The hottest Mathematics Substack posts right now

And their main takeaways

Meta: 5,000

lcamtuf’s thing • 8366 implied HN points • 27 Feb 25

Reaching 5,000 subscribers is a big deal for a project that went against the usual trends. It's great to see growth, even if it seems small compared to others.
Writing a newsletter is unique because you don't get much direct feedback from readers. It's interesting to see who signs up or leaves but hard to know what they really think.
Three articles worth revisiting cover complex topics: discrete Fourier transforms, fractals, and core concepts in electronic circuits. They offer in-depth discussions that are easy to understand, even for beginners.

AlphaGeometry2: Impressive accomplishment, but still a long path ahead

Marcus on AI • 3161 implied HN points • 17 Feb 25

🕹 Technology Artificial Intelligence Machine Learning Computer Science Mathematics Research

AlphaGeometry2 is a specialized AI designed specifically for solving tough geometry problems, unlike general chatbots that tackle various types of questions. This means it's really good at what it was built for, but not much else.
The system's impressive 84% success rate comes with a catch: it only achieves this after converting problems into a special math format first. Without this initial help, the success rate drops significantly.
While AlphaGeometry2 shows promising advancements in AI problem-solving, it still struggles with many basic geometry concepts, highlighting that there's a long way to go before it can match high school students' understanding in geometry.

All paths point downhill

arg min • 218 implied HN points • 31 Oct 24

🕹 Technology Algorithms Optimization Data science Machine Learning Mathematics

In optimization, there are three main approaches: local search, global optimization, and a method that combines both. They all aim to find the best solution to minimize a function.
Gradient descent is a popular method in optimization that works like local search, by following the path of steepest descent to improve the solution. It can also be viewed as a way to solve equations or approximate values.
Newton's method, another optimization technique, is efficient because it converges quickly but requires more computation. Like gradient descent, it can be interpreted in various ways, emphasizing the interconnectedness of optimization strategies.

Rubik’s abstract polytopes

A Piece of the Pi: mathematics explained • 18 implied HN points • 03 Mar 25

🔬 Science Mathematics Topology Geometry Puzzles Education

Rubik's Cube can be made in different shapes, like a tetrahedron or dodecahedron, instead of just the classic cube. These variations have their own names, like the Megaminx for the dodecahedron.
A new study explains how to think about these puzzles in more dimensions, counting how many different ways they can be turned or rearranged. This includes understanding shapes like the hypercube.
The math behind a Rubik's Cube shows interesting patterns, like counting sticks and pieces, and this can help us understand all kinds of shapes and designs better.

Basic Linear Algebra Subprogramming

arg min • 178 implied HN points • 29 Oct 24

🕹 Technology Computing Mathematics Optimization Data science Algorithms

Understanding how optimization solvers work can save time and improve efficiency. Knowing a bit about the tools helps you avoid mistakes and make smarter choices.
Nonlinear equations are harder to solve than linear ones, and methods like Newton's help us get approximate solutions. Iteratively solving these systems is key to finding optimal results in optimization problems.
The speed and efficiency of solving linear systems can greatly affect computational performance. Organizing your model in a smart way can lead to significant time savings during optimization.

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A use-theory of testing

arg min • 436 implied HN points • 24 Oct 24

🔬 Science Statistics Research Mathematics Data Analysis Theory

Statistical tests are designed to help separate real signals from random noise. It's not just about understanding what they mean, but what they can do in practical situations.
Many people misuse statistical tests, which can lead to misunderstandings about their purpose. Communities should establish clear guidelines on how to use these tests correctly.
The main function of statistical tests is to regulate opinions and decisions in various fields like tech and medicine. They help ensure that important standards are met, rather than just preventing errors.

Great scientists follow intuition and beauty, not rationality

The Intrinsic Perspective • 33817 implied HN points • 30 Dec 24

🔬 Science Physics Mathematics Philosophy History Psychology

Great scientists often rely on their gut feelings and a sense of beauty rather than just cold hard logic. This mix of intuition leads to important discoveries.
Famous scientists aren't just rational thinkers; they have quirky beliefs and passions that drive their creativity. This uniqueness helps them come up with groundbreaking ideas.
There's a complex balance between formal science and the imaginative, intuitive side. Embracing both can push the boundaries of what we understand about the universe.

The Shape of Stats to Come

arg min • 634 implied HN points • 10 Oct 24

🚌 Education Statistics Optimization Philosophy Data Analysis Mathematics

Statistics often involves optimizing methods to get the best results. Many statistical techniques can actually be viewed as optimization problems.
Choosing a statistical method isn't just about the math—it's also based on beliefs about reality. This philosophical side is important but often overlooked.
There's a danger in relying too much on tools and models we can solve. Sometimes, we force the data to fit our preferred methods instead of being open to the actual complexities.

The Making of a Gene Circuit

Asimov Press • 290 implied HN points • 16 Feb 25

🔬 Science Biology Genetics Mathematics Technology Engineering

The repressilator is a simple gene circuit that helps scientists understand how to control living cells. It's made of three genes that work together in a loop to create a rhythmic 'on-off' signal.
Michael Elowitz and his team proved that you could design circuits in living cells, which opened the door to synthetic biology. This means we can now program cells to perform specific tasks.
Modern advancements have built on the repressilator, allowing us to create complex gene circuits that can mimic computing processes inside cells. This shows how biology can become a tool for engineering and technology.

A 15-minute intro to involute gears

lcamtuf’s thing • 6938 implied HN points • 17 Nov 24

🕹 Technology Engineering Manufacturing Design Mathematics Mechanics

Involute gears are used in many everyday items like toys and cars. Their special shape helps them work smoothly and efficiently.
These gears have specific properties that reduce friction and vibrations, allowing them to transfer motion without problems. This ensures they work together seamlessly.
Understanding the design of involute gears helps in creating various gear types, even unique shapes, making it easier to innovate in mechanical engineering.

Designed Interactions

arg min • 257 implied HN points • 15 Oct 24

🚌 Education Statistics Optimization Mathematics Machine Learning

Experiment design is about choosing the right measurements to get useful data while reducing errors. It's important in various fields, including medical imaging and randomized trials.
Statistics play a big role in how we analyze and improve measurement processes. They help us understand the noise in our data and guide us in making our experiments more reliable.
Optimization is all about finding the best way to minimize errors in our designs. It's a practical approach rather than just seeking perfection, and we need to accept that some questions might remain unanswered.

Inverse Problems

arg min • 515 implied HN points • 03 Oct 24

🔬 Science Mathematics Statistics Physics Engineering Computer Science

Inverse problems help us create images or models from measurements, like how a CT scan builds a picture of our insides using X-rays.
A key part of working with inverse problems is using linear models, which means we can express our measurements and the related image or signal in straightforward mathematical terms.
Choosing the right functions to handle noise and image characteristics is crucial because it guides how the algorithm makes sense of the data we collect.

Turing's Work on the Riemann Hypothesis

Cantor's Paradise • 379 implied HN points • 24 Jan 25

🔬 Science Mathematics Computing History Technology

Alan Turing is famous for his work in computer science and cryptography, but he also made important contributions to number theory, specifically the Riemann hypothesis.
The Riemann hypothesis centers on a mathematical function which helps in understanding the distribution of prime numbers, and it remains unproven after over 160 years.
Turing created special computers to help calculate values related to the Riemann hypothesis, showing his deep interest in the question of prime numbers and mathematical truth.

An Inversion Cookbook

arg min • 297 implied HN points • 04 Oct 24

🚌 Education Mathematics Statistics Optimization Regression Data science

Using modularity, we can tackle many inverse problems by turning them into convex optimization problems. This helps us use simple building blocks to solve complex issues.
Linear models can be a good approximation for many situations, and if we rely on them, we can find clear solutions to our inverse problems. However, we should be aware that they don't always represent reality perfectly.
Different regression techniques, like ordinary least squares and LASSO, allow us to handle noise and sparse data effectively. Tuning the right parameters can help us balance accuracy and manageability in our models.

Counting dimer tilings

A Piece of the Pi: mathematics explained • 36 implied HN points • 21 Feb 25

🔬 Science Mathematics Combinatorics Physics Statistics Theory

Dimer tilings involve arranging domino-shaped pieces on grids, and how many ways you can arrange them can vary based on the layout. For example, on a 3x3 grid with one space empty, there are 18 different arrangements.
If at least one dimension of a rectangle is even, it's possible to cover it completely with dimers. However, if both dimensions are odd, it's impossible to cover them without leaving gaps.
There are mathematical patterns and theorems, like Gomory's Theorem, that help understand how to tile grids with dimers. These principles can show when tiling is possible based on the arrangement and color of squares.

Trò chơi của Diophantus

Thái | Hacker | Kỹ sư tin tặc • 2037 implied HN points • 27 Jun 24

🔬 Science Mathematics History Physics Astronomy Data science

The game of Diophantus, an ancient Greek mathematician, has had a lasting impact on cryptography and internet security, with the basis of elliptic curve cryptography originating from his mathematical puzzles.
Diophantus's famous book 'Arithmetica' went missing for centuries but resurfaced to contribute to the advancements in mathematics, leading to significant discoveries like Fermat's Last Theorem.
The study of elliptic curves, inspired by concepts like Kepler's study of ellipses, has become a central focus in mathematics, intersecting various branches like number theory, algebra, and geometry, and even impacting modern technology such as Bitcoin security.

The Monty Hall problem: The golden goat variation

Simplicity is SOTA • 131 implied HN points • 03 Feb 25

🔬 Science Mathematics Probability Bayesian Logic Puzzles

The Monty Hall problem has a new twist, focusing on a valuable goat instead of a car. In this version, knowing which goat is valuable affects your choice.
Using Bayes' theorem can help calculate the probabilities in this variation. After a goat is revealed, you can reassess your chances to make a better decision.
The essential lesson is to update your beliefs with new information. Recognizing how new clues impact your choices is key to making smarter decisions.

Genomic prediction of IQ is modern snake oil

The Infinitesimal • 1298 implied HN points • 06 Jul 24

🔬 Science Genetics Biology Psychology Mathematics Research

Genetic tests claiming to predict IQ are not reliable. They often rely on complex methods that mostly just lead to guesswork.
The accuracy of these genetic predictions is very low, explaining only a tiny fraction of variations in IQ scores. In fact, other factors like age and social environment play a much bigger role.
Many of these predictions confuse people about how genetics really work. It's important to understand that these scores should be treated more like entertainment than serious assessments.

The maximum number of SETs for 12 cards

A Piece of the Pi: mathematics explained • 90 implied HN points • 10 Feb 25

🔬 Science Mathematics Linear Algebra Game Theory

The game SET uses 81 cards that have four qualities: quantity, shading, color, and shape. Players look for sets of three cards where each quality is either all the same or all different.
SET can be understood through linear algebra, where each card is represented as a four-dimensional vector. If the vectors for three cards add up to zero, they form a valid set.
Recent research showed that with 12 cards, a maximum of 14 sets can be formed, and they provided proofs for similar results with fewer cards. This reveals interesting mathematical properties of the game.

The Origins of Deutsche Physik (1910s)

Cantor's Paradise • 363 implied HN points • 06 Jan 25

🔬 Science Physics History Politics Philosophy Mathematics

The conflict in the physics community during the 1910s was largely about differing views on science and the influence of World War I. German physicists felt pressure to defend national pride while dealing with the rise of theoretical physics led by figures like Einstein.
There was a significant clash between experimentalists and theorists, with older physicists struggling to accept new ideas. Many were uncomfortable with Einstein's theories because they felt too abstract and removed from traditional experimental methods.
As political tensions grew, the conflict transformed into overt anti-Semitism, particularly targeting Einstein. Some physicists expressed nationalistic and racial ideologies, which later aligned with the Nazi agenda.

The Gold Coin game

Infinitely More • 7 implied HN points • 12 Feb 25

🎭️ Culture Games Strategy Logic Philosophy Mathematics

The Gold Coin game involves two players trying to get a valuable gold coin while moving other coins on a board. Players can either move a coin or take the leftmost coin in their turn.
The game is strategic and requires understanding the best moves to win. Knowing winning moves can make the game easier to navigate.
Practicing the game with a partner helps improve your skills and understanding of the rules, making it more enjoyable to play.

John F. Nash Jr.'s Letter to J. Robert Oppenheimer (1957)

Cantor's Paradise • 205 implied HN points • 17 Jan 25

🎭️ Culture Science Mathematics History Philosophy Education

John F. Nash Jr. was very bold in reaching out to famous scientists like Einstein and von Neumann. He wasn't afraid to discuss his ideas with them, even at a young age.
Nash had limited formal education in physics but still engaged deeply with complex ideas. He wasn't shy about diving into new topics and sharing his thoughts.
His interactions with these great minds show that having confidence and curiosity can lead to meaningful discussions, even with experts in the field.

Analyze FIR filters using just high-school algebra

filterwizard • 39 implied HN points • 23 Sep 24

🕹 Technology Signal Processing Mathematics Algebra Engineering

FIR filters have a finite impulse response, meaning they only remember a limited amount of past input. This makes them predictable and stable, especially for applications needing fast settling times.
You can think of FIR filter coefficients as a polynomial, which allows you to use algebra to analyze and create filters. This approach helps in understanding how changing coefficients affects the filter's behavior.
By factoring the polynomial of an FIR filter, you can create smaller filters that combine to produce the same overall effect. This technique allows for a deeper exploration of filter design, giving you more control over the filter's characteristics.

4.4.1 How Do Transmission Lines Work?

Fields & Energy • 319 implied HN points • 14 Aug 24

🔬 Science Physics Electromagnetism Engineering Technology Mathematics

Transmission lines work by sending electrical signals through wires, where one wire gets a negative charge and the other gets a positive charge. This creates electric fields that help move energy along the line.
To avoid signal loss and distortion, it's important to balance the electric and magnetic energies in transmission lines. If they are not balanced, the signal can get messed up over long distances.
Oliver Heaviside developed key equations that describe how signals travel through transmission lines. His work highlighted the importance of using both electric and magnetic energies to achieve clear signal propagation.

My Class About Causal Inference

By Reason Alone • 42 implied HN points • 13 Feb 25

🚌 Education Teaching Learning Causal Inference Social Sciences Mathematics

Teaching causal inference helps students understand the relationship between cause and effect in social sciences. It's important to make complex ideas relatable to engage younger audiences.
Using visual aids, like graphs, can enhance understanding of complicated topics, especially in a classroom setting. Students can connect better with the material when it’s presented visually.
Recommended readings and real-world examples, like the draft lottery, can spark curiosity in students. Sharing interesting studies can help them see the relevance of these concepts in everyday life.

Science giveth and science taketh away

Razib Khan's Unsupervised Learning • 366 implied HN points • 17 Dec 24

🔬 Science Evolution Genetics Mathematics Natural History

Science has advanced a lot since Darwin's time, but we often miss the wonder and excitement that comes with these discoveries. It seems like people today are less amazed by scientific progress than they used to be.
Darwin proposed that evolution happens through natural selection, but he didn’t fully explain how traits are passed down. Later scientists combined genetics with evolution to better understand how traits vary across generations.
Today, understanding evolution requires recognizing four main forces: mutation, migration, selection, and drift. These forces shape the genetic diversity that fuels evolution.

The Story of the Telegrapher's Equations

Fields & Energy • 259 implied HN points • 16 Aug 24

🔬 Science Physics Mathematics History Technology Engineering

Oliver Heaviside was a young scientist who created the Telegrapher's Equations in 1876. His work helped connect theories of electromagnetism to practical applications in telecommunication.
Before Heaviside, the diffusion model was the main idea for how signals traveled. Heaviside improved this by showing that signals could travel as waves instead of just spreading out slowly.
The development of these equations was influenced by earlier mathematicians like Fourier and scientists like Lord Kelvin. Heaviside's contribution built on their ideas and advanced the understanding of signal transmission over long distances.

Now synthesize your FIR filters using high-school algebra

filterwizard • 19 implied HN points • 27 Sep 24

🕹 Technology Engineering Software Mathematics Signal Processing Computer Science

You can create FIR filters by breaking them down into smaller parts using simple math. This makes it easier to understand how each piece works together.
The sharp notches or deep points in a filter's response happen because of certain factors in the polynomial. Each notch can be traced back to specific frequencies based on these factors.
To improve a filter's performance, you can add more mathematical pieces to make the response smoother in certain areas. This way, you can customize how the filter behaves at different frequencies.

The Birth of Transmission Line Theory

Fields & Energy • 279 implied HN points • 09 Aug 24

🔬 Science Physics History Mathematics Electromagnetism Engineering

The first Transatlantic Telegraph Cable in 1858 was crucial for developing transmission line theory. It helped researchers understand how to send messages over long distances.
Lord Kelvin created an early model for long cables, focusing on how to evenly spread resistance and capacitance. This helped explain why the first cable failed.
Oliver Heaviside later added the concept of inductance to the equations, which improved the understanding of transmission lines even further.

This week: Bacteria anticipate the seasons! Placebos activate the same neurons as anesthesia! And don’t touch radio towers with hot dogs.

Niko McCarty • 79 implied HN points • 07 Sep 24

🔬 Science Biology Neuroscience Physics Mathematics Environmental Science

Bacteria can sense changes in seasons and adapt to prepare for colder weather. This helps them survive better when temperatures drop.
Placebos work by activating the same brain neurons as pain relief drugs like anesthesia. This shows how our mind can influence our body’s responses.
A fun fact: touching a hot dog to a radio tower can turn it into a speaker. Just a quirky reminder to be careful with food and electronics!

The game of Nim

Infinitely More • 10 implied HN points • 07 Feb 25

🚌 Education Mathematics Games Logic Strategy Learning

The game of Nim is based on a smart mathematical strategy that lets informed players almost always win against those who don't know the trick.
In Nim, players take turns removing coins from piles, and the goal is to take the last coin to win.
Anyone, even kids, can learn the winning strategy and easily beat more experienced players who don't know it.

Identifying bottlenecks in networks

A Piece of the Pi: mathematics explained • 48 implied HN points • 03 Feb 25

🔬 Science Mathematics Networks Physics Statistics Logic Analysis

Bottlenecks in networks are crucial points that can slow down communication or movement. Identifying these points helps understand how the entire network functions.
Networks can be made up of different regions that are linked by these bottlenecks. Recognizing connections between these regions is important for overall analysis.
Knowing where the bottlenecks are can help improve the efficiency of networks, whether in transportation or social connections. This can lead to better planning and resource allocation.

The CAP Theorem of Clustering: Why Every Algorithm Must Sacrifice Something

Confessions of a Code Addict • 529 implied HN points • 29 Oct 24

🕹 Technology Algorithms Data science Software Engineering Mathematics Research

Clustering algorithms can never be perfect and always require trade-offs. You can't have everything, so you have to choose what matters most for your project.
There are three key properties that clustering should ideally have: scale-invariance, richness, and consistency, but no algorithm can achieve all three simultaneously.
Understanding these sacrifices helps in making better decisions when using clustering methods. Knowing what to prioritize can lead to more effective data analysis.

Turning a triangle into a square

A Piece of the Pi: mathematics explained • 115 implied HN points • 11 Jan 25

🔬 Science Mathematics Geometry Puzzles Research

Henry Dudeney showed in 1902 that you can cut an equilateral triangle into four pieces and rearrange them into a square with the same area. This is a fun example of how shapes can transform while keeping their total area the same.
The Wallace–Bolyai–Gerwien theorem explains how you can rearrange two shapes with the same area into each other through cutting, but Dudeney's method is unique because the pieces stay connected during the transformation.
Recent research proved that you can't turn a triangle into a square using fewer than four pieces without flipping any. This shows how specific and tricky these geometric dissections can be.

How to Measure Molecules

Asimov Press • 367 implied HN points • 17 Nov 24

🔬 Science Physics Chemistry History Experimentation Mathematics

In the late 19th century, Lord Rayleigh measured the size of a single molecule using simple materials like oil and water. This clever experiment showed how basic observations can lead to important scientific discoveries.
Benjamin Franklin also made significant observations about oil on water in the 18th century, but he didn't calculate the size of molecules. His work laid the groundwork for future scientists like Rayleigh.
Rayleigh's experiment demonstrated that you don’t always need complex tools to make groundbreaking discoveries. Even simple experiments can provide valuable insights that inspire later research.

A Nonconstructive Existence Proof of Aligned Superintelligence

Transhuman Axiology • 99 implied HN points • 12 Sep 24

🕹 Technology AI Superintelligence Computing Robotics Mathematics

Aligned superintelligence is possible, despite some people thinking it isn't. This idea shows proof that it can exist without needing complicated construction.
Desirable outcomes for AI mean producing results that people think are good. We define these outcomes based on what humans can realistically accomplish.
While the concept of aligned superintelligence exists, it faces challenges. It's hard to create, and even if we do, we can't be sure it will work as intended.

“Math is hard” — if you are an LLM – and why that matters

Marcus on AI • 4782 implied HN points • 19 Oct 23

🔬 Science Mathematics Artificial Intelligence Machine Learning

Even with massive data training, AI models struggle to truly understand multiplication.
LLMs perform better in arithmetic tasks than smaller models like GPT but still fall short compared to a simple pocket calculator.
LLM-based systems generalize based on similarity and do not develop a complete, abstract, reliable understanding of multiplication.

Chess and the number e

A Piece of the Pi: mathematics explained • 163 implied HN points • 16 Dec 24

🔬 Science Mathematics Combinatorics Statistics Game Theory Logic

The number e, around 2.718, plays a big role in math, especially in combinatorial problems like derangements. This is when items are arranged so that none are in their original position.
In chess, setting up nonattacking rooks can be related to derangements. The chance that none of them land on the main diagonal equals about 36.8%, which links back to the number e.
Recent studies have also looked at how many safe squares remain on a chessboard when placing random pieces. As more pieces are added, the proportion of safe squares follows certain patterns connected to e.

Tormented by an Urn

rachaelmeager • 535 implied HN points • 04 Jun 24

🚌 Education Teaching Learning Mathematics Statistics Philosophy

The Polya urn model, though simple at first glance, reveals the complexity of statistics and emphasizes the importance of understanding problems deeply before attempting to solve them.
Teaching and learning in math are not just about facts; they require creativity and passion to engage students, much like how poets perceive deeper meanings in their art.
There is a strong connection between the arts and sciences, where both disciplines can benefit from understanding each other, and students should learn foundational concepts in both to grasp the complexities of the world.

The mathematics of the game Waffle

A Piece of the Pi: mathematics explained • 48 implied HN points • 22 Jan 25

🔬 Science Mathematics Game Theory Algorithms

Waffle is a fun word game where you need to form six five-letter words in a grid. You can swap letters to find the right words based on clues given.
To solve Waffle, you must figure out the words first, then how to rearrange the letters, and finally do it using the least number of swaps.
Group theory is useful for solving Waffle puzzles because it helps to find ways to rearrange the letters efficiently, especially when dealing with repeated letters.