A Pythagorean-like theorem for tetrahedra, and its generalization to higher dimensions. Illustrated with Python code.| John D. Cook
The cross ratio of four points A, B, C, D is defined by where XY denotes the length of the line segment from X to Y. The idea of a cross ratio goes back at least as far as Pappus of Alexandria (c. 290 – c. 350 AD). Numerous theorems from geometry are stated in terms of the cross ratio. For example, the cross […] The post Cross ratio first appeared on John D. Cook.| John D. Cook
Apollonius of Perga (c. 262 BC – c. 190 BC) discovered a theorem that generalizes the Pythagorean theorem but isn’t nearly as well known. Let ABC be a general triangle, and let D be the midpoint of the segment AB. Let a be the length of the side opposite A and b the length of the side […] The post An ancient generalization of the Pythagorean theorem first appeared on John D. Cook.| John D. Cook
Exactly one month ago I wrote about sampling points from a triangle. This post will look at the three dimensional analog, sampling from a tetrahedron. The generalization to higher dimensions works as well. Sampling from a triangle In the triangle post, I showed that a naive approach doesn’t work but a variation does. If the […] The post Random samples from a tetrahedron first appeared on John D. Cook.| John D. Cook
‘István Dukai is a Budapest-based graphic and visual artist whose works evoke the traditions of constructivism, minimal art, op art, and the Bauhaus. Behind the discipline of form lies a profound sensitivity and deep respect for materials. His art is guided by a reductionist approach where every line, curve, and colour has its own place and purpose.’ The post The Nature of Cyberspace: István Dukai’s New Exhibition at Hotel Clark appeared first on Hungarian Conservative.| Hungarian Conservative
Given the anti-clockwise points of a properly formed, non-self-intersecting, not-necessarily-convex polygon, render it as a filled ASCII art polygon. Input A series of at least 3 (x,y) pairs representing the points. It's up to you whether the first point is repeated as the last point to close the loop. x coordinates are integers in the range 1-78. coordinates are integers in the range 1-38. Output An 80x40 text field representing the range 0-79 x 0-39 (0,0 at the top left), consisting of spac...| Recent Questions - Code Golf Stack Exchange
Today, another guest post, this time from a homeschooling mom. — Henri Algebra and Geometry Greetings! I hope this email finds you well. I know you don’t know us, but I wanted to personally…| Henri's Math Education Blog
(This challenge exists to extend sequence A276272 in the On-Line Encyclopedia of Integer Sequences, and perhaps create a new OEIS sequence1.) This is a code-challenge, which will have you write code to compute as many terms of this sequence as possible. --- Background A polytet is a kind of polyform which is constructed by gluing regular tetrahedra together face-to-face in such a way that none of the interiors of the tetrahedra overlap. These are counted up to rotation in 3-space but not refl...| Recent Questions - Code Golf Stack Exchange
There are mathematical operations of all kinds with the property that doing the operation twice is tantamount to not doing anything at all. Such operations are called involutions, and you can find them all over the place in math: taking the negative of a number, taking the reciprocal of a number, rotating an object by […]|
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
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
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
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
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
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
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
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
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
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
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 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
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
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
