A simple geometry question that fools almost everyone
(theguardian.com)
A triangle and a rectangle walked into a pub
A triangle and a rectangle walked into a pub
The largest sofa you can move around a corner
(quantamagazine.org)
A new proof reveals the answer to the decades-old “moving sofa” problem. It highlights how even the simplest optimization problems can have counterintuitive answers.
A new proof reveals the answer to the decades-old “moving sofa” problem. It highlights how even the simplest optimization problems can have counterintuitive answers.
Generating Voronoi diagrams using Fortune's algorithm
(redpenguin101.github.io)
Generating Voronoi Diagrams using Fortune’s Algorithm
Generating Voronoi Diagrams using Fortune’s Algorithm
Gold-Medalist Performance in Solving Olympiad Geometry with AlphaGeometry2
(arxiv.org)
We present AlphaGeometry2, a significantly improved version of AlphaGeometry introduced in Trinh et al. (2024), which has now surpassed an average gold medalist in solving Olympiad geometry problems.
We present AlphaGeometry2, a significantly improved version of AlphaGeometry introduced in Trinh et al. (2024), which has now surpassed an average gold medalist in solving Olympiad geometry problems.
Building a Mesh Using Spherical Embedding
(andrews.wiki)
When trying to build a 3D model of an object in the real world, the goal is most often to construct a connected mesh of triangles or quadrilaterals that represents that object's surface.
When trying to build a 3D model of an object in the real world, the goal is most often to construct a connected mesh of triangles or quadrilaterals that represents that object's surface.
Quaternions and spherical trigonometry
(wordpress.com)
Hamilton’s quaternion number system is a non-commutative extension of the complex numbers, consisting of numbers of the form where are real numbers, and are anti-commuting square roots of with , , . While they are non-commutative, they do keep many other properties of the complex numbers:
Hamilton’s quaternion number system is a non-commutative extension of the complex numbers, consisting of numbers of the form where are real numbers, and are anti-commuting square roots of with , , . While they are non-commutative, they do keep many other properties of the complex numbers:
Gabriel's Horn
(wikipedia.org)
A Gabriel's horn (also called Torricelli's trumpet) is a type of geometric figure that has infinite surface area but finite volume.
A Gabriel's horn (also called Torricelli's trumpet) is a type of geometric figure that has infinite surface area but finite volume.
Rafael Araujo's 20 Mesmerizing Geometrical Masterpieces (2024)
(abakcus.com)
Artist Rafael Araujo expresses his love of nature through geometry, intertwining mathematical precision with the organic beauty found within the natural world.
Artist Rafael Araujo expresses his love of nature through geometry, intertwining mathematical precision with the organic beauty found within the natural world.
Mathematicians discover new way for spheres to 'kiss'
(quantamagazine.org)
A new proof marks the first progress in decades on important cases of the so-called kissing problem. Getting there meant doing away with traditional approaches.
A new proof marks the first progress in decades on important cases of the so-called kissing problem. Getting there meant doing away with traditional approaches.
Mathematician solves the moving sofa problem
(phys.org)
A mathematician at Yonsei University, in Korea, claims to have solved the moving sofa problem.
A mathematician at Yonsei University, in Korea, claims to have solved the moving sofa problem.
The Year in Math
(quantamagazine.org)
Landmark results in geometry and number theory marked an exciting year for mathematics, at a time when advances in artificial intelligence are starting to transform the subject’s future.
Landmark results in geometry and number theory marked an exciting year for mathematics, at a time when advances in artificial intelligence are starting to transform the subject’s future.
The Hexagonal Tiling Honeycomb
(arxiv.org)
The hexagonal tiling honeycomb is a beautiful structure in 3-dimensional hyperbolic space.
The hexagonal tiling honeycomb is a beautiful structure in 3-dimensional hyperbolic space.
Optimality of Gerver's Sofa
(arxiv.org)
We resolve the moving sofa problem by showing that Gerver's construction with 18 curve sections attains the maximum area $2.2195\cdots$.
We resolve the moving sofa problem by showing that Gerver's construction with 18 curve sections attains the maximum area $2.2195\cdots$.
An Aperiodic Monotile (2023)
(uwaterloo.ca)
An aperiodic monotile, sometimes called an "einstein", is a shape that tiles the plane, but never periodically. In this paper we present the first true aperiodic monotile, a shape that forces aperiodicity through geometry alone, with no additional constraints applied via matching conditions.
An aperiodic monotile, sometimes called an "einstein", is a shape that tiles the plane, but never periodically. In this paper we present the first true aperiodic monotile, a shape that forces aperiodicity through geometry alone, with no additional constraints applied via matching conditions.
3D Space Can Be Tiled with Corner-Free Shapes
(hackaday.com)
Tiling a space with a repeated pattern that has no gaps or overlaps (a structure known as a tessellation) is what led mathematician [Gábor Domokos] to ponder a question: how few corners can a shape have and still fully tile a space?
Tiling a space with a repeated pattern that has no gaps or overlaps (a structure known as a tessellation) is what led mathematician [Gábor Domokos] to ponder a question: how few corners can a shape have and still fully tile a space?
A binary tree of all Pythagorean triples
(richardt.io)
I sketch how the stereographic projection of the Stern–Brocot tree forms an ordered binary tree of Pythagorean triples, which can be used to compute best approximations of turn angles of points on the circle and finally trigonometric functions.
I sketch how the stereographic projection of the Stern–Brocot tree forms an ordered binary tree of Pythagorean triples, which can be used to compute best approximations of turn angles of points on the circle and finally trigonometric functions.
The geometry of data: the missing metric tensor and the Stein score [Part II]
(christianperone.com)
I’m writing this second part of the series because I couldn’t find any formalisation of this metric tensor that naturally arises from the Stein score (especially when used with learned models), and much less blog posts or articles about it, which is surprising given its deep connection between score-based generative models, diffusion models and the geometry of the data manifold.
I’m writing this second part of the series because I couldn’t find any formalisation of this metric tensor that naturally arises from the Stein score (especially when used with learned models), and much less blog posts or articles about it, which is surprising given its deep connection between score-based generative models, diffusion models and the geometry of the data manifold.
2-adic numbering for binary tilings
(11011110.github.io)
I’ve posted quite a bit about the binary tiling of the hyperbolic plane recently, including what you get when you shrink its vertical edges, a related “nowhere-neat” tessellation, the connection to Smith charts and Escher, a method to 3-color its tiles, a half-flipped variation of the tiling, and its applications in proving that folding origami is hard. But I thought there might be room for one more post, in honor of the Wikipedia binary tiling article’s new Good Article Status.
I’ve posted quite a bit about the binary tiling of the hyperbolic plane recently, including what you get when you shrink its vertical edges, a related “nowhere-neat” tessellation, the connection to Smith charts and Escher, a method to 3-color its tiles, a half-flipped variation of the tiling, and its applications in proving that folding origami is hard. But I thought there might be room for one more post, in honor of the Wikipedia binary tiling article’s new Good Article Status.
Counterintuitive Properties of High Dimensional Space (2018)
(eecs.berkeley.edu)
Our geometric intuition developed in our three-dimensional world often fails us in higher dimensions. Many properties of even simple objects, such as higher dimensional analogs of cubes and spheres, are very counterintuitive. Below we discuss just a few of these properties in an attempt to convey some of the weirdness of high dimensional space.
Our geometric intuition developed in our three-dimensional world often fails us in higher dimensions. Many properties of even simple objects, such as higher dimensional analogs of cubes and spheres, are very counterintuitive. Below we discuss just a few of these properties in an attempt to convey some of the weirdness of high dimensional space.
Curly-Cue: Geometric Methods for Highly Coiled Hair
(cs.yale.edu)
We present geometric methods for generating shapes that are characteristic of highly coiled hair.
We present geometric methods for generating shapes that are characteristic of highly coiled hair.
An n-ball Between n-balls
(arnaldur.be)
There is a geometric thought experiment that is often used to demonstrate the counterintuitive shape of high-dimensional phenomena. This article is an interactive visual journey into the construct in the thought experiment, and the mathematics behind it.
There is a geometric thought experiment that is often used to demonstrate the counterintuitive shape of high-dimensional phenomena. This article is an interactive visual journey into the construct in the thought experiment, and the mathematics behind it.
Python for Inversive and Hyperbolic Geometry
(psu.ac.th)
The Python code available on this page is a collection of classes and support functions for visualizing inversive and hyperbolic geometry, with the hyperbolic examples utilizing the Poincaré disc model.
The Python code available on this page is a collection of classes and support functions for visualizing inversive and hyperbolic geometry, with the hyperbolic examples utilizing the Poincaré disc model.
Physicists Reveal a Quantum Geometry That Exists Outside of Space and Time
(quantamagazine.org)
A decade after the discovery of the “amplituhedron,” physicists have excavated more of the timeless geometry underlying the standard picture of how particles move.
A decade after the discovery of the “amplituhedron,” physicists have excavated more of the timeless geometry underlying the standard picture of how particles move.
Mathematicians discover new class of shape seen throughout nature
(nature.com)
Mathematicians have described1 a new class of shape that characterizes forms commonly found in nature — from the chambers in the iconic spiral shell of the nautilus to the way in which seeds pack into plants.
Mathematicians have described1 a new class of shape that characterizes forms commonly found in nature — from the chambers in the iconic spiral shell of the nautilus to the way in which seeds pack into plants.
Mathematicians discover new class of shape seen throughout nature
(nature.com)
Mathematicians have described1 a new class of shape that characterizes forms commonly found in nature — from the chambers in the iconic spiral shell of the nautilus to the way in which seeds pack into plants.
Mathematicians have described1 a new class of shape that characterizes forms commonly found in nature — from the chambers in the iconic spiral shell of the nautilus to the way in which seeds pack into plants.
Mathematicians define new class of shape seen throughout nature
(nature.com)
Mathematicians have described1 a new class of shape that characterizes forms commonly found in nature — from the chambers in the iconic spiral shell of the nautilus to the way in which seeds pack into plants.
Mathematicians have described1 a new class of shape that characterizes forms commonly found in nature — from the chambers in the iconic spiral shell of the nautilus to the way in which seeds pack into plants.