A new study explores how positive geometry could unify quantum physics and cosmology, offering a path toward a Theory of Everything.| The Debrief
September is here, and so is the school year for my US friends. Start a habit of doing some problem solving in your math classes this year! Here is the 1990 Calendar of Problems from 35 years ago f…| Reflections and Tangents
I was playing around with some ideas for a nice geometry puzzle involving circles on my iPad when my daughter saw it and said it looked like fire. I then added some background elements like the sun, the cloud, the sky, and the land, and now it looks like a tsunami of fire on a sunny day! Anyway, here’s the question. What fraction of the image is covered in fire? And more importantly, can you find it with as little calculation as possible? --- Clarifications: All curves in the fire are circu...| Recent Questions - Puzzling Stack Exchange
Here's a circular chip with three evenly spaced notches. Two such chips can be attached as shown, where the "boundary circle" of each chip goes through the center of the other, and the planes of the two chips are orthogonal. I have been trying to make a closed loop out of such chips but the best I have so far is the following (made in Geogebra). The two red circles at the top intersect nontrivially and the angle between them is about 130º, so this certainly doesn't lead to a closed loop, but...| Recent Questions - Puzzling Stack Exchange
Qwirkle tiles are identically-sized squares. Each has two attributes: one of 6 symbols and one of 6 colors. There are 3 tiles for each combination of symbol and color, thus 6×6×3 = 108 tiles. We want to fit these on a square grid (at most one tile at each position), while respecting standard Qwirkle constraints: Each horizontal or vertical segment of consecutive non-empty grid squares has tiles with either the same symbol and different colors, or the same color and different symbols. It fol...| Recent Questions - Puzzling Stack Exchange
Wow the summer is flying by, and it is back-to-school time already in many parts of the US. If you’re about to start your academic year, here is the August 2010 Calendar of Problems from 15 y…| Reflections and Tangents
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
A circle is a ‘self-referential’ kind of thing, we might say – a circle is a self-referential kind of thing because every point on its circumference has the very same relationship to the centre, which is also the circle, just...| The Negative Psychologist
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
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
h/t Benny Sudakov The Ramsey number R(ℓ,k) is the smallest integer n such that in any two-coloring of the edges of the complete graph on n vertices, , by red and blue, there is either a red (a complete graph … Continue reading →| Combinatorics and more
Note: this article| mathesis
Spotting my first daisy of the year always fills me with joy as it signals the fruitful promise of Summer. Often overlooked as a garden weed, the humble daisy possesses great beauty and also vast wisdom for those with eyes to see. Join me as I explore the daisy’s symbolism, folklore and medicinal properties alongside … More Ode to the Dazzling Daisy| The Muse in the Mirror
I’m all for finding activities that make math review actually fun for students! And when it comes to teaching geometry, with all the definitions and formulas that 4th and 5th... The post The No-Prep Geometry Boom Cards You NEED for Your Students appeared first on Alyssa Teaches.| Alyssa Teaches
I did it again.| ars
Let me get back to the idea of using the asterism as a Fediverse logo.| ars
Fresnel ellipsoids are regions of space in which obstacles will cause destructive interference for radio transmission. Post explains why ellipsoids.| John D. Cook
In this post, I discuss a remarkable new paper Cubulating the sphere with many facets by Sergey Avvakumov and Alfredo Hubard Abstract: For each we construct cube complexes homeomorphic to the -sphere with vertices in which the number of facets … Continue reading →| Combinatorics and more
A brief update: Since Friday June 13 Israel has been engaged in a direct war with Iran. This follows two major missiles attacks of Iran against Israel in April and October 2024, as well as Iran’s central role in the … Continue reading →| Combinatorics and more
Last time, we considered how to represent algebraically the division of a line segment in a given ratio. At the end, we touched on a subject I recalled discussing extensively almost four years ago: that such a “division” can be either internal (inside the segment, as you’d expect) or external (elsewhere on the line containing …Internal and External Division of a Segment Read More »| The Math Doctors
Gardener's method for drawing an ellipse and how it could be useful in computer graphics| John D. Cook
It is well known that we can make a cardioid by drawing straight lines inside a circle. Simply choose one point on the circle’s boundary to be the base point. Then, connect points on the circle to the points twice as far away from the base point (measured along the circumference). If there are n... Read More| David Richeson: Division by Zero
Add these Snowman Shape Puzzles to your lesson plan on shapes for preschoolers today! In this preschool shape activity, your students will match the snowman shapes to real life 2D shapes. Upgrade your preschool winter theme toolbox today! Read the full post: Simple Snowman Real Life 2D Shapes Puzzles| Kool Kids Games
Spring is here, and I’m sharing the May Calendar for some problem solving enjoyment in the last stretch of your school year. This is the May 2019 Calendar of Problems from 6 years ago for you…| Reflections and Tangents
I am surprised that nobody actually did a mathematical analysis of the variation model in OpenType. They are simply reusing the concepts| Typeof.net
Equations and Python code for going back and forth between quaternion and matrix representations of rotations.| John D. Cook
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
Welcome to April, and I hope you’re ready to spring into some problem solving! Here is the April 2011 Calendar of Problems from 14 years ago for you and your students to try. I have the last …| Reflections and Tangents
Using Dynamic Technology to Build Understanding ◊ Episode 5: Marvelous Midsegments ◊ Welcome to Episode 5, where we take the scenic route in our geometry journey. What happens when we find midpoints of the sides of various polygons and connect them with a segment (known as a midsegment)? Marvelous properties show up; read on for … Continue reading Go For Geometry! 5→| Reflections and Tangents
Normalising flows for PDE learning. Figure 1 Lipman et al. (2023) seems to be the origin point, extended by Kerrigan, Migliorini, and Smyth (2024) to function-valued PDEs. Figure 2: An illustration of our FFM method. The vector field (in black) transforms a noise sample drawn from a Gaussian process with a Matérn kernel (at ) to the function (at ) via solving a function space ODE. By sampling many such , we define a conditional path of measures approximately interpolating between and the f...| The Dan MacKinlay stable of variably-well-consider’d enterprises
Diffusion models for PDE learning. Figure 1 Slightly confusing terminology, because we are using diffusion models to learn PDEs, but the PDEs themselves are often used to model diffusion processes. Also sometimes the diffusion models that do the modelling aren’t actually diffusive, but are based on Poisson flow generative models. Naming things is hell. 1 Classical diffusion models TBD 2 Poisson Flow generative models These are based on non-diffusive physics but also seem to be used to simu...| The Dan MacKinlay stable of variably-well-consider’d enterprises
Figure 1 Placeholder. Levers for Biological Progress - by Niko McCarty In order for 50-100 years of biological progress to be condensed into 5-10 years of work, we’ll need to get much better at running experiments quickly and also collecting higher-quality datasets. This essay focuses on how we might do both, specifically for the cell. Though my focus in this essay is narrow — I don’t discuss bottlenecks in clinical trials, human disease, or animal testing — I hope others will take o...| The Dan MacKinlay stable of variably-well-consider’d enterprises
This is an open problem about “variable values” in variable fonts. Consider the variable value model we discussed in previous post| Typeof.net
Defining a mechanism to represent polynomials in OtVar’s mechanism is possible, if one axis value could be assigned to multiple axes. Fro| Typeof.net
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
In my last blog piece I wrote about my attempt to find a longer straight line in Great Britain that doesn't cross a public road than the one identified by Ordnance Survey in their 2019 blog. I did this using their OS Open Roads dataset, and I excluded public roads because that's what I think makes most sense. I found a longer line in a different area, though I definitely wouldn't recommend trying to walk it but I would recommend watching this video. I wasn't quite satisfied that my previous l...| Stats, Maps n Pix
In my current project I’m doing a fair amount of geometry, and one small problem I needed to solve a while back was finding whether a point is inside a cylinder. The accepted answer for this on math.s| Luke Plant's home page
More thoughts on diffusion guidance, with a focus on its geometry in the input space.| Sander Dieleman
This week’s Fiddler is an optimization problem about fitting particles in a box. You have three particles inside a unit square that all repel one another. The energy between each pair of particles is $1/r$, where $r$ is the distance between them. To be clear, the particles can be anywhere inside the square or on … Continue reading "Particles in a box" The post Particles in a box first appeared on Book Proofs.| Book Proofs
Update: Let me mention a ninth paper that just appeared on the arXive. IX. … and the optimal sofa for the moving sofa problem is … Gerver’s sofa. Optimality of Gerver’s Sofa…| Combinatorics and more
I was flipping through Owen Jones’s Grammar of Ornament a couple months ago, and my eye was caught by this handsome pattern I had not noticed before. This is Jones’s Plate XLII, in the chapter on designs from the Alhambra in Granada, Spain. He calls it, “Part of the ceiling of the Portico of the … Continue reading Constructing the Pattern on the Sala de la Barca Ceiling→| The Wandering Cartographer
(This is the math geek part about the Sala de la Barca ceiling. For instructions on constructing the pattern with compass and straightedge, go over to Part 1.) In the process of figuring out how to draw this pattern, I ran into a lot of questions, and had to do more than a little math … Continue reading The Mathematics of the Pattern on the Sala de la Barca Ceiling→| The Wandering Cartographer
Now that we know our way around the pattern (go back to Part 1), it should be fairly straightforward to construct with a compass and straightedge. But be aware: any pattern that requires you to construct a pentagon is an advanced challenge. They are trickier to make than squares or hexagons. Here’s what we want … Continue reading Constructing Bourgoin’s Figure 171 – Part 2→| The Wandering Cartographer
Just veering off into geometry here…. In November I was watching Eric Broug, an Islamic geometric design guru, give a talk online at an Islamic art conference, and I noticed that behind him they were projecting an interesting pattern on the scrim. I froze the video and grabbed a screenshot… What the heck is this? … Continue reading Constructing Bourgoin’s Figure 171 – Part 1→| The Wandering Cartographer
Please Note: This post may contain affiliate links. Please read my disclosure (link) for more info. Apart from classroom assessments, offline practices are often effective in cultivating crucial concepts in the learner, ensuring the real-life essence in them. Monopoly, for instance, is a game of business and money transactions. Not only for this, some board games may ensure ... Read more| Number Dyslexia
This note describes a method for fixing Thunderbird when it hangs while indexing. I’m posting this in case it’s useful to folks and for my own reference. It took me a quite a while to figure it out. So you have been using Thunderbird for your email for a while, on Windows, Linux, Mac, FreeBSD, and so on, for let’s say a couple of decades. If you copy your profile around much, you are bound to hit an indexing issue where Thunderbird is unable to index your archives. You open up Activity ...| decuser’s blog
Proof in Geometry Many years ago, when I was still teaching high school, I added a Teaching Proof page to my website, which included a bit of philosophizing and links to the relevant parts of the site. If you’re looking for ideas and materials on this topic, you should definitely check out that page. I’ve… Continue reading Proof in High School→| Henri's Math Education Blog
This is an update of a post from 2013, when Didax published my book Working with Pentominoes. You can still buy the book. It is geared to grades 4-8, though I used some of the content in high schoo…| Henri's Math Education Blog
This week’s Fiddler is about a generalized notion of “radius”. For a circle with radius $r$, its area is $\pi r^2$ and its circumference is $2\pi r$. If you take the derivative of the area formula with respect to $r$, you get the circumference formula! Let’s define the term “differential radius.” The differential radius $r$ … Continue reading "When is a triangle like a circle?" The post When is a triangle like a circle? first appeared on Book Proofs.| Book Proofs
This is a note where I begin to develop my own geometry based on the example of Euclid, but with an eye towards addressing its shortcomings - what hubris?! But seriously, I’m pretty sure my geometry will pale in comparison. The purpose of the exploration is not to show off, but to learn more about the world and in this regard, will be interesting and educational.| decuser’s blog
This note sets up a series of related notes pertaining to my explorations in Geometry and by extension, Maths. The explorations are my work in trying to make sense of the world through math. They are presented here, in part, to motivate me to express my thoughts in a more organized fashion than I might otherwise, and in part to share in the hopes that some small few might benefit or wish to chat about things. So, enjoy and if you do, feel free to email me about it.| decuser’s blog
In my previous blog, I described a lesson I taught based on measuring the area and perimeter of my foot. Here I describe what happened when I returned to the class to have the students think more about the data they recorded and the mathematics it revealed. The post Part 2: Untangling Area and Perimeter appeared first on MARILYN BURNS MATH.| MARILYN BURNS MATH
Bless me Father for I have sinned. It’s been four and a half years since my last curriculum article. My entire teaching career has been spent navigating a student ability gap that got noticeably wider sometime before the pandemic. The strongest kids are getting better, the weaker ones spent middle school not learning a thing […]| educationrealist
Here’s the thing: I only have one prep this year. The whole year. I have a lot of credentials. I can teach any subject except science. (OK, language and PE as well, but they’re not real…| educationrealist
I’ve been looking through my saved puzzles again and I found this nice little one in the maths newsletter from Chris Smith (@aap03102): It’s a nice little question that took me some thinking about. First I considered the half squares with hypotenuse 2. As these are isoceles RATs, that means their side length is rt2 […]| cavmaths
When two lines are crossed with a line called transversal than the pair of angles formed on the outer but on the opposite side of the transversal are called alternate exterior angles. If two lines being crossed are parallel lines than the Alternate exterior angles are parallel. Source| Earth's Lab
Eccentricity can be defined as how much a conic section deviates from being circular. A circle has an eccentricity of 0, this means that eccentricity shows how un-circular a curve is. Higher the eccentricity, lesser the curve. Eccentricity of different curves: A circle has an eccentricity of Source| Earth's Lab
Add these Snowman Shape Puzzles to your lesson plan on shapes for preschoolers today! In this preschool shape activity, your students will match the snowman shapes to real life 2D shapes. Upgrade your preschool winter theme toolbox today!| Kool Kids Games
NP-Incompleteness:| www.kuniga.me
NP-Incompleteness:| www.kuniga.me
(Cross-posted from Talking Math with Your Kids) Two and a half years ago, I was developing Which One Doesn’t Belong? (before Stenhouse had signed on to publish it). I went on a tour of elemen…| Overthinking my teaching
Behind the scenes - how I'm making redblobgames.com - interactive explanations of game development algorithms and math| simblob.blogspot.com
Behind the scenes - how I'm making redblobgames.com - interactive explanations of game development algorithms and math| simblob.blogspot.com