ln(2)

, où p dénote la proportion de particules bleues dans la moitié supérieure. L’entropie varie entre 0 et 1, la valeur maximale 1 étant atteinte pour p = 1/2.

Cette vidéo, en deux parties, montre d’abord le mouvement des particules en accéléré, puis dans une version ralentie d’un facteur 3.

The Rayleigh-Taylor instability affects a layer of heavier fluid initially situated above a layer of lighter fluid. Under the influence of gravity, the fluids tend toward a more stable configuration, with the positions of the fluids reversed. The instability involves the motion of the interface between the two fluids, which is initially in an unstable equilibrium.

In this simulation, the effect is illustrated using particles that interact via a Lennard-Jones potential. The blue particles are 10 times heavier than the red particles. The displayed temperature is a measure of the average kinetic energy, while the entropy is given by

-(pln(p) + (1-p)ln(1-p)) / ln(2)

, where p denotes the proportion of blue particles in the upper half. Entropy varies between 0 and 1, with the maximum value of 1 reached when p = 1/2.

This two-part video first shows the motion of the particles in fast motion, then in a version slowed down by a factor of 3.

An unfolding of a brief moment in the braiding of the 3-sphere

This gentle, coloured pencil drawing came out of ten months of drawing studies, several attempts at large interpretations and many long Zoom sessions with knot theorist Scott Carter. It is based on Scott’s and Saiichi Kamada’s 3-fold branched cover of the 3-sphere branched over the trefoil knot. Below is the story of its conception.

We hope to first welcome the General Assembly to Dublin, in a historic venue operated by one of the oldest charities in Ireland, where any surplus from room hire costs will be reinvested back into STEM activity for schoolchildren, and into our wider mathematical societal mission.

We will then welcome you all to Glasgow, where we will deliver the first ‘2km congress’ in which all our mathematical venues are within the 250m radius of the publicly owned Scottish Event Campus (SEC), and all our wider hotels, bars, restaurants, and social activity venues are within a 2km radius. To us, the ICM is fundamentally about people, and our specific aim is to use the geography and facilities within the city to maximise the interactions and collaborations between participants by minimising dispersal. Our ‘2km congress’ concept will create a relaxing congress experience, within a green and leafy city in which the climate in July is ideal for research mathematics.

Our bid presents an ICM that will feel welcoming, in which the most important mathematical interactions of all, those between small groups of people, will find a natural home and can flourish. Through our Developing Countries fund, and through demonstrating the viability of our 2km congress model, we also seek to provide a template for the future. Our congress model can be delivered in many more cities around the world, widening the pool of future ICM venues.

From the shared cultural experience of Dublin and Glasgow, to the mathematics, ties and connections made in the light of the long summer evenings, and above all to the importance we place on people, we very much hope to welcome you in 2030.

The International Congress of Mathematicians (ICM) is the largest conference for the topic of mathematics. It meets once every four years, hosted by the International Mathematical Union (IMU).

The Fields Medals, the IMU Abacus Medal (known before 2022 as the Nevanlinna Prize), the Gauss Prize, the Leelavati Prize (since 2010) and the Chern Medal are awarded during the congress's opening ceremony. Each congress is memorialized by a printed set of Proceedings recording academic papers based on invited talks intended to be relevant to current topics of general interest. Being invited to talk at the ICM has been called "the equivalent ... of an induction to a hall of fame".

German mathematicians Felix Klein and Georg Cantor are credited with putting forward the idea of an international congress of mathematicians in the 1890s.

The University of Chicago, which had opened in 1892, organized an International Mathematical Congress at the Chicago World's Fair in 1893, where Felix Klein participated as the official German representative.

The first official International Congress of Mathematicians was held in Zürich in August 1897. The organizers included such prominent mathematicians as Luigi Cremona, Felix Klein, Gösta Mittag-Leffler, Andrey Markov, and others. The congress was attended by 208 mathematicians from 16 countries, including more than 100 from Switzerland or Germany, around 20 from each of France, Italy, and Austria-Hungary, 13 from the Russian Empire and 7 from the US. Only four were women: Iginia Massarini, Vera Schiff [ru], Charlotte Scott, and Charlotte Wedell.

During the 1900 congress in Paris, France, David Hilbert announced his famous list of 23 unsolved mathematical problems, now termed Hilbert's problems. Moritz Cantor and Vito Volterra gave the two plenary lectures at the start of the congress.

https://en.wikipedia.org/wiki/International_Congress_of_Mathematicians

This work visualizes the surface swept by a vortex filament evolving from a square loop under the vortex filament equation, which models the self-induced motion of vortices in an ideal, inviscid fluid such as smoke rings or bubble vortices. While these flows usually produce smooth motion, the equation admits well-defined evolution even for polygonal vortex filaments with corners. These evolve periodically and generate trajectories with multifractal properties. The rendered surface reveals the geometry of this motion, exposing layered, Romanesco-like patterns that emerge from a single local rule. How much of this intricate geometry can be captured in real experiments with vortex rings remains an open and exciting question.

https://gallery.bridgesmathart.org/exhibitions/2026-jmm-art-exhibition/jiri-minarcik

https://minarcik.com/

1 = 1

1 = (-1) + 2

1 = (-2) + 3

1 = (-3) + 4

1 = (-4) + 5 ... And so on...

By adding term by term to all these equations, we get: 1 + 1 + 1 + 1 + 1 + 1 + ... = 1 + (-1) + 2 + (-2) + 3 + (-3) + 4 + (-4) + 5 + ...

In the expression on the right, all the terms cancel out in pairs, giving: 1 + 1 + 1 + 1 + 1 + 1 + ... = 0

The expression on the left, consisting of an infinite sum of terms equal to 1, tends to infinity. Thus, 0 is equal to infinity.

And yet 0 is not equal to infinity. So where is the mistake?

Tableau exposé dès l'ouverture de la salle Pi du Palais de la découverte à Paris, en 1937.

Il s'agit d'une copie réalisée à partir d'une photo d'un tableau d'Emile Borel prise lors d'un de ses cours à l'université. Le support utilisé est un panneau de bois recouvert de peinture à tableau, sur laquelle la photo était reproduite à la peinture blanche.

À l'époque où l'éducation nationale usait de la craie, les enfants qui défilaient au Palais de la découverte avaient de la craie dans les poches et écrivaient sur le tableau de Borel. On pouvait effacer leurs bêtises sans effacer la peinture blanche reproduisant l'écriture de Borel.

This painting was on display when the Pi Room at the Palais de la Découverte in Paris opened in 1937.

It is a copy based on a photograph of a painting by Émile Borel taken during one of his university lectures. The support used is a wooden panel covered with blackboard paint, onto which the photograph was reproduced in white paint.

Back when the national education system used chalk, the children who visited the Palais de la Découverte would have chalk in their pockets and write on Borel’s painting. Their scribbles could be erased without removing the white paint that reproduced Borel’s writing.

Palais de la découverte 1937

Hauteur : 2 m ; Largeur : 4 m. Peinture à tableau sur bois.

We are delighted to announce that the Fifteenth Congress of the European Society for Research in Mathematics Education (CERME15) will take place from 8th to 12th February 2027. It will be preceded by the YERME day for Young Researchers from 7th to 8th February. The conference will take place in Bratislava (Slovakia) and will be hosted by Comenius University, jointly organised by the Faculty of Mathematics, Physics and Informatics and the Faculty of Education. CERME is a distinctive conference designed to foster communication, cooperation, and collaboration among researchers in mathematics education. Its main feature is that attendees participate in one of 31 Thematic Working Groups (TWGs) over several sessions on a common theme in mathematics education research. The TWGs cover a wide range of themes and provide an opportunity for researchers to work together. Plenary events will focus on two themes in European research in mathematics education, that is, research on inclusive mathematics education and research on and with mathematics teachers.

...

In response to Jaffe and Quinn [math.HO/9307227], the author discusses forms of progress in mathematics that are not captured by formal proofs of theorems, especially in his own work in the theory of foliations and geometrization of 3-manifolds and dynamical systems.

This essay on the nature of proof and progress in mathematics was stimulated by the article of Jaffe and Quinn, “Theoretical Mathematics: Toward a cultural synthesis of mathematics and theoretical physics”. Their article raises interesting issues that mathematicians should pay more attention to, but it also perpetuates some widely held beliefs and attitudes that need to be questioned and examined.

The article had one paragraph portraying some of my work in a way that diverges from my experience, and it also diverges from the observations of people in the field whom I’ve discussed it with as a reality check.

After some reflection, it seemed to me that what Jaffe and Quinn wrote was an example of the phenomenon that people see what they are tuned to see. Their portrayal of my work resulted from projecting the sociology of mathematics onto a one-dimensional scale (speculation versus rigor) that ignores many basic phenomena.

Responses to the Jaffe-Quinn article have been invited from a number of mathematicians, and I expect it to receive plenty of specific analysis and criticism from others. Therefore, I will concentrate in this essay on the positive rather than on the contranegative. I will describe my view of the process of mathematics, referring only occasionally to Jaffe and Quinn by way of comparison.

In attempting to peel back layers of assumptions, it is important to try to begin with the right questions

...

Submission history

From: William P. Thurston

[v1] Fri, 1 Apr 1994 00:00:00 UTC

« Art du trait, rigueur de la forme, poésie des couleurs. Carrément carré, les carrés dans le carré. Quel est le nombre de carrés dans ce carré ? Le nombre d'intersections ? Pour quel résultat selon la suite de Fibonacci ? »

Donné à l'IHP par Gine Delauney en 2016.

Créateur : Gine Delauney

Date : 2013

Format : Hauteur : 80 cm ; Largeur : 80 cm Huile sur toile

Tipping points are important--and well-named--features of many complex systems, including financial and ecological systems. At a tipping point, a small change in conditions in the system can result in drastic changes overall (just as a small change in the weight of one end of a see-saw can reverse positions). Most such systems are studied using mathematical models based on collections of differential equations. The variables in the equations are related, leading to feedback in the system and potentially substantial changes, such as economic collapse. Research is now being done to recognize tipping points in hopes that something can be done before it is too late.

Some catastrophic changes have occurred when Earth’s natural systems have been disrupted. One happened over 200 million years ago when more than 90% of the planet’s species became extinct. Mathematics helped in the formulation of a new theory for the cause of the die-off: a methane-producing microbe that thrived on nickel produced by active volcanoes in Siberia. Faster-than-exponential growth in carbon levels at the time pointed to a biological trigger, and computational genomics showed that that strain of microbe came into existence at about the same time as the extinctions. This is a case in which a tipping point was in fact about the size of a point.

https://www.ams.org/publicoutreach/mathmoments/mm127-tipping-point-podcast

If you are sailing and the wind picks up, at what point will your boat capsize? Abrupt and often irreversible transitions can be observed in a wide variety of systems including ecological communities, complex disease and social networks, to name a few. These so-called ’critical transitions’ are typically brought on by a gradual change in external conditions that quietly bring the system into the vicinity of a tipping point. Despite the seemingly unpredictable nature of an approaching tipping point, there are certain mathematical features that a system exhibits on this journey, which can provide clues as to the risk of an imminent critical transition. This talk will introduce the audience to such features along with a more general discussion on the role of mathematics in understanding complex systems and their embedded tipping points. Whether it’s to prevent disasters or predict reactions, a better understanding of the science behind tipping points is essential to understanding both natural and artificial phenomena.

https://www.ted.com/talks/thomas_bury_the_mathematics_of_tipping_points

In the 1960s, the Soviet climatologist and mathematician Mikhail Budyko set out to investigate the potential future of a planet on the brink of nuclear Armageddon. He started by looking some 600 million years into the past.

Back then, some scientists claimed, the ancient planet was an iced-over snowball. Most researchers considered that a crackpot theory. Ice over the equator? Please. But Budkyo developed a mathematical model to back it up. If sea ice had been able to expand past a critical latitude, he suggested, then its reflective surface would have returned more sunlight to space. This would have kicked off an out-of-control feedback loop: The planet would cool further and ice would build up until it spread everywhere. The Earth would, in other words, tip from one equilibrium into a different one, reaching a new stable — and frozen — state.

Budyko’s investigation was motivated by a pressing question: If the global climate had tipped dramatically and catastrophically in the past, could humans tip it in the present? He and others feared what would happen if the United States and the Soviet Union launched their nuclear arsenals. “They realized, look, if we block the sun for sufficiently long, we’re going to just destroy life on the planet,” said Valerio Lucarini (opens a new tab), a statistical physicist studying the Earth’s climate at the University of Leicester. “Not even the cockroaches will survive.”

The missiles didn’t launch. But it turned out that nukes weren’t necessary for humans to tip the climate. By the time Budyko was building his Snowball Earth models, it was clear that atmospheric carbon dioxide was rising, and with it global temperatures.

Since then, mathematicians have uncovered the potential for abrupt and radical shifts in Earth’s climate — known popularly as tipping points. The loss of sea ice could cause the oceans to absorb more of the sun’s heat, crossing a threshold (opens a new tab) that kicks off runaway ice melt and rising seas. The Amazon rainforest could wither into a savanna (opens a new tab); coral reefs could bleach ghost-white (opens a new tab); a major current in the Atlantic Ocean might go slack (opens a new tab) and fail to deliver warmth to Europe, turning Scotland into Siberia.

Tipping points often capture the worst-case scenarios of climate models: the reorganization of the world we know, and the human civilization we’ve built within it, into a new equilibrium state — an unimaginable, frightening unknown.

Yet the math of tipping points is fraught with uncertainty. The Earth is certainly warming, and the effects of that warming, if left unchecked, will be dire. But tipping points are subtler phenomena. Slight changes in the assumptions a mathematical model is built on can cause tipping points to unfold very differently or even slip away entirely. And in most cases, scientists are armed with relatively little data, making it challenging to understand the chaotic nature of tippable climate systems, much less predict where they’re headed.

The uncertainty is so great that some scientists question whether it’s useful to talk about tipping points at all. Perhaps these vague apocalyptic visions only cause confusion and distraction (opens a new tab). If the scenarios are both terrifying and abstract, they might make people decide it’s just not worth the effort to fight climate change.

Mathematicians following in Budyko’s footsteps want to change the way we think about tipping points, and to translate them into meaningful information. “Natural systems don’t obey theorems — fortunately, or else the world would be a very boring place to live,” Lucarini said. But, he added, it’s urgent to find more nuanced ways to bridge the “big gray zone” between math and reality.

Mathematicians can’t change certain factors, like how little data they have to go on or how vulnerable model outcomes might be to their assumptions. But they’ve been studying tipping point behavior — in one form or another, and in all kinds of complex systems — for more than a century. In doing so, they’ve learned valuable lessons about what tipping points can and can’t reveal about Earth’s climate, and about how close to reality they should try to get. “You can live happily in the mirror world of math. It’s beautiful,” said Marten Scheffer (opens a new tab), a complex systems theorist at Wageningen University. “But applying it to reality is a minefield.”

https://www.quantamagazine.org/the-math-of-climate-change-tipping-points-20250915/

A lake that used to be clear, with a rich vegetation and a diverse aquatic life, suddenly becomes turbid, with much less vegetation and only bottom dwelling fish remaining. It turns out that the change comes from increased nutrient loading, but when the runoff leading to the nutrient inflow is reduced, the lake doesn’t become clear again – it remains murky.

A dry land area with patchy vegetation becomes completely barren after an especially dry season, but when normal rain patterns return, it remains a desert.

An entire planet that used to have varied climate zones, ranging from tropical areas to icecaps near the poles, freezes over completely, perhaps due to variations in the solar energy output, with all oceans frozen except near some thermal vents and all continents covered by thick ice sheets. When the solar output increases again, the planet remains in its frozen state.

These are examples of transitions of ecological systems past “tipping points” – the subject of a fascinating talk given by Mary Lou Zeeman on March 28 of this year in the Carriage House lecture hall of the Mathematical Association of America (MAA) in Washington, DC. Mary Lou, one of six children of the well known British mathematician Sir Christopher Zeeman, is a professor of mathematics at Bowdoin College and works on dynamical systems, with applications in ecology and biology. The Carriage House auditorium was full when she gave her talk. The audience included students, residents of the Washington area who are interested in science, and local mathematicians – just the ecological mix that the MAA lecture series tries to achieve.

There is a commonality to all these scenarios that can be described with mathematical methods from bifurcation theory. Mary Lou used the “Snowball Earth” scenario of the third example to illustrate this. According to geological evidence, this “mother of all tipping points” actually occurred on Earth not just once, but several times about 600 million years ago. Each complete glaciation lasted many millions of years and ended only when carbon dioxide in the atmosphere accumulated due to volcanic emissions to levels which were much higher than today, leading to a monstrous greenhouse effect and a rapid transition from “snowball” to “hothouse” Earth. Mary Lou presented a fairly simple energy balance model that is capable of explaining the fact that both a moderate and a frozen climate state are possible and stable on the same planet, with the same solar output. These different climate states are possible since a planet with a moderate climate tends to have a low albedo (most of the sunlight is absorbed by oceans and continents and keeps the planet warm) while a frozen planet has a high albedo (sunlight is reflected back by ice packs and snowfields, keeping the planet cold). The model is flexible enough to explain also the transitions between “snowball” and “hothouse” states. Intriguingly, the so-called Cambrian explosion, during which many of today’s animal phyla first appeared, happened not long after these snowball episodes.

Relatively simple mathematical models offer common explanations of such multiple stable states. Transitions between such states tend to be rapid and surprising, which is a scary thought. Mathematical insights can also lead to better detection mechanisms for such transitions and even suggest experiments to assess the resilience of an ecological system against random perturbations. For example, near such a transition point, such a system will return to its stable state more slowly after a perturbation, and its response to such a perturbation will also show more variance. Mary Lou specifically pointed to the work of Marten Scheffer and his co-authors on early warning signs for such critical transitions (Nature 2009, Science 2012).

The mathematical sciences therefore can contribute to decision support for managers and policymakers. The speaker suggested that when ecological systems are observed and managed for sustainability, such a goal should include resilience. In mathematical terms, this means one should not just identify stable equilibrium states but also understand the “size” of their basin of attraction and their sensitivity to changes of external parameters.

And here’s another term that I remember from this talk: Mathematical scientists should show “interdisciplinary courage” and instill this in their students. This includes not just a willingness to learn the language and problems of another discipline. In the privacy of their office, mathematicians are already used to dead ends and unsuccessful attempts before coming up with good ideas. As members of interdisciplinary research teams, they also need to risk having “bad ideas in public”. That’s a resilience that all of us should acquire.

http://mpe.dimacs.rutgers.edu/2013/04/09/mathematics-of-tipping-points/

That’s the thing about the Beatles: despite being perhaps the most beloved rock band of all time, they’re as well known for their individual personalities as their collective work.

“People have favourites: John or Paul, or perhaps George or Ringo,” says Beatles fanatic and Dalhousie Math professor Jason Brown. “And they cling strongly to what that person’s contribution was to the Beatles.”

Which can get tricky at times, given that the vast majority of the band’s songs are credited to “Lennon-McCartney” regardless of which of the two may have written it. As well, especially on earlier Beatles material, there was often a great deal of collaboration and editing done between the two songwriters — and many cases where the recollections of exactly who wrote what differ.

The question of authorship is one of the many Beatles-related ponderings that have tickled Dr. Brown’s mathematical mind over the years. Now, together with Harvard statistician Mark Glickman, he’s put together a study aiming to settle some long-standing Beatles debates — with math.

The math behind the music

The study, the results of which were first shared late last month at the annual JSM (Joint Statistical Meetings) conference, is an exercise in stylometry: using statistical techniques to determine authorship. With the help of Harvard student Ryan Song, Drs. Brown and Glickman analyzed the entire Beatles catalogue up through 1966’s Revolver, going through the scores and recordings for every song. (“There seems to be a change in styles after Revolver,” says Dr. Brown, noting as well that Lennon-McCartney songwriting partnership became a bit more disparate. “So it seemed like a good breaking point.”)

What the researchers were looking for were songwriting patterns that disguised McCartney or Lennon’s work: things that each of them did, consciously or unconsciously, that left fingerprints on the songs. The fact that there are about 70 or so Beatles songs (or portions of songs) from that era on which the authorship isn’t disputed, based on interviews over the years, gave the researchers a starting point to identify those patterns and, from there, analyze songs where there’s some debate.

Take, for example, “In My Life,” a Beatles classic from Rubber Soul which ranks 23rd on Rolling Stone’s list of the 500 greatest songs of all time. Sung by Lennon, “In My Life” is a song where the recollections of the two songwriters differ as to how it was written. Lennon (who was killed in 1980) credited McCartney with the song’s harmony part and the middle-eight, but McCartney has claimed in the past that he wrote the majority of the music himself.

The math sides with Lennon on this one: the analysis found just a 0.018 per cent probability that McCartney wrote the music for “In My Life.”

“That particular song generates a bit of emotion on both sides because it’s considered such a great song,” says Dr. Brown. “But there are other songs like ‘The Word,’ where many believe it’s a Lennon song, but our study shows it to be almost certainly written by McCartney.”

Finding the pattern

So what sort of patterns distinguish the two songwriters?

“Paul, perhaps because of his larger vocal range, often has big skips in terms of melody notes,” Dr. Brown says, offering one particular example. “For instance, in ‘Love Me Do’ — the end of the chorus — Paul sings the top line and then jumps down an octave. In ‘Eleanor Rigby’ he does the opposite thing. John doesn’t do that to the same degree.

“On the other hand, John liked, chord-wise, to go between what’s called the tonic — the base chord of the key — to the relative minor. He does it in ‘Run for Your Life,’ ‘It’s Only Love’ — it’s a typical move for him. I think it’s a sense of ambiguity in the key: is it the major or the relative minor? He liked that sort of ambiguity in his songwriting.”

This isn’t the first time Dr. Brown has applied his mathematical expertise to unlock long-argued Beatles lore. Ten years ago, he earned international attention for his work using a mathematical calculation called Fourier transform to try and identify how the oft-imitated, never-quite-duplicated opening chord of “A Hard Day’s Night” was played. (The answer, he found, was a piano note buried in the mix.) He also took to the pages of Guitar Player magazine to write about George Harrison’s solo on that same song, and how the math shows it was initially performed at half-speed and sped up to match the rest of the recording.

His newest project has been generating headlines around the world over the past week — in Canada, the U.S., Britain, Australia and beyond. (The day he spoke with Dal News he had three other interviews lined up.) And if you think that all of this effort to unlock Beatles’ secrets takes some of the fun out of rock ‘n’ roll, Dr. Brown couldn’t disagree more: he finds, in the math, even more evidence of the band’s greatness.

“For a lot of people, the music is so eternal and so fresh that it’s like it’s just yesterday — to pardon the pun,” he says with a laugh, when asked why his research on the Beatles seems to generate such attention. “The Beatles hold a special place in people’s hearts and minds. I’m still captivated by the brilliance of their songwriting, like no other band. Even after all this analysis, it still excites me.”

https://www.dal.ca/news/2018/08/09/dal-mathematician-helps-sort-out-beatles-songwriting-disputes.html

The music of most hit songs is pretty well known, but sometimes there are mysteries. One question that remained unanswered for over forty years is: What instrumentation and notes make up the opening chord of the Beatles’ “A Hard Day’s Night”? Mathematician Jason Brown–a big Beatles fan–recently solved the puzzle using his musical knowledge and discrete Fourier transforms, mathematical transformations that help decompose signals into their basic parts. These transformations simplify applications ranging from signal processing to multiplying large numbers, so that a researcher doesn’t have to be “working like a dog” to get an answer.

Brown is also using mathematics, specifically graph theory, to discover who wrote “In My Life,” which both Lennon and McCartney claimed to have written. In his graphs, chords are represented by points that are connected when one chord immediately follows another. When all songs with known authorship are diagrammed, Brown will see which collection of graphs–McCartney’s or Lennon’s–is a better fit for “In My Life.” Although it may seem a bit counterintuitive to use mathematics to learn more about a revolutionary band, these analytical methods identify and uncover compositional principles inherent in some of the best Beatles’ music. Thus it’s completely natural and rewarding to apply mathematics to the Fab 2^2^

https://www.ams.org/publicoutreach/mathmoments/mm73-beatles-podcast.

Hear how Jason Brown solved a mystery about the opening chord of the Beatles' "A Hard Day's Night."

https://www.ams.org/samplings/mathmoments/podcast-mom-beatles.mp3

Pythagoras was many wonderful things. A delirious mystic. A benevolent cult leader. A bean-hating vegetarian. A real person (maybe).

One thing he was not: the guy who gave us the Pythagorean Theorem.

So why does he get his name on it? I cry foul. I cry “no more.” I cry, “Let us band together and vote on a better name for this ancient theorem! Not because it will actually result in a name change, but because it’s a fun debate!”

Who’s with me?!

I submit for your consideration the following names, from Hambrecht and the other clever folks in his thread:

1. The Three Squares Theorem. Although we perceive it as a claim about numbers, for most mathematical cultures, this was a claim about shapes. To wit: if you affix squares to the sides of a right triangle, the two smaller areas add up to equal the largest.

2. The Babylonian Formula. Give credit where credit is due! As Hambrecht says, this name “has the hint of far-away times and places… Through millenia and continents, this piece of math connects us to strange, alien people, yet so much our equals.” He calls it “fuel for children’s imaginations.” Even more important, as an astute observer points out: it can be abbreviated as “Baby Formula.”

3. The Distance Theorem. The theorem’s most ubiquitous use is in finding distances, especially in higher dimensions.

4. The Huey Lewis Theorem. Proposed (or, pun-posed) by Susan Burns, because, and I quote: “It’s hyp to be square…or is that b-squared?”

5. The Adrakhonic Theorem, because that’s what it’s called in Neal Stephenson’s novel Anathem (which I just added to my reading list).

6. Squaring the Triangle. Olaf Doschke’s suggestion, with a ring of the famously impossible “squaring the circle.”

7. The Sum of Squares Theorem. Descriptive, clear.

8. Garfield’s Theorem. Because if we’re just naming it after a random dude who supplied a proof, why not pick an assassinated U.S. president?

9. Theorem 3-4-5. After the most famous Pythagorean triple.

10. Euclid, Book I, Proposition 47. “Like chapter and verse in the mathematical bible,” explains George Jelliss.

11. The Hypotenuse Theorem. Because it’s all about that longest side.

12. The Right Theorem. Because it’s all about that right angle. (Also, because it’s right.)

13. The Distance/Area Theorem. Because it’s all about multiple things at once.

14. The Benjamin Watson Theorem. Because of this heroic, historic tackle, brought to my attention by Fawn Nguyen in her appearance on My Favorite Theorem.

Now, we could leave it there. We could say, “This has been a fruitful discussion. Let’s call it a day!” We could say, “Obviously a random blog post isn’t going to succeed in renaming the most famous theorem in mathematics, so let’s go home and eat raisins and watch sitcom reruns like the human mediocrities we are.”

But I say no! I say it’s time for a referendum!

What say you, good people of the internet? What is the best name for this fundamental theorem of geometry?

Other ideas are, of course, welcome in the comments below.

Amalia Pica makes sculptures, installations, performances, and drawings that address a correspondingly broad array of themes. She favors found objects and commonplace materials to create her pieces, whose concerns range from language and communication, to history and politics, or to the ways in which our childhood experiences shape our adult imaginations.

As a primary school student in Argentina, Pica was taught set theory as expressed in Venn diagrams, though, as she notes, only a few years before she would not have been. She recalls that the ban of set theory by Argentina’s dictatorship in the 1970s occurred just as group assembly was also deemed subversive. Pica speculates that set theory was prohibited because it was seen as the mathematical expression of a gathering. With this work, she literally shines a light on the absurdity of the injunction. A caption on the wall provides historical context for the work.

https://www.coleccioncisneros.org/collections/gallery/contemporary/artwork/venn-diagrams-under-the-spotlight

Pica's interest in the relationship between text and image is evident in Venn Diagrams (under the Spotlight), which consists of two colored circles of light cast from theater spotlights to form a Venn diagram. The Argentine government banned this diagram from being taught in classrooms in the 1970s, as it was thought to be an incendiary model of social collaboration. "The two circles of light are nothing but forms until the caption situates them historically, cluing you to their perception as subversive in the context of Argentine dictatorship in the 1970s. I’m interested in the ideas that we project onto images and objects: how they resist as much as accommodate them."

https://en.wikipedia.org/wiki/Amalia_Pica

A Venn diagram is a widely used diagram style that shows the logical relation between sets, popularized by John Venn (1834–1923) in the 1880s. The diagrams are used to teach elementary set theory, and to illustrate simple set relationships in probability, logic, statistics, linguistics and computer science. A Venn diagram uses simple closed curves on a plane to represent sets. The curves are often circles or ellipses.

History

Venn diagrams were introduced in 1880 by John Venn in a paper entitled "On the Diagrammatic and Mechanical Representation of Propositions and Reasonings" in the Philosophical Magazine and Journal of Science, about the different ways to represent propositions by diagrams. The use of these types of diagrams in formal logic, according to Frank Ruskey and Mark Weston, predates Venn but are "rightly associated" with him as he "comprehensively surveyed and formalized their usage, and was the first to generalize them".

Diagrams of overlapping circles representing unions and intersections, such as Borromean rings, were already in frequent use in the Middle Ages. However, the extent to which these types of diagrams can be considered precursors to Venn diagrams is disputed. Euler diagrams, which are similar to Venn diagrams but do not necessarily contain all possible unions and intersections, were named after the mathematician Leonhard Euler in the 18th century. However, these diagrams, which are considered the precursors of Venn diagrams, can also be clearly traced back to the 16th century. Pioneers in this tradition of Euler diagrams included Erhard Weigel (1625–1699) and his students Johann Christoph Sturm (1635-1703) and Gottfried Wilhelm Leibniz (1646–1716). Christian Weise (1642–1708) is also worth mentioning, whose student Johann Christian Lange worked intensively on these diagrams. Euler further developed these diagrams, and Immanuel Kant (1724–1804) and his students popularized them in the 19th century.

Venn did not use the term "Venn diagram" and referred to the concept as "Eulerian Circles". He became acquainted with Euler diagrams in 1862 and wrote that Venn diagrams did not occur to him "till much later", while attempting to adapt Euler diagrams to Boolean logic. In the opening sentence of his 1880 article Venn wrote that Euler diagrams were the only diagrammatic representation of logic to gain "any general acceptance".

Venn viewed his diagrams as a pedagogical tool, analogous to verification of physical concepts through experiment. As an example of their applications, he noted that a three-set diagram could show the syllogism: 'All A is some B. No B is any C. Hence, no A is any C.'

Charles L. Dodgson (Lewis Carroll) includes "Venn's Method of Diagrams" as well as "Euler's Method of Diagrams" in an "Appendix, Addressed to Teachers" of his book Symbolic Logic (4th edition published in 1896). The term "Venn diagram" was later used by Clarence Irving Lewis in 1918, in his book A Survey of Symbolic Logic.

In the 20th century, Venn diagrams were further developed. David Wilson Henderson showed, in 1963, that the existence of an n-Venn diagram with n-fold rotational symmetry implied that n was a prime number. He also showed that such symmetric Venn diagrams exist when n is five or seven. In 2002, Peter Hamburger found symmetric Venn diagrams for n = 11 and in 2003, Griggs, Killian, and Savage showed that symmetric Venn diagrams exist for all other primes. These combined results show that rotationally symmetric Venn diagrams exist, if and only if n is a prime number.

Venn diagrams and Euler diagrams were incorporated as part of instruction in set theory, as part of the new math movement in the 1960s. Since then, they have also been adopted in the curriculum of other fields such as reading. With the work of Sun-Joo Shin, Venn diagrams have been recognized as a logical system equivalent to symbolic logic. Similar methods were then adopted in mathematics and subsequently in computer science.

https://en.wikipedia.org/wiki/Venn_diagram

When doing research in computer vision or image processing, it's useful to have a test image or two. Writing programs that reduce noise, alter brightness, or enhance edges is all very well and good, but without test images, we can't know if they work. Early on in vision science, the acquisition of images was hard, and there were a handful of images everyone used. This was partly due to expediency (not everyone had access to a scanner) and partly due to comparability (we want to be able to see the results of each algorithm on the same image or set of images). Today, nearly everyone has a digital camera as part of the device in their pocket, in the 70s and 80s such devices simply didn't exist.

At the very beginning of the discipline that's now become computer vision, sometime in the early 70s - probably early 1973 - a researcher was looking for a test image. Alexander Sawchuk (now a professor at the University of Southern California) scanned one for this researcher: it was the centrefold of a Playboy magazine, and that scan has now entered into our scientific culture in a way the originators could never have imagined. The woman is wearing a floppy hat, and is gazing over her bare shoulder towards the camera. As a 512 pixel squared image, the woman's body is cropped at the shoulders; the suggestion of nudity is there, but the image itself is decidedly safe for work. Her name is Lena Söderberg. The researcher’s name is lost to internet history, as is the name of the person who actually brought the porn to work.

Lena (sometimes spelled Lenna, which is the anglicisation of her name used by Playboy) has been called “The First Lady of the Internet”. Her image has been printed on a chip, shrunk to the size of a human hair, blurred, sharpened, and undoubtedly enhanced. She appeared on the cover of the journal Optical Engineering in 1991, and the model herself was invited to a conference of the Image Science and Technology society in 1997. Her picture is on the wall, tastefully framed, in at least one reputable computer vision lab. For computer vision researchers, her image is everywhere.

In defence of the use of Lena, Hutchinson said “... the image mixes areas of light and dark, fuzzy and sharp, detailed and flat—providing a stiff test for an image processing algorithm”. (I am unsure whether the word stiff in that sentence is a deliberate double-entendre. In a sense, I hope it is: if we're going to have scholarly articles about the use of pornography in our science, let them at least try to be funny.) Interestingly, Hutchinson's article talks about issues with Lena, from the copyright (Playboy were, for a time, not best pleased by the widespread distribution of one of their images) to the equalities issues. Many journal editors, whilst uncomfortable with the idea of such images appearing in their journals, were even less comfortable with the idea of banning such images.

But, as Hutchinson's article was published in 2001, we can surely assume that things have changed – that was over ten years ago. Our discipline is becoming more aware of the issues of minorities: the fact that women in technology can feel isolated is something that everyone working in the field probably knows by now. The usage of Lena must have gone down. The controversial nature of the image (there are discussion threads on the topic stretching back years, on some social media platforms) must make people think twice. Surely people realise that women, as a minority in the field, might be put off by statements like “... the Lena image is a picture of an attractive woman. It is not surprising that the (mostly male) image processing research community gravitated toward an image that they found attractive.”

Right?

Wrong. At ISISPA2013, the IEEE International Symposium on Image and Signal Processing and Analysis, during a single day of the talks I saw her six times. She's included in the standard OpenCV download (at least twice – as a test image in the both the c++ and python directories). On the machine I'm using to type this blog post, despite never having used Lena in a piece of research, and never having deliberately downloaded her, I've just done a search, and I'm amazed. I've got 18 copies (excluding cached thumbnails and the directory of files I've gathered whilst researching this article). It seems that whenever you download a piece of vision code, Lena still comes along for the ride.

As a vision researcher, I'm used to being the odd one out: at the first workshop I ever spoke at I was the only woman in the room. Generally, that doesn't bother me. I'm not entirely sure what I think about Lena though – having decided she would be a good topic for an article, I read up on the history, and to be honest, I still find it a bit bizarre. But despite my avowedly feminist stance, I’m somehow unable to get that annoyed about it.

The fact that there's a historic Playboy image at pretty much every conference I go to, and on the walls of my colleague's labs, and downloaded with every single image processing library I use, well... on the one hand, it's part of that drip-drip-drip of strangeness that comes from working in a male-dominated field, where the topics of conversation and the general attitude can be a little disconcerting. But on the other hand, with changing cultural attitudes, and the effect the internet has had on pornography, the entire centrefold (yes, you can easily find it online if you look) seems very tame indeed by today's standards. And the crop that is used in image processing research is, well... I've developed quite an affection for the picture. It's one of the quirks of computer science. So when I was asked what picture we should use to illustrate this blog post, there was only one choice.

Of course, we really had to use a Lena image on this page (although subject to Gaussian Blur with a 60px by 60px kernel). You can probably still see that it’s just a shoulder and a head in a floppy hat: what’s all the fuss about? Well, we all know what it represents and we all know where it comes from. And sometimes, there’s more to an image than meets the eye.

https://www.software.ac.uk/blog/how-photo-playboy-became-part-scientific-culture

In image processing, a Gaussian blur (also known as Gaussian smoothing) is the result of blurring an image by a Gaussian function (named after mathematician and scientist Carl Friedrich Gauss).

It is a widely used effect in graphics software, typically to reduce image noise and reduce detail. The visual effect of this blurring technique is a smooth blur resembling that of viewing the image through a translucent screen, distinctly different from the bokeh effect produced by an out-of-focus lens or the shadow of an object under usual illumination.

Mathematically, applying a Gaussian blur to an image is the same as convolving the image with a Gaussian function. This is also known as a two-dimensional Weierstrass transform. By contrast, convolving by a circle (i.e., a circular box blur) would more accurately reproduce the bokeh effect.

Since the Fourier transform of a Gaussian is another Gaussian, applying a Gaussian blur has the effect of reducing the image's high-frequency components; a Gaussian blur is thus a low-pass filter.

A halftone print rendered smooth through Gaussian blur

https://en.wikipedia.org/wiki/Gaussian_blur

We present the Rhythm Circle, an interactive open-source web environment that allows users to generate and manipulate rhythms using the circular representation. In this environment, three rhythms can be looped simultaneously, starting and ending at the same time via three concentric circles, each corresponding to a drum element: a snare drum, a kick drum and a hi-hat. The rotation speed, corresponding to the tempo, is adjustable, and there is an option to export the generated rhythms in midi format. The platform includes preset rhythms, offering binary and ternary patterns that correspond, in this case, to circle subdivisions into 16 and 12 equal parts. One of the advantages of this web environment is the possibility of transforming the different rhythms, not only by activating or deactivating the onsets on the three respective circles, but also by applying musical transformations. The first musical transformation is the change of the subdivision of a circle, for example, from 16 to 17. The second musical transformation is the rotation of a circle, which corresponds to a time shifting of the musical rhythm. A usage counter was implemented, and in just a few months, the platform recorded over 30,000 visits from users worldwide

https://hal.science/hal-05054439/file/2025_ICMC.pdf

I recently developed the Rhythm Circle web application, an open-source environment designed to generate and manipulate rhythms using the circular representation.

This provides an original visualization of rhythm and might enable musicians, both professional and amateur, to explore new musical concepts based on the circular representation.

https://www.paullascabettes.com/home

“Circles in a Circle” is a compact and closed composition. Kandinsky began a thoughtful study of the circle as an artistic unit starting from this painting. In his letter to Galka Scheyer he wrote, “it is the first picture of mine to bring the theme of circles to the foreground.” The outer black circle, as if the second frame for a picture, encourages us to focus on the interaction between the inside circles, and two intersecting diagonal stripes enhance the effect, adding a perspective to the composition.

1923

Geometric abstraction

Oil on canvas

38.9 × 37.6" (98.7 × 95.6 cm)

https://www.wassilykandinsky.net/work-247.php

A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. The distance between any point of the circle and the centre is called the radius. The length of a line segment connecting two points on the circle and passing through the centre is called the diameter. A circle bounds a region of the plane called a disc.

The circle has been known since before the beginning of recorded history. Natural circles are common, such as the full moon or a slice of round fruit. The circle is the basis for the wheel, which, with related inventions such as gears, makes much of modern machinery possible. In mathematics, the study of the circle has helped inspire the development of geometry, astronomy and calculus.

Prehistoric people made stone circles and timber circles, and circular elements are common in petroglyphs and cave paintings. Disc-shaped prehistoric artifacts include the Nebra sky disc and jade discs called Bi.

The Egyptian Rhind papyrus, dated to 1700 BCE, gives a method to find the area of a circle. The result corresponds to ⁠256/81⁠ (3.16049...) as an approximate value of π.

Book 3 of Euclid's Elements deals with the properties of circles. Euclid's definition of a circle is:

A circle is a plane figure bounded by one curved line, and such that all straight lines drawn from a certain point within it to the bounding line, are equal. The bounding line is called its circumference and the point, its centre.

— Euclid, Book I, Elements

In Plato's Seventh Letter there is a detailed definition and explanation of the circle. Plato explains the perfect circle, and how it is different from any drawing, words, definition or explanation. Early science, particularly geometry and astrology and astronomy, was connected to the divine for most medieval scholars, and many believed that there was something intrinsically "divine" or "perfect" that could be found in circles.

In 1880 CE, Ferdinand von Lindemann proved that π is transcendental, proving that the millennia-old problem of squaring the circle cannot be performed with straightedge and compass.

With the advent of abstract art in the early 20th century, geometric objects became an artistic subject in their own right. Wassily Kandinsky in particular often used circles as an element of his compositions.

https://en.wikipedia.org/wiki/Circle

Marguerite's Theorem (French: Le Théorème de Marguerite) is 2023 French-Swiss drama film co-written and directed by Anna Novion [fr]. It is about a female mathematics student at ENS whose career is upended when an error is discovered in her work.[4][5] It premiered on 22 May 2023 at the 76th Cannes Film Festival. It was distributed in France on 1 November 2023.

Marguerite is a young and brilliant mathematician, the only girl in her class at the ENS, entirely devoted to her passion. The day an error is discovered in her thesis, she is devastated. In a dizzy spell, she leaves the school, wiping out the past. She then dives into the real world, discovers autonomy, befriends the young Noa, and has sex for the first time. Matured by her experiences, it is in this new momentum that she manages to find a correct proof of her theorem.

https://en.wikipedia.org/wiki/Marguerite%27s_Theorem

Goldbach's conjecture is one of the oldest and best-known unsolved problems in number theory and all of mathematics. It states that every even natural number greater than 2 is the sum of two prime numbers.

Letter from Goldbach to Euler dated 7 June 1742 (Latin–German)

The conjecture has been shown to hold for all natural numbers less than 4×10^18^, but remains unproven despite considerable effort.

https://en.wikipedia.org/wiki/Goldbach%27s_conjecture

According to Hardy (1999, p. 19), "It is comparatively easy to make clever guesses; indeed there are theorems, like 'Goldbach's Theorem,' which have never been proved and which any fool could have guessed." Faber and Faber offered a $1000000 prize to anyone who proved Goldbach's conjecture between March 20, 2000 and March 20, 2002, but the prize went unclaimed and the conjecture remains open.

https://mathworld.wolfram.com/GoldbachConjecture.html

Le Théorème de Marguerite (Marguerite’s Theorem), as explained by Anna Novion

What is the starting point of your feature film?

I remember a period when I was seriously ill. I had to stay home for a long time to get better. I was around 20 and remember being isolated from the world. When I had recovered, I felt that a distance had been created with the carefree attitude of young people of my age. When I start writing a feature film, I always rely on an emotion I’ve felt and the idea is to transpose it. The atmosphere of elite schools and the way students live in a vacuum, devoted to their work, seemed a good way of talking about this isolation.

Why did you choose to focus on the world of mathematics?

I first met with literary scholars from elite institutions. But they didn’t strike me as all that isolated. On the contrary, they were rather open to the world. I found mathematics students far more inspiring. And then there was my encounter with Ariane Mézard, which was decisive. She’s one of the rare French women mathematicians and the two of us really hit it off. The way she presented mathematics to me really provoked something within me. She discussed it in a really artistic way. She spoke to me of everything that animates me in my profession.

Which is?

Passion, necessity, difficulty, tenacity, relentlessness… I realized that there was a real parallel to be drawn between mathematics and artistic creation. What connects mathematics and directing is the risk and the passion that sometimes leads us to work for years without knowing if our work is going to amount to anything. It’s a very personal film that evokes my relationship to creation. I also wanted to talk about what it’s like to be a woman in a masculine milieu. I felt this pressure connected to the fact of being a sort of exception that pushes us to have to prove that we belong. Marguerite, my character, considers herself as a kind of anomaly. She feels this competition all the more so as she is the only woman.

How did you prepare for the film?

I spent four months at the École Normale Supérieure school meeting with mathematicians. I didn’t want them to say that I had just skimmed over the subject. With this ambition: how to make mathematicians at work captivating on the screen when you’re not from this world. The idea was to show to what point doing mathematics means working all the time.

How did you collaborate with Ella Rumpf?

She worked four hours a week over four months with Ariane Mézard. We quickly realized that it was useless for her to have the mathematics that she was going to write explained to her. She thus learned the formulas by heart. We needed it to appear extremely natural on the screen. It’s an American-style role, where we worked a lot on the posture of the character: her way of walking, her way of expressing herself, the speed of her speech and the singularity of her gaze on others.

Visually, light floods the crescendo of the film. Can you explain this aesthetic choice?

The character goes through many stages on the way to her flourishing. There are also several stages in the light and in the framing of shots. Her life is very structured at the beginning, so the shots are rather geometric and monochromatic. And little by little, with the irrational in her life that she’s going to discover, including the world of feelings, light and colour appear.

https://www.festival-cannes.com/en/2023/le-theoreme-de-marguerite-marguerite-s-theorem-as-explained-by-anna-novion/

This picture displays the process for the 64 first prime numbers:

{2,3,5,7,11,13,17,19,23,29,,31,37,41,43,47,53,59,61,67,71,,73,79,83,89,97,101,103,107,109,113,,127,131,137,139,149,151,157,163,167,173,,179,181,191,193,197,199,211,223,227,229,,233,239,241,251,257,263,269,271,277,281,,283,293,307,311}

-top row of the picture- with the following colors regarding the numbers:

0 = Dark Yellow, 1 = Cyan, 2 = Light Yellow,

when all other numbers -{3,5,7,11,...}- are Dark Red...

According to the the Gilbreath Conjecture the left-hand side column must be Cyan ('1') except the upper square that is Light Yellow ('2', the first prime number).

http://www.lactamme.polytechnique.fr/Mosaic/images/GILB.22.D/display.html

Gilbreath's conjecture is a conjecture in number theory regarding the sequences generated by applying the forward difference operator to consecutive prime numbers and leaving the results unsigned, and then repeating this process on consecutive terms in the resulting sequence, and so forth. The statement is named after Norman L. Gilbreath who, in 1958, presented it to the mathematical community after observing the pattern by chance while doing arithmetic on a napkin. In 1878, eighty years before Gilbreath's discovery, François Proth had published the same observations.

https://en.wikipedia.org/wiki/Gilbreath's_conjecture

Proth-Gilbreath Conjecture

Beat the Andrew Odlyzko Record G(Pi(1.1x10^14^))=635 (1993)

G(Pi(10^14^))=693 on october 5 2025 at 20:59:18

G(Pi(1.145x10^14^))=744 on 10/27/2025 at 12:10:26

1-Definition:

This conjecture was stated in 1958 by Norman L. Gilbreath but published earlier in 1878 by François Proth. It is related to the prime numbers and to the sequences generated by taking the absolute value of the difference between each prime number and its successor and then repeating this process ad infinitum:

The conjecture states that the first value of each line is 1 (except the first one where it is a 2 -the only even prime number-) and was studied by Andrew Odlyzko in 1993. He did check it for all prime numbers less than 10^13^.

On sunday 10/05/2025 20:45 (Paris time, France) I did succeed to check it up to 10^14^ and on tuesday 10/07/2025 02:25 pm (East Time), Simon Plouffe (Canada) did the same. Moreover he did confirm the maximal value (693) of the G(Pi(x)) function with x ∈ [2,10^14^] that was anticipated on 09/25/2025.

2-The Theory:

Obviously one cannot check the Proth-Gilbreath Conjecture for there is an infinity of prime numbers. Only a demonstration can solve this unless a counter-example is discovered, that is a line not starting with a '1' (except the first one).

Let pn be the prime numbers:

p1 = 2

p2 = 3

p3 = 5

etc...

Let's define the suite dk(n):

d0(n) = pn for all n such as n > 0

dk(n) = |dk-1(n) - dk-1(n+1)| for all k such as k > 0 and for all n such as n > 0

Then one must check that:

dk(1) = 1 for all k such as k > 0

Due to the finite limits of computers, it is impossible to exhaustively check this property. Fortunately Andrew Odlyzko noticed that if for a certain N there exists K such that:

dK(1) = 1

dK(n) ∈ {0,2} for all n such as 0 < n < N+1

then:

dk(1) = 1 for all k such as K-1 < k < N+K

Let's call G(N) the smallest k (if it exists) such that:

dj(1) = 1, 0 < j < k+1

dk(n) ∈ {0,2} for all n such as 0 < n < N+1

A trivial reasoning shows that G(N) does exist for all N and that the process can be stopped as soon as there are only '0's, '1's and '2's on the current line of rank k. For example:

More at https://www.lactamme.polytechnique.fr/Mosaic/descripteurs/GilbreathConjecture.01.Ang.html

Verification and attempted proofs

Several sources write that, as well as observing the pattern of Gilbreath's conjecture, François Proth released what he believed to be a proof of the statement that was later shown to be flawed. However, Zachary Chase disputes this, writing that although Proth called the observation a "theorem", there is no evidence that he published a proof, or false proof, of it.

[...] but the conjecture remains an open problem. Instead of evaluating n rows, Odlyzko evaluated 635 rows and established that the 635th row started with a 1 and continued with only 0s and 2s for the next n numbers. This implies that the next n rows begin with a 1.

Simon Plouffe has announced a computational verification for the primes up to 10^14^.

https://en.wikipedia.org/wiki/Gilbreath's_conjecture

In probability theory, Buffon's needle problem is a question first posed in the 18th century by Georges-Louis Leclerc, Comte de Buffon :

The a needle lies across a line, while the b needle does not.

Suppose we have a floor made of parallel strips of wood, each the same width, and we drop a needle onto the floor. What is the probability that the needle will lie across a line between two strips?

Buffon's needle was the earliest problem in geometric probability to be solved; it can be solved using integral geometry. The solution for the sought probability p, in the case where the needle length l is not greater than the width t of the strips, is p = 2/π * l/t .

This can be used to design a Monte Carlo method for approximating the number π, although that was not the original motivation for de Buffon's question. The seemingly unusual appearance of π in this expression occurs because the underlying probability distribution function for the needle orientation is rotationally symmetric.

https://en.wikipedia.org/wiki/Buffon%27s_needle_problem

Needle : A really weird way to find Pi

Discovery of Calculus in the 17th century opened a new window for estimating ⁡π more precisely. Since then, we've seen plenty of methods to do it. Some of them are pretty weird but simple enough to understand. Buffon's Needle Problem is one of them. This problem was first posed by the French Naturalist Georges-Louis Leclerc, Comte de Buffon in 1733. It goes like this:

Suppose you are given a floor with equally spaced parallel lines on it. If we drop a needle on the floor, what is the probability that the needle will lie across a line?

Let's say you've thrown n needles (or a single needle for n times or whatever). m number of needles cross/intersect with a line (or m times if you have one needle). The probability P that a needle crosses the line:

P = m/n

Buffon's Needle Problem. The green needles are the ones crossing the parallel lines

It may seem a bit odd how ⁡π is related to this problem. Hang on tight. I'll explain.

To solve this problem, we'll require some basic knowledge of Probability and Integral Calculus. Let's assume that the spacing between two consecutive parallel lines is D, the length of the needle is L. Now, let the distance from the middle point of a needle on the floor to its closest line be x and the acute angle between the line and needle be θ (See figure above). All needles are of equal length.

Since we are considering x to be the smallest distance from the center of the needle to any one of the lines, it can vary only within 0 and D/2. And since θ is an acute angle, it can take any value between 0 and π/2. Using trigonometry, we can find out that the vertical component of length L is L/2 * sin(θ). The needle will cross the line if x is less than L/2 * sin(θ).

Okay. Now let's visualize possible outcomes in a graph. Let the horizontal axis be θ and the vertical axis be x. So a rectangle with sides π/2 and D/2 represents all possible outcomes in this experiment.

Horizontal axis refers to θ and vertical axis refers to x. The rectangle represents all possible outcomes

Now I am going to shade all those points in the rectangle that represent the events where a needle crosses a line according to the two conditions above. Think about it. It's just the area under the curve D/2 * sin(θ) where 0 ≤ θ ≤ π .

The shaded area indicates the probability that a needle will cross a line

The probability we are looking for would be the ratio of the area under the sine curve and the rectangle.

Area of the rectangle:

Area under the sine curve:

Ratio of the areas:

Rearranging, we get

In 1901, Italian mathematician Mario Lazzarini performed Buffon's needle experiment and, tossing a needle 3408 times, obtained π up to six decimal points correctly!

https://tahsin314.github.io/writings/maths/buffons_needle.html

Finally, since Buffon’s needle problem is interested in estimating π – and since I have mostly focused on the error in this method instead – I thought it might be fun to take a look at the the empirical distribution of $\hat{\pi}$ for one of my experimental setups. The following figure shows how this distribution varies as a function of the number of needles thrown for the l = h case. The data are plotted using a dot-plot, while box-plots have been overlaid on top.

https://www.paulkepley.com/2021-05-13-AsymptoticBuffon/

Here's an interesting quote from the correspondence of Isaac Newton:

This is from the 2nd letter that Newton wrote to Leibniz (via Oldenburg) in 1677. He was responding to some questions from Leibniz about his method of infinite series and came close to revealing his "fluxional method" (i.e., calculus), but then decided to conceal it in the form of an anagram. After describing his methods of tangents and handling maxima and minima, he wrote

The foundations of these operations is evident enough, in fact; but because I cannot proceed with the explanation of it now, I have preferred to conceal it thus: 6accdae13eff7i3l9n4o4qrr4s8t12ux. On this foundation I have also tried to simplify the theories which concern the squaring of curves, and I have arrived at certain general Theorems.

The anagram expresses, in Newton's terminology, the fundamental theorem of the calculus: "Data aequatione quotcunque fluentes quantitates involvente, fluxiones invenire; et vice versa", which means "Given an equation involving any number of fluent quantities to find the fluxions, and vice versa." Arranging the characters in his Latin sentence in alphabetical order (and assuming he counted the dipthong "ae" as a separate character, and u's and v's are counted as the same character), the number of occurrences of each character are as follows

This agrees with Newton's anagram

except that I count nine t's instead of eight. Possibly Newton's original Latin spelling used one fewer t's, although I can't see which one of them could plausibly be omitted. It could also be that the anagram has been incorrectly copied, but it agrees with the version in both Westfall's and Christianson's biographies, as well as the transcription of Newton's letter contained in Calinger's Classics of Mathematics. Another possibility is that Newton simply mis-counted. This isn't as implausible as it might seem at first, since there is a well-known psychological phenomenon of overlooking the second letter in short connective words (like the f in "of") when quickly counting the number of occurrences of a certain letter in a string of text. It's very easy, when counting the number of t's in Newton's Latin phrase, to neglect the "t" in the word "et".

Ironically, neither Leibniz nor Newton had published anything on calculus at the time this letter was exchanged, although both are believed to have been in possession of the calculus, so if Newton had just come right out with a complete and explicit statement of his calculus he would have placed Leibniz in a very difficult position, and would have established his own priority beyond doubt (since the letter passed through Oldenburg). Instead, Newton's very protectiveness and secrecy caused him to lose whatever unambiguous claim to priority he might have had (and led to an acrimonious priority dispute that embittered both his and Leibniz's later lives).

On a very deep level it was natural for Newton to express himself in anagrams, because he seems to have regarded "contrived obscurity" (in Domson's words) as an essential feature of God's design for the world, and Newton adopted this mode of operation in his own work. Recall that he said he had made the Principia "designedly abstruse" (to avoid being baited by "little smatterers" in mathematics). Even more pointedly, Newton spent many years attempting to interpret the prophesies in the Bible, which he believed were presented in deliberately disguised form so that their meaning could only be inferred by solving them like puzzles. Interestingly, he had disdain for people who tried to unravel prophesies of future events. In his view, this was misguided and doomed to failure. He believed the encoded prophesies were intended to be understood only after the fact. In his "Observations upon the Prophecies of the Apocalypse of St. John" he wrote

The folly of interpreters has been to foretell times and things by this Prophecy, as if God designed to make them Prophets. By this rashness they have not only exposed themselves, but brought Prophecy also into contempt. The design of God was much otherwise. He gave this [Revelations] and the Prophecies of the Old Testament, not to gratify men's curiosities by enabling them to foreknow things, but that after they were fulfilled they might be interpreted by the event, and [that] his own Providence, not the Interpreters, be then manifested thereby to the world. For the event of things predicted many ages before, will then be a convincing argument that the world is governed by Providence.

We might say that Newton saw the biblical prophesies serving exactly the same function as his anagram on fluxions and fluents, i.e., it is presented in a form that cannot be interpreted to reveal its meaning before the fact, but after the facts have transpired and the solution of the puzzle is found, it can serve as irrefutable evidence of the Providence of the creator.

https://www.mathpages.com/home/kmath414/kmath414.htm

In musical tuning and harmony, the Tonnetz (German for 'tone net') is a conceptual lattice diagram representing tonal space first described by Leonhard Euler in 1739. Various visual representations of the Tonnetz can be used to show traditional harmonic relationships in European classical music.

A modern rendering of the Tonnetz. The A minor triad is in dark blue, and the C major triad is in dark red. Interpreted as a torus, the Tonnetz has 12 nodes (pitches) and 24 triangles (triads).

Euler's Tonnetz

Euler's Tonnetz, pictured at left, shows the triadic relationships of the perfect fifth and the major third: at the top of the image is the note F, and to the left underneath is C (a perfect fifth above F), and to the right is A (a major third above F). Gottfried Weber, Versuch einer geordneten Theorie der Tonsetzkunst, discusses the relationships between keys, presenting them in a network analogous to Euler's Tonnetz, but showing keys rather than notes. The Tonnetz itself was rediscovered in 1858 by Ernst Naumann in his Harmoniesystem in dualer Entwickelung., and was disseminated in an 1866 treatise of Arthur von Oettingen. Oettingen and the influential musicologist Hugo Riemann (not to be confused with the mathematician Bernhard Riemann) explored the capacity of the space to chart harmonic modulation between chords and motion between keys. Similar understandings of the Tonnetz appeared in the work of many late-19th century German music theorists.

The appeal of the Tonnetz to 19th-century German theorists was that it allows spatial representations of tonal distance and tonal relationships. For example, looking at the dark blue A minor triad in the graphic at the beginning of the article, its parallel major triad (A-C♯-E) is the triangle right below, sharing the vertices A and E. The relative major of A minor, C major (C-E-G) is the upper-right adjacent triangle, sharing the C and the E vertices. The dominant triad of A minor, E major (E-G♯-B) is diagonally across the E vertex, and shares no other vertices. One important point is that every shared vertex between a pair of triangles is a shared pitch between chords - the more shared vertices, the more shared pitches the chord will have. This provides a visualization of the principle of parsimonious voice-leading, in which motions between chords are considered smoother when fewer pitches change. This principle is especially important in analyzing the music of late-19th century composers like Wagner, who frequently avoided traditional tonal relationships.

https://en.wikipedia.org/wiki/Tonnetz

Harmonic Trajectories in the Tonnetz

Introduction

The Neo-Riemannian Tonnetz is a way of visualizing musical relationships between chords. It was developed by music theorists to help understand how chords can transition smoothly from one to another.

Imagine a grid or network of interconnected points. Each point represents a different chord. The horizontal lines connect chords that are closely related, while the vertical lines connect chords that share similar tonal qualities.

The Tonnetz is based on the idea that chords can be transformed or changed into one another through small movements. These transformations are represented by diagonal lines on the Tonnetz. For example, a chord can be transformed into another chord by changing one note at a time, moving in a specific direction on the grid.

By studying the Tonnetz, musicians and theorists can analyze chord progressions and see how different chords are related to each other. It provides a visual representation of the harmonic possibilities and helps to explain the underlying structure of music.

In simple terms, the Neo-Riemannian Tonnetz is a grid that shows how chords in music are connected and can be transformed smoothly from one to another. It helps musicians and theorists understand how chords fit together and how they can create pleasing transitions in music.

https://emmanouil-karystinaios.github.io/post/tonnetz/