The hottest Mathematics Substack posts right now

And their main takeaways
Category
Top Science Topics

A Linear Algebra Trick for Computing Fibonacci Numbers Fast

Confessions of a Code Addict • 158 HN points • 05 Nov 23
🔬 Science Mathematics
  1. A linear algebra technique can be applied to compute Fibonacci numbers quickly with a logarithmic time complexity.
  2. Efficient algorithms like repeated squaring can compute powers of matrices in logarithmic time, improving performance for Fibonacci number calculations.
  3. A closed form expression using the golden ratio offers a direct method to compute Fibonacci numbers, showing different approaches with varied performance.

Not so fast, Mr. Fourier!

lcamtuf’s thing • 119 HN points • 12 Mar 24
🔬 Science Mathematics
  1. The discrete Fourier transform (DFT) is a crucial algorithm in modern computing, used for tasks like communication, image and audio processing, and data compression.
  2. DFT transforms time-domain waveforms into frequency domain readings, allowing for analysis and manipulation of signals like isolating instruments or applying effects like Auto-Tune in music.
  3. Fast Fourier Transform (FFT) optimizes DFT by reducing the number of necessary calculations, making it more efficient for large-scale applications in computing.

Fun with Serial and Parallel Sum Reductions

CPU fun • 121 implied HN points • 22 Feb 24
🔬 Science Mathematics
  1. Floating point arithmetic can be more complex than expected, especially due to limited mantissa bits, affecting the accuracy of calculations.
  2. Complaining about OpenMP reductions giving 'the wrong answer' is misguided; the issue likely existed in the serial code and is now being exposed.
  3. Changing the type of the accumulator to 'double' can help resolve issues with floating point arithmetic and accuracy during sum reductions.

Tiling the plane with congruent polygons

A Piece of the Pi: mathematics explained • 18 implied HN points • 29 Jun 25
🔬 Science Mathematics
  1. You can't cover a flat surface with regular pentagons because their angles don't fit together perfectly. The angle of a pentagon is 108°, and it's not a number that evenly divides into 360°.
  2. However, there are other shapes, like certain hexagons and quadrilaterals, that can tile the plane without any gaps. These shapes can fit together nicely to fill space.
  3. Tiling is a fun way to explore patterns and geometry, showing how shapes can interact in creative and mathematical ways. It leads to interesting discoveries in both art and mathematics.

Counting dimer tilings

A Piece of the Pi: mathematics explained • 36 implied HN points • 21 Feb 25
🔬 Science Mathematics
  1. 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.
  2. 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.
  3. 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.
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The tactical variation of the fundamental theorem

Infinitely More • 12 implied HN points • 10 Aug 25
🚌 Education Mathematics
  1. In some finite games, either one player has a winning tactic or both players can draw the game. This rule applies to many familiar games like tic-tac-toe and Connect Four.
  2. Not all games follow this winning or drawing pattern. Some games, like the Chocolatier's game, can have different outcomes.
  3. There is a way to tell which games have a winning tactic or drawing tactics by looking at how much information is available on the board. More information can lead to clearer winning strategies.

The devil is old

Wednesday Wisdom • 113 implied HN points • 21 Feb 24
🕹 Technology Mathematics
  1. Experience and age often bring wisdom, knowledge, and a unique perspective.
  2. In technology, while tools and capabilities have evolved, fundamental principles like people dynamics, math, and physics remain constant.
  3. Despite advancements, people still struggle with basic math, concurrent programming, and effective communication in group settings.

How we might have viewed the continuum hypothesis as a fundamental axiom necessary for mathematics

Infinitely More • 20 implied HN points • 22 May 25
📖 Philosophy Mathematics
  1. The continuum hypothesis (CH) is about understanding different sizes of infinity, particularly if there's a number between natural numbers and real numbers. Many assume its truth or falsehood is needed for math.
  2. If early mathematicians had been clearer about infinitesimals and different types of numbers, they might have accepted CH as a key part of math and calculus, making hyperreal numbers a standard concept.
  3. Whether CH is true or false is not just a technical question; it reflects deeper philosophical views about the nature of mathematics and how we interpret infinity and set theory.

Boomerang tilings

Infinitely More • 15 implied HN points • 25 Jun 25
🔬 Science Mathematics
  1. Boomerangs are special shapes called nonconvex quadrilaterals. They can be used to explore interesting questions about tiling.
  2. The main question is whether a convex polygon can be tiled completely using just a few boomerangs. This is a challenging mathematical problem.
  3. Finding a solution to this problem requires careful thought and may not be easy. Just because one attempt fails, it doesn’t mean that it can’t be done at all.

Infinite Connect Four

Infinitely More • 15 implied HN points • 17 Jun 25
🎭️ Culture Mathematics
  1. Connect Four is a game where players try to get four of their coins in a row, either horizontally, vertically, or diagonally. The game shows that the first player has a winning strategy if they play perfectly.
  2. The concept of an infinite version of Connect Four allows for interesting variations, like playing on an infinite board. This leads to questions about how long winning chains players can aim to create.
  3. With infinite possibilities, players might aspire to create very long winning combinations, even infinite chains. This expands the game beyond traditional limits and invites deeper strategic thinking.

Repunits and prime numbers

A Piece of the Pi: mathematics explained • 42 implied HN points • 18 Nov 24
🔬 Science Mathematics
  1. Repunits are numbers made only of the digit 1 and can appear in different bases. For example, the number 31 can be written as 111 in base 5 and base 2.
  2. Mersenne primes are special numbers of the form 2^p - 1 that can be prime, where p is also a prime. However, it's rare for these to actually be prime numbers.
  3. One interesting link is between Mersenne primes and perfect numbers, which are those that equal the sum of their divisors. Each Mersenne prime corresponds to a perfect number, like how 31 corresponds to the perfect number 496.

Take my Infinity final exam!

Infinitely More • 35 implied HN points • 21 Dec 24
🚌 Education Mathematics
  1. The Cantor-Hume principle connects with Euclid's principle, and there are different views on whether they agree or conflict. It's a topic worth exploring in depth.
  2. Understanding potential and actual infinity is important in calculus, especially when dealing with infinite series. This distinction affects how we solve mathematical problems.
  3. The continuum hypothesis and the axiom of choice raise interesting questions in philosophy and mathematics, showing how these concepts challenge our understanding of infinity and choice.

The zigzag theorem

Infinitely More • 33 implied HN points • 04 Jan 25
🚌 Education Mathematics
  1. The zigzag theorem states that when you create a zigzag pattern in a rectangle, the triangles formed below this pattern take up exactly half the area of the rectangle.
  2. Even if the zigzag lines sometimes move backward without crossing, the triangles will still cover half the rectangle's area due to how the bases and heights of the triangles are calculated.
  3. This theorem is interesting because it holds true even if the zigzag involves an infinite number of lines.

What is the essential structure of the complex numbers?

Infinitely More • 38 implied HN points • 10 Nov 24
📖 Philosophy Mathematics
  1. There are different ways to think about complex numbers, like focusing on their algebraic or topological structures. Each viewpoint gives us unique insights into how complex numbers behave.
  2. Mathematicians don't all agree on what the essential structure of complex numbers is, leading to multiple interpretations. It shows us that understanding math can be quite flexible.
  3. The paper identifies four main perspectives on complex numbers, which can help clarify the discussions around their nature and engage with broader philosophical questions in mathematics.

The Mathematical Capabilities of ChatGPT.

Deep-Tech Newsletter • 39 implied HN points • 17 Feb 23
🕹 Technology Mathematics
  1. Recently published research suggests that ChatGPT's mathematical abilities are below those of an average mathematics graduate student.
  2. There is skepticism that large language models like ChatGPT will lead to Artificial General Intelligence due to their poor mathematical reasoning performance.
  3. ChatGPT has been subject to criticisms and shortcomings, with some considering it less innovative and revolutionary compared to expectations.

QF Mathematics Mentorship

Quantum Formalism • 59 implied HN points • 16 Jun 22
🚌 Education Mathematics
  1. QF Mathematics Mentorship offers free mentorship from PhD-level mathematicians for researchers and aspiring open-source quantum computing contributors.
  2. The program aims to mentor individuals on advanced branches of mathematics needed for research or open-source projects.
  3. Preference is given to projects that can benefit from utilizing abstract branches of mathematics otherwise inaccessible to non-mathematicians.

Log Transformations for efficient multiplication between lots of positive numbers [Technique Tuesdays]

Technology Made Simple • 39 implied HN points • 02 Nov 22
🔬 Science Mathematics
  1. Log transformations can be used for efficient multiplication between large numbers by converting the problem into addition of logs, making it more manageable.
  2. Logs have interesting properties that make them useful for handling computations with very large or very small numbers.
  3. Using log transformations is a clever math technique that is commonly used in fields like AI, Big Data, and Machine Learning to handle large computations.

The Parks puzzle

A Piece of the Pi: mathematics explained • 36 implied HN points • 11 Nov 24
🚌 Education Mathematics
  1. The Parks puzzle is a game where you place trees on a grid with specific rules, similar to Sudoku. Each row, column, and park needs a certain number of trees without them being next to each other.
  2. While checking if a proposed solution is correct is easy, finding that solution can be quite complex. Researchers found that the Parks puzzle belongs to a group of difficult problems called NP-complete.
  3. The puzzle can be used to model logical operations like AND and OR. This means it has connections to computer science concepts and can help explore complex problems.

Math for kids outside of the Calculus Sequence

Kids Who Love Math • 111 HN points • 07 Aug 23
🚌 Education Mathematics
  1. There's a clear path from arithmetic to calculus in math education, but kids who advance too quickly may face challenges in a traditional school setting.
  2. Instead of just accelerating through the math curriculum, consider enrichment to explore topics outside the typical sequence like statistics, probability, and mathematical finance.
  3. Parents can support their kids in exploring enrichment math by learning alongside them, finding tutors or math circles, and utilizing resources like books and educational videos.

The Structure of Intuition

NonTrivial • 19 implied HN points • 28 May 23
🔬 Science Mathematics
  1. Mathematics and rigor go hand in hand, helping to explain the world with precision and clarity.
  2. The cost of precision lies in the potential loss of context and connection to reality.
  3. Intuition, rooted in analogy-making, offers a deeper connection to reality than mathematics alone.

Zero is the Universal Bayesian Prior

Bram’s Thoughts • 19 implied HN points • 14 Jul 23
🔬 Science Mathematics
  1. Estimating values in multiple dimensions can be more accurate by making them slightly smaller towards zero.
  2. Using a Bayesian prior of zero in analysis can be counterproductive and arbitrary.
  3. Consider using realistic Bayesian priors, like room temperature, for more reasonable estimates.

Links from July

nic thinks about things • 19 implied HN points • 01 Aug 23
🕹 Technology Mathematics
  1. Actuaries have the lowest divorce rate, while Gaming managers and Bartenders have the highest.
  2. Creating a "physical" camera in Blender feels like simulating physics.
  3. Improving indoor air quality is a cost-effective way to enhance health and cognition.

How I once won the lottery

inexactscience • 19 implied HN points • 06 Sep 23
🚌 Education Mathematics
  1. Sticking to one choice in a lottery doesn't change your odds, which stay at 1 in 24 no matter what. It seems like it should matter, but it really doesn't.
  2. If a lottery is unfair and avoids streaks, choosing the same number can actually be a better strategy because it decreases your risk of never winning.
  3. Many people fall for the gambler's fallacy, thinking just because a number hasn't won in a while, it should win soon. But in a fair lottery, each draw is independent and has the same odds.

The Camel Principle: Why Adding Zero is the Most Powerful Trick in Mathematics

The Palindrome • 1 implied HN point • 12 Jan 26
🚌 Education Mathematics
  1. The camel principle is the idea that you can add zero in clever ways to transform problems, and that tiny trick can unlock big simplifications.
  2. Adding zero is essential because it helps rewrite expressions, simplify derivations, and connect different methods across mathematics and machine learning.
  3. A practical workshop can teach these foundations by building linear regression from scratch, covering vectors, vectorized code, optimization, and gradient descent with notebooks and recordings for practice.

On going first

Infinitely More • 12 implied HN points • 30 May 25
📖 Philosophy Mathematics
  1. In many games, going first can give a big advantage, which isn't always fair. It's important to find ways to balance the game for both players.
  2. Rules like the 'swap rule' in games like Hex can help make things fair, but they may not always work perfectly for every game.
  3. For games like Go, figuring out the right value to give the second player can be tricky, and the common values used might not actually be the best.

Math for Computer Science [Math Mondays]

Technology Made Simple • 59 implied HN points • 26 Apr 22
🕹 Technology Mathematics
  1. Focus on Calculus for software development: Understand precalc topics like functions, transformation, and algebra well.
  2. Importance of Probs and Stats: Learn to think in a Bayesian context, focus on probabilistic thinking.
  3. Value of Linear Algebra: Grasp foundational concepts, computational side less important for traditional software development.

Interpretability

Infinitely More • 28 implied HN points • 30 Nov 24
📖 Philosophy Mathematics
  1. In math, we can understand one idea by using another. It's like using different languages to explain the same thing.
  2. Sometimes, when we translate ideas back and forth, we lose some meaning, similar to playing a game of telephone.
  3. To make this work, we create special objects in a new system that can help us relate and understand the original idea better.

Measure Theory and Functional Analysis

Quantum Formalism • 39 implied HN points • 24 Aug 22
🚌 Education Mathematics
  1. First problem set is for reviewing integration theory - try it before the course starts!
  2. Second problem set is meant to be challenging, introducing new mathematical content.
  3. Course will provide detailed solutions to assignments, so don't get discouraged by difficult problems.

105 trillion digits of pi

A Piece of the Pi: mathematics explained • 60 implied HN points • 15 Mar 24
🔬 Science Mathematics
  1. The number pi has now been calculated to 105 trillion decimal places using the Chudnovsky algorithm over 75 days.
  2. Ramanujan's formula for pi has been expanded and improved upon over the years, with the Chudnovsky brothers developing a formula that computes pi to 13 decimal places.
  3. Bellard's formula and the BBP formula provide ways to compute specific digits of pi without having to calculate all earlier digits, making validations faster and more efficient.

Rubik’s abstract polytopes

A Piece of the Pi: mathematics explained • 18 implied HN points • 03 Mar 25
🔬 Science Mathematics
  1. 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.
  2. 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.
  3. 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.

Let's cut some cake

Sunday Letters • 79 implied HN points • 20 Mar 22
🚌 Education Mathematics
  1. To share something fairly, one person should cut it while the other picks their piece. This way, both care about fairness.
  2. In team discussions or disagreements, break down decisions into parts to find common ground and make it easier for everyone to agree.
  3. Using a math-based approach can help settle arguments quickly and fairly, showing that cooperation can work better than fighting.