Hacker News with Generative AI: Geometry

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 3 days ago | 17 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 6 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 8 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 19 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 34 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 35 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 43 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 47 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 54 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 55 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 62 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 62 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 66 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 71 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 77 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 106 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 107 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 117 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 117 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 121 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 122 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 131 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 133 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.
Machine Learning, Data Science, Generative Models, Geometry
64 points by perone 140 days ago | 7 comments
Why 4D geometry makes me sad [video] (youtube.com)
Mathematics, Geometry, Video, Personal Opinion
100 points by surprisetalk 145 days ago | 44 comments
The Shape That Could Replace Space-Time –- Maybe [Amplituhedron] [video] (youtube.com)
Physics, Mathematics, Theoretical Physics, Geometry, Video
5 points by peter_d_sherman 149 days ago | 0 comments
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.
Mathematics, Geometry, Computer Science, Hyperbolic Geometry
49 points by lapnect 155 days ago | 4 comments
Students with 'impossible' proof of Pythagorean Theorem discover more solutions (livescience.com)
Mathematics, Education, Geometry
6 points by okasaki 156 days ago | 1 comments
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.
Mathematics, Geometry
248 points by nabla9 171 days ago | 67 comments
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.
Hair, Computer Science, Geometry, 3D Modeling
184 points by cainxinth 174 days ago | 29 comments