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.
Mathematicians Find Proof to 122-Year-Old Triangle-to-Square Puzzle (scientificamerican.com)
Mathematicians Find Proof to 122-Year-Old Triangle-to-Square Puzzle
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.
'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.
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!
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.
A simple geometry question that fools almost everyone (theguardian.com)
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.
Generating Voronoi diagrams using Fortune's algorithm (redpenguin101.github.io)
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.
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.
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:
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.
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.
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.
Mathematician solves the moving sofa problem (phys.org)
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.
The open problem -- the "moving sofa problem" -- has possibly just been solved! (twitter.com)
The Hexagonal Tiling Honeycomb (arxiv.org)
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 $2.2195\cdots$.
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.
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?
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.
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.
Why 4D geometry makes me sad [video] (youtube.com)
The Shape That Could Replace Space-Time –- Maybe [Amplituhedron] [video] (youtube.com)
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.
Students with 'impossible' proof of Pythagorean Theorem discover more solutions (livescience.com)
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.
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.