In which I attempt maths and experience regret.| Diagram Monkey
It’s been over three years since my last post on this blog and I have sometimes been asked, understandably, whether the project I announced in my previous post was actually happening. The ans…| Gowers's Weblog
October is here, and with it the crisp fall air here in the northeast US. No matter where you are, you can get a breath of fresh air with some problem solving in your math classes! Here is the 2015…| Reflections and Tangents
CUDA PTX ldmatrix Instruction and Its CuTe Wrapper| Lei Mao's Log Book
Learning times tables doesn’t have to be a battle. With modern teaching approaches backed by cognitive science, children can master multiplication facts while developing genuine mathematical understanding. This comprehensive guide reveals the most effective, research-based strategies for teaching times tables that transform struggling learners into confident mathematicians. Why Traditional Times Tables Methods Fail (And What ... Read more The post The Complete Guide to Teaching Times Tables...| Special Education and Inclusive Learning
Faculty in The University of Tulsa’s Department of Mathematics are creating hands-on research opportunities for undergraduate students through a unique initiative. Professor Richard Redner, Ph.D., and Assistant Professor Nicholas Cox-Steib, Ph.D., have created a research initiative centered on zero-one matrices – binary matrices that while widely applicable in fields such as signal processing, computer science, […]| The University of Tulsa
A collection of simple, hands-on math activities suitable for 2- and 3- year old toddlers and preschoolers. Great for home or preschool.| Childhood101
Gerardus Mercator warped the world to flatten it. He created a conformal map that helped early navigators hold courses of constant bearing across vast oceans.| Galileo Unbound
Is category theory being co-opted? These authors think so: • Esteban Montero and Brandon Baylor, Category theory is being co-opted, Holon Substack, 7 September 2025. Category theory offers a p…| Azimuth
Using Dynamic Technology to Build Understanding ◊ Episode 7: Pythagorean Squares◊ Today is a “perfect square day” since the date is 9-16-25 (or 16-9-25 if you live somewhere that puts the day first…| Reflections and Tangents
When you talk to a large language model (LLM), it feels like the model understands meaning. But under the hood, the system relies on numbers, vectors, and math to find the relationships between words and sentences. One of the most important tools tha...| freeCodeCamp.org
When Turkish courts sentenced crypto exchange founder Faruk Özer to 11,196 years in prison, they created a mathematical anomaly that reveals fascinating patterns about justice, time, and the limitations of human-scale punishment. Join the Atlas Monkey crew as we decode this temporal enigma through the lens of Clockweave's computational justice algorithms.| Seuros Blog - Navigation Logs from the Ruby Nebula
Designing Tilers for Data Access| Lei Mao's Log Book
Science has made great strides in modeling space, time, mass and energy. Yet little attention has been paid to the precise representation of the information ubiquitous in nature. Introduction to Evolutionary Informatics fuses results from complexity modeling and information theory that allow both meaning and design difficulty in nature to be measured in bits. Built on the foundation of a series of Read More ›Source| Books – Discovery Institute
Star Wars Day is here again and it’s a terrific time to integrate Star Wars| Kelly's Classroom Online
Host page for the first Symposium on the Platonic Space| Forms of life, forms of mind
Preschool math printables and simple math activities for preschoolers make it easy to add hands-on learning to your day. These low-prep printables help kids build key math skills in a way that feels fun and natural. You’ll find a collection of free printable math activities for preschoolers, plus ideas for adding play-based learning into your... The post Free Preschool Math Printables appeared first on Stay At Home Educator.| Stay At Home Educator
Preschool math games bring energy and movement into everyday learning, and math activities for preschoolers help little ones build math skills in ways that feel easy and natural. When kids play games, they stay curious and involved, which means they practice skills without feeling pressure. With print-and-go resources and online math games for preschoolers, it’s... The post The Best Free Preschool Math Games appeared first on Stay At Home Educator.| Stay At Home Educator
Does teaching preschool math numbers feel more like work than fun? Let’s flip that around! With the right math activities for preschoolers, learning numbers can be full of giggles and hands-on play. Even worksheets can be fun, because we’ve added a little bit of tracing, some possible coloring, and a whole lot of playdough! Grab... The post Preschool Math Numbers Playdough Mats appeared first on Stay At Home Educator.| Stay At Home Educator
Thomas Bloom’s erdosproblems.com site hosts nearly a thousand questions that originated, or were communicated by, Paul Erdős, as well as the current status of these questions (about a third of which are currently solved). The site is now a couple years old, and has been steadily adding features, the most recent of which has been […]| What's new
Morava, a member of the Hopkins faculty for nearly four decades, remembered for his warmth, brilliance| The Hub
Sometimes, when I write a post about AI, I’ve been sitting on an idea for a long time. I’ve talked to experts, I’ve tried to understand the math, I’ve honed my points and cleared away clutter. This is not one of those times. The ideas in this post almost certainly have something deeply wrong with […]| 4 gravitons
When bad news gets me down, I often get insomnia. I wake up in the middle of the night, start thinking about how we’re all doomed, and can’t easily stop. To break out of these doom loop…| Azimuth
Visit the post for more.| Frame-Poythress.org
Did you own a personal computer 40 years ago? How many teachers and students had access to computers in 1983. What if I told you that a 400-page book focused on teaching computer programming to children sold more than 100,000 copies way back then? Well, it’s true. Former engineer, elementary school teacher, and member of […] The post Classic Book Now Available Online (free) appeared first on The Daily Papert.| The Daily Papert
Documents and recollections from the early days of 1:1 computing in schools. The post Laptops appeared first on The Daily Papert.| The Daily Papert
A month or two ago I wrote this post which expressed my frustration with various issues around private datasets as a way of measuring the mathematical abilities of language models. More generally I was frustrated about the difficulty of being … Continue reading →| Xena
Undergraduate mathematicians usually have a hard time defining functions from quotients in Lean, because they have been taught a specific model for quotients in their classes, which is not the model that Lean uses. This post is an attempt to … Continue reading →| Xena
So I'm two months into trying to teach a proof of Fermat's Last Theorem to a computer. We already have one interesting story, which I felt was worth sharing. Continue reading →| Xena
A huge amount happened in the Lean theorem prover community in 2023; this blog post looks back at some of these events, plus some of what we have to look forward to in 2024. Modern mathematics I personally am a … Continue reading →| Xena
I previously had put together analysis that utilized the full name and date of birth information from the Virginia Registered Voter List (“RVL”) in order to look for duplicate registrations, either exact matches or by using a string distance measure (the Levenshtein distance) to accommodate for typos, abbreviations, and mis-spellings. Just prior to the start […]| Digital Poll Watchers (dot) Org
Deriving Inverse Layout Mathematically| Lei Mao's Log Book
Creating Tiled Layouts Using Blocked Product and Raked Product| Lei Mao's Log Book
Update: See my daily math streak progress here 👈 I recently passed 100 days of practicing math every single day 💯 I’ve wanted to beef up my math chops for a while, but I needed a good reason that would justify the time investment. Plus, it’s always easier to learn when you have a clear […]| Gabe Mays
In Part 10 we saw that, loosely speaking, the theory of a hydrogen atom is equivalent to the theory of a massless left-handed spin-½ particle in the Einstein universe—a static universe where space is a 3-sphere. Today we’ll ‘second quantize’ both of these equivalent theories and get new theories that again are equivalent. ‘Second quantization’ […]| Azimuth
The poet Blake wrote that you can see a world in a grain of sand. Today we’ll see a universe in an atom! We’ll see that states of the hydrogen atom correspond to states of a massless spin-½ particle in the Einstein universe—a closed, static universe where space is a 3-sphere. The rotational symmetries of […]| Azimuth
Today I want to make a little digression into the quaternions. We won’t need this for anything later—it’s just for fun. But it’s quite beautiful. We saw in Part 8 that if we take the spin of the electron into account, we can think of bound states of the hydrogen atom as spinor-valued functions on […]| Azimuth
Now comes the really new stuff. I want to explain how the hydrogen atom is in a certain sense equivalent to a massless spin-½ particle in the ‘Einstein universe’. This is the universe Einstein believed in before Hubble said the universe was expanding! It has a 3-sphere for space, and this sphere stays the same […]| Azimuth
Did you know that Lawvere did classified work on arms control in the 1960s, back when he was writing his thesis? Did you know that the French government offered him a job in military intelligence? The following paper should be interesting to applied category theorists—for a couple of different reasons: • Bill Lawvere, The category […]| Azimuth
I’ve explained a cool way to treat bound states of the hydrogen atom as wavefunctions on a sphere in 4-dimensional space. But so far I’ve been neglecting the electron’s spin. Now let’s throw that in too! This will wind up leading us in some surprising directions. So far I’ve just been reviewing known ideas, but […]| Azimuth
I’ve been working with Adittya Chaudhuri on some ideas related to this series of blog articles, and now our paper is done! • John Baez and Adittya Chaudhuri, Graphs with polarities. Abst…| Azimuth
Prof. Olivia Caramello, Coordinator of the Centre for Topos Theory and its Applications (CTTA), gave an invited lecture in Shanghai| Istituto Grothendieck
Preschool math number recognition and counting are essential skills in our preschoolers’ early math development. Children, without even realizing it, start developing mathematical skills from the moment they begin to notice numbers all around them. These early experiences with numbers help develop number sense naturally. Preschoolers learn numbers best by doing! Using games, songs, and... The post Preschool Math Numbers: Ways to Build Counting Skills appeared first on Stay At Home Educator.| Stay At Home Educator
This is a puzzle with both the computer-puzzle tag and the no-computers tag. We have the following list of five fractions: 11/5, 30/77, 1/11, 21/2, 5/7 Starting with an integer x, we perform the| Puzzling Stack Exchange
A polycube is a three dimensional generalisation of polyomino, i.e., it is formed by gluing unit cubes along their faces. Given a cube of side n, one can easily dissect it into n congruent polycube...| Puzzling Stack Exchange
Recently, I got an IKEA LILLABO train set for my daughter. It has twelve curved segments, each is 1/8th of a circle of radius 1, two straight segments of length 1, and a bridge of length 2 (in top ...| Puzzling Stack Exchange
Elucidating CuTe Inner Partition and Local Tile| Lei Mao's Log Book
Elucidating CuTe Outer Partition and Local Partition| Lei Mao's Log Book
Last week had a special math-y date: 7-24-25, which is a Pythagorean triple. This means that the sum of the squares of the smaller two numbers equals the square of the largest number. Let’s d…| Reflections and Tangents
Here's an interesting question: given a target integer n,| mathesis
Note: this article| mathesis
There has been an extraordinary participation for the conference "Mathematics as an artistic experience" held on 11 July 2025 in Paris at the Hermite Amphitheatre of the Henri Poincaré Institute, and organised by the Grothendieck Institute in collaboration with the IHP and the MICS Laboratory of Centrale Supélec (Paris-Saclay University).| Istituto Grothendieck
In ordinary math, the infinite decimal .999… is defined to be the limit of the terminating decimals .9, .99, .999, …; that is, it’s defined to be the real number that the fractions 9/10, 99/1…| mathenchant.wordpress.com
We piled into the two cars and headed north to the World Heritage city of Évora for the third conference on Theoretical and Computational Algebra. Arriving in my room, the welcome message on the television informed me that the temperature … Continue reading →| Peter Cameron's Blog
I was looking for information about mathematics programs at other liberal arts colleges, so I put together this collection of links. I thought others might find it helpful, so I’m posting it here. (I actually asked ChatGPT to assemble this list. It did an OK, but not great, job. About 70% of the links were... Read More| David Richeson: Division by Zero
Recently I was made aware of the work Ed Solomon had been doing with data from the 2020 Colorado Cast Vote Records (CVRs), and I’ve taken some time to replicate and validate some of his data observations. I don’t always agree with Ed, but I wanted to take some time and verify the facts of the matter for myself.| Digital Poll Watchers (dot) Org
MuSA_RT 3.1 is out! This latest version features RealityKit-based graphics, support for audio file input, and the ability to stream result analysis to other MuSA_RT clients for local rendering.| ARJF
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Nina found the official solution to Dudeney’s “Digital Difficulties” (her article on it is shared here and here). The guess on how they solved it was confirmed in the Strand solut…| Win Vector LLC
Some years ago I speculated that it would nice if a certain mathematical object existed, and even nicer if it were to satisfy an ordinary differential equation of a special sort. I was motivated by a particular physical question, and it seemed very natural to me to imagine such an object... So natural that I was sure that it must already have been studied, the equation for it known. As a result, every so often I'd go down a rabbit hole of a literature dig, but not with much success because it...| Asymptotia
Not too long ago, my math block started the same way every day: students walked in, grabbed their notebooks, and started a warm-up. Some finished quickly.| Region 13's Blog
In another blow to public schools, a majority of House members voted on May 7 to lower the high school mathematics graduation requirements for all public schools. Because the new, lower standards fall below the minimum admissions requirements for UNC system schools, the bill also directs the UNC Board of Governors to lower its standards […]| Public Schools First NC
At a quick glance: 32 is greater than 10. 31/32 is about 0.96875, not near pi ~ 3.141593. 31/10 = 3.1 is a worse approximation of pi than 22/7 ~ 3.142857.| Win Vector LLC
Every summer, the School of Education hosts a Math Teacher Academy, in which math teachers from all over the state can come and learn about techniques that will best help their students succeed. This year, it will be June 23-25 from 9 a.m. to 3 p.m.| The Baylor Lariat - The official student news source
Welcome to the 239th installment of the Carnival of Mathematics, a roundup of interesting mathematics content from April 2025. The Carnival is a monthly post hosted by a different blog each month. Since this is the 239th Carnival, I’ll begin with some interesting facts about the number 239. Fun Facts About 239 239 is a … Continue reading Carnival Of Mathematics→| Reflections and Tangents
I recently watched a great interview of the mathematician and Fields medalist (2022) Hugo Duminil-Copin by Science étonnante (aka David Louapre). At some point, there was an interesting discussion on the role of AI in the discovery of new mathematical findings. I think the arguments generalize far beyond mathematics: AI as a creative sparring partner, AI as a way to have an inner, interactive dialogue with oneself, AI as a tool to automate the boring parts of the process, human fallibility a...| blog.mathieuacher.com
When you teach at a small high school, like I do, you where a lot of different hats. There’s a ton of stuff that needs to get done and are fewer people to do it. Naturally, everybody is asked…| lazy 0ch0
Can chance childhood thoughts shape the whole life? Maybe a lifetime studying the nuanced complexity of humanity alongside the elusive simplicity of mathematics comes down to The Woodentops.| Alan Dix
Using Dynamic Technology to Build Understanding ◊ Episode 6: Quadrilaterals ◊ In Episode 5, we took a brief look at trapezoids, and that got me thinking about the family of quadrilaterals: all the wonderful four-sided polygons that students learn about in geometry. How might Dynamic Geometry (DG) help our learners investigate these shapes? Constructing Parallelograms … Continue reading Go For Geometry! 6→| Reflections and Tangents
So much of mathematics is focused on the end point: the answer, the solution, the proof. We attach so much emphasis on the destination — the more efficient the better. We place such high value on t…| How I teach maths.
A deep dive into DeepSeek’s Multi-Head Latent Attention, including the mathematics and implementation details. The layer is recreated in Julia using Flux.jl.| liorsinai.github.io
A paper from last week with high press visibility that makes claims to climate1 applicability is titled: Topology shapes dynamics of higher-order networks The higher-order Topological Kuramoto dyna…| GeoEnergy Math
The Ultimate Guide to AI Math Solvers: Boosting Your Math Skills with Technology Introduction to AI Math Solvers Mathematics has always been a subject that challenges students, often requiring extra practice and assistance to master. With the rapid advancements in technology, Artificial Intelligence (AI) has stepped in to bridge the gap, offering innovative solutions that […]| Interactive Mathematics - Mathematics, learning, computing, travel - and what...
We are currently seeking a Research Fellow in Eco-epidemiology/Human Ecology to join our team at Flinders University. The successful candidate will develop spatial eco-epidemiological models for the populations of Indigenous Australians exposed to novel diseases upon contact with the first European settlers in the 18th Century. The candidate will focus on: The ideal candidate willContinue reading "New job posting: Research Fellow in Eco-Epidemiology & Human Ecology"| Global Ecology @ Flinders
I love John Mason and although I’ve even quoted it myself a few times, I think I don’t wholeheartedly agree with the statement “it is the ways of thinking that are rich, not the task itself&#…| How I teach maths.
Root Finding| degenerateconic.com
There’s a pretty thought experiment that’s sometimes attributed to Democritus though it’s actually due to a later popularizer of the atomic hypothesis1 and it goes like this: Suppose we use the wor…| mathenchant.wordpress.com
Humans are lazy when thinking about infinity. Usually it doesn’t matter, but sometimes, our imprecision comes with big philosophical implications.| Steve Patterson
For the last several years, I’ve been on the hunt. I’ve been searching for an explanation for the popularity of irrational beliefs. People casually accept contradictions into their worldview; they are convinced that paradoxes exist. I’ve been trying to understand why. Their arguments frequently end up appealing to mistaken interpretations of quantum physics or the […]| Steve Patterson
Using Dynamic Technology to Build Understanding ◊ Episode 2: Draw Vs. Construct ◊ When using Dynamic Geometry technology platforms, a careful distinction must be made between a figure that is DRAWN…| Reflections and Tangents
Stephen Wolfram discusses understanding proofs discovered using automated theorem proving. Based on Wolfram’s proof of the simplest axioms of logic of Boolean algebra.| writings.stephenwolfram.com
The Martinez-Rueda algorithm computes boolean operations between polygons. It can be used for polygon intersections (polygon clipping), unions, differences a...| liorsinai.github.io
THIS POST IS CURRENTLY UNDER CONSTRUCTION| Matthew N. Bernstein
Structural Causal Models (SCMs) and Do-Calculus are foundational to causal reasoning and inference. SCMs formalize causal relationships through mathematical ...| Rehan Guha -Portfolio & Blog
Just in time for the holidays, Colleen Robichaux and I wrote this paper on positivity of Schubert coefficients. This paper is unlike any other paper I had written, both in the content and the way w…| Igor Pak's blog
What happens to ground truth when finding a factoid or photo no longer means consulting an archive but generating one from scratch? That’s the question that drives “Honey, AI Shrunk the Archive,” an essay I wrote for the forthcoming anthology New Directions in Digital Textual Studies.| Still Water Lab
[Needs cleanup… just dumping here for now.] Mark Jason Dominus tweeted and later blogged about this puzzle: From the four numbers [6, 6, 5, 2], using only the binary operations [+, -, *, /], form the number 17. When he tweeted the first time, I thought about it a little bit (while walking from my […]| The Lumber Room
In his wonderful On Proof and Progress in Mathematics, Thurston begins his second section “How do people understand mathematics?” as follows: This is a very hard question. Understanding is an individual and internal matter that is hard to be fully aware of, hard to understand and often hard to communicate. We can only touch on […]| The Lumber Room
(TODO: Learn and elaborate more on their respective histories and goals.) The formula (reminded via this post), a special case at of was found by Leibniz in 1673, while he was trying to find the area (“quadrature”) of a circle, and he had as prior work the ideas of Pascal on infinitesimal triangles, and that […]| The Lumber Room
1. Alphabet Suppose we have an alphabet of size . Its generating function (using the variable to mark length) is simply , as contains elements of length each. 2. Words Let denote the class of all words over the alphabet . There are many ways to find the generating function for . 2.1. We have […]| The Lumber Room
A long time ago, Diophantus (sort of) discussed integer solutions to the equation (solutions to this equation are called Pythagorean triples). Centuries later, in 1637, Fermat made a conjecture (now called Fermat’s Last Theorem, not because he uttered it in his dying breath, but because it was the last one to be proved — in ~1995) […]| The Lumber Room
Found via G+, a new physical experiment that approximates $latex \pi$, like Buffon’s needle problem: The Pi Machine. Roughly, the amazing discovery of Gregory Galperin is this: When a ball of…| The Lumber Room
If you’re a practitioner of machine learning, then there is little doubt you have seen or used an algorithm that falls into the general category of kernel methods. The premier example of such methods is the support vector machine. When introduced to these algorithms, one is taught that one must provide the algorithm with a kernel function that, intuitively, computes a degree of “similarity” between the objects you are classifying. In practice, one can get pretty far with only this under...| Matthew N. Bernstein
The dot product is a fundamental operation on two Euclidean vectors that captures a notion of similarity between the vectors. In this post, we’ll define the dot product and offer a number of angles for which to intuit the idea captured by this fundamental operation.| Matthew N. Bernstein
Is there anything new in Enumerative Combinatorics? Most experts would tell you about some interesting new theorems, beautiful bijections, advanced techniques, connections to other areas, etc. Most…| Igor Pak's blog
Math problems allow students to learn new concepts and strengthen problem-solving skills. But many learners feel confused or frustrated if they can’t find the correct solution. A math problem solver is a handy tool that helps students doublecheck their work and identify errors.| SquareCirclez
When a tech reporter for the NY Times outsourced her decisions for a week to ChatGPT, she complained that “AI made me basic.” But it turns out the math behind generative AI can lead to results that are blandly average or wildly inaccurate.| Still Water Lab
Previously: Covering Sieves. We’ve seen an intuitive description of presheaves as virtual objects. We can use the same trick to visualize natural transformations. A natural transformation can be drawn as a virtual arrow between two virtual objects corresponding to two presheaves and . Indeed, for every , seen as an arrow , we get an […]| Bartosz Milewski's Programming Cafe
Previously: Sheaves as Virtual Objects. In order to define a sheaf, we have to start with coverage. A coverage defines, for every object $latex u$, a family of covers that satisfy the sub-coverage …| Bartosz Milewski's Programming Cafe