Hacker News with Generative AI: Geometry

Electromagnetism as a Purely Geometric Theory (iopscience.iop.org)
This research article derives a nonlinear generalization of Maxwell's equations from a variational approach, when the action measures the variability of the metric tensor.

Physics, Electromagnetism, Mathematics, Geometry

148 points by andyjohnson0 6 days ago | 64 comments

Monsky's Theorem (mathmondays.com)
For which $n$ can you cut a square into $n$ triangles of equal area?

Mathematics, Geometry, Puzzles

48 points by hyperbrainer 6 days ago | 13 comments

Crows can recognize geometric regularity (phys.org)
A trio of animal physiologists at the University of Tübingen, in Germany, has found that at least one species of crow has the ability to recognize geometric regularity.

Animals, Science, Cognition, Crows, Geometry

178 points by wglb 9 days ago | 81 comments

65537-gon (wikipedia.org)
In geometry, a 65537-gon is a polygon with 65,537 (216 + 1) sides. The sum of the interior angles of any non–self-intersecting 65537-gon is 11796300°.

Geometry, Mathematics, Polygons

14 points by luu 11 days ago | 3 comments

A crow's math skills include geometry (npr.org)
Crows are able to look at a handful of four-sided shapes and correctly distinguish those that exhibit geometric regularity from those that don't, according to a provocative new study.

Animal Intelligence, Science, Geometry, Cognitive Abilities

21 points by domofutu 12 days ago | 0 comments

Actually drawing some ovals – that are not ellipses (2017) (medium.com)
In the last part I hopefully made it clear why you wouldn’t want to use an actual ellipse when making a real object, curves constructed from multiple fixed radius arcs are much more useful and look just the same.

Computer Graphics, Design, Geometry, Programming

43 points by todsacerdoti 27 days ago | 19 comments

Mathematicians Find Proof to 122-Year-Old Triangle-to-Square Puzzle (scientificamerican.com)
Mathematicians Find Proof to 122-Year-Old Triangle-to-Square Puzzle

Mathematics, Geometry, Puzzles, History

12 points by IdealeZahlen 30 days ago | 0 comments

Abel Prize Awarded to Japanese Mathematician Who Abstracted Abstractions (nytimes.com)
Masaki Kashiwara, a Japanese mathematician, received this year’s Abel Prize, which aspires to be the equivalent of the Nobel Prize in math. Dr. Kashiwara’s highly abstract work combined algebra, geometry and differential equations in surprising ways.

Mathematics, Awards, Japan, Algebra, Geometry

12 points by donohoe 31 days ago | 0 comments

'Once in a Century' Proof Settles Math's Kakeya Conjecture (quantamagazine.org)
The deceptively simple Kakeya conjecture has bedeviled mathematicians for 50 years. A new proof of the conjecture in three dimensions illuminates a whole crop of related problems.

Mathematics, Conjecture, Proof, Geometry

104 points by pseudolus 42 days ago | 29 comments

The three-dimensional Kakeya conjecture, after Wang and Zahl (wordpress.com)
There has been some spectacular progress in geometric measure theory: Hong Wang and Joshua Zahl have just released a preprint that resolves the three-dimensional case of the infamous Kakeya set conjecture!

Mathematics, Conjecture, Geometry

47 points by robinhouston 58 days ago | 1 comments

Geometric Algebra (bivector.net)
Clifford's Geometric Algebra enables a unified, intuitive and fresh perspective on vector spaces, giving elements of arbitrary dimensionality a natural home.

Mathematics, Geometry, Linear Algebra

200 points by agnishom 58 days ago | 61 comments

A simple geometry question that fools almost everyone (theguardian.com)
A triangle and a rectangle walked into a pub

Math, Puzzles, Geometry

16 points by mywacaday 67 days ago | 14 comments

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.

Mathematics, Optimization, Puzzles, Geometry

185 points by abe94 71 days ago | 90 comments

Generating Voronoi diagrams using Fortune's algorithm (redpenguin101.github.io)
Generating Voronoi Diagrams using Fortune’s Algorithm

Algorithms, Computer Graphics, Geometry, Data Structures

204 points by redpenguin101 77 days ago | 19 comments

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.

Artificial Intelligence, Mathematics, Geometry, Olympiads

64 points by hnhn34 78 days ago | 5 comments

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.

3D Modeling, Computer Graphics, Geometry, Algorithms

55 points by CarpeQueso 85 days ago | 6 comments

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:

Mathematics, Algebra, Number Theory, Geometry

124 points by t55 86 days ago | 48 comments

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.

Geometry, Mathematics, Paradox

5 points by lorepieri 90 days ago | 0 comments

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.

Art, Geometry, Nature

126 points by NoRagrets 95 days ago | 18 comments

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.

Mathematics, Geometry, Science

161 points by isaacfrond 100 days ago | 71 comments

Mathematician solves the moving sofa problem (phys.org)
A mathematician at Yonsei University, in Korea, claims to have solved the moving sofa problem.

Mathematics, Geometry, Puzzles

4 points by wglb 130 days ago | 2 comments

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.

Mathematics, Geometry, Number Theory, Artificial Intelligence

15 points by isaacfrond 130 days ago | 0 comments

The open problem -- the "moving sofa problem" -- has possibly just been solved! (twitter.com)

Mathematics, Computer Science, Geometry

12 points by NavinF 140 days ago | 4 comments

The Hexagonal Tiling Honeycomb (arxiv.org)
The hexagonal tiling honeycomb is a beautiful structure in 3-dimensional hyperbolic space.

Mathematics, Geometry, Hyperbolic Space

44 points by belter 141 days ago | 9 comments

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 $2.2195\cdots$.

Mathematics, Geometry, Optimization

131 points by petters 144 days ago | 36 comments

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.

Mathematics, Geometry, Computer Science, Research

59 points by phaedryx 145 days ago | 19 comments

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?

Mathematics, Geometry, 3D Printing

16 points by Tomte 154 days ago | 7 comments

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.

Mathematics, Geometry, Binary Trees, Pythagorean Triples

181 points by hakmem 156 days ago | 19 comments

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.