Hacker News with Generative AI: Geometry

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
157 points by isaacfrond 2 days ago | 69 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 31 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 32 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 42 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 42 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 46 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 47 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 56 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 58 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 65 days ago | 7 comments
Why 4D geometry makes me sad [video] (youtube.com)
Mathematics, Geometry, Video, Personal Opinion
100 points by surprisetalk 70 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 75 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 80 days ago | 4 comments
Students with 'impossible' proof of Pythagorean Theorem discover more solutions (livescience.com)
Mathematics, Education, Geometry
6 points by okasaki 81 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 96 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 99 days ago | 29 comments
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.
Mathematics, Geometry, Visualizations
242 points by Hugsun 101 days ago | 56 comments
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.
Python, Geometry, Visualization
77 points by thunderbong 112 days ago | 13 comments
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.
Quantum Physics, Physics, Mathematics, Geometry
42 points by pseudolus 115 days ago | 12 comments
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.
Mathematics, Nature, Geometry
272 points by pseudolus 117 days ago | 63 comments
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.
Mathematics, Nature, Geometry, Biology
48 points by signa11 119 days ago | 4 comments
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.
Mathematics, Nature, Geometry
23 points by gnabgib 120 days ago | 2 comments
Metallic Mean (wikipedia.org)
Mathematics, Geometry, Golden Ratio
7 points by downboots 121 days ago | 1 comments
Why Gauss wanted a heptadecagon on his tombstone (scientificamerican.com)
Mathematics, History, Science, Geometry
297 points by sohkamyung 124 days ago | 62 comments
Soft cells and the geometry of seashells (academic.oup.com)
A central problem of geometry is the tiling of space with simple structures.
Geometry, Biology, Science
36 points by pabo 129 days ago | 0 comments
A universal triangulation for flat tori (2022) (arxiv.org)
A result due to Burago and Zalgaller (1960, 1995) states that every orientable polyhedral surface, one that is obtained by gluing Euclidean polygons, has an isometric piecewise linear (PL) embedding into Euclidean space $\mathbb{E}^3$. A flat torus, resulting from the identification of the opposite sides of a Euclidean parallelogram, is a simple example of polyhedral surface. In a first part, we adapt the proof of Burago and Zalgaller, which is partially non-constructive, to produce PL isometric embeddings of flat tori. Our implementation produces embeddings with a huge number of vertices, moreover distinct for every flat torus. In a second part, based on another construction of Zalgaller (2000) and on recent works by Arnoux et al. (2021), we exhibit a universal triangulation with 2434 triangles which can be embedded linearly on each triangle in order to realize the metric of any flat torus.
Mathematics, Geometry, Topology
4 points by coley_moke 136 days ago | 1 comments
What is Spin? A Geometric explanation [video] (youtube.com)
Physics, Geometry, Video, Education
4 points by chii 184 days ago | 0 comments
Shortest distance between two points is not always a straight line (metaquestions.me)
Mathematics, Geometry, Logic
37 points by ofou 200 days ago | 15 comments
Banach–Tarski Paradox (wikipedia.org)
Mathematics, Geometry, Paradoxes
62 points by tontonius 206 days ago | 75 comments
Desperately Seeking Squircles (figma.com)
Design, Geometry, Figma
15 points by kjeetgill 207 days ago | 4 comments