Tootfinder

Hyper-Catalan and Geode Recurrences and Three Conjectures of Wildberger
Dean Rubine
https://arxiv.org/abs/2507.04552 https://arxiv.or…

Hyper-Catalan and Geode Recurrences and Three Conjectures of Wildberger
The hyper-Catalan number $C[m_2,m_3,m_4,\ldots]$ counts the number of subdivisions of a roofed polygon into $m_2$ triangles, $m_3$ quadrilaterals, $m_4$ pentagons, etc. Its closed form has been known since Erdélyi and Etherington, 1940. In 2025, Wildberger and Rubine showed its generating sum $\mathbf{S}[t_2,t_3,t_4,\ldots]$ is a zero of the general geometric univariate polynomial. We use that to derive a recurrence for hyper-Catalans, which expresses each in terms of other hyper-Catalans with…

Decremental Greedy Polygons and Polyhedra Without Sharp Angles
David Eppstein
https://arxiv.org/abs/2507.04538 https://arxiv.org/pdf/…

Decremental Greedy Polygons and Polyhedra Without Sharp Angles
We show that the max-min-angle polygon in a planar point set can be found in time $O(n\log n)$ and a max-min-solid-angle convex polyhedron in a three-dimensional point set can be found in time $O(n^2)$. We also study the maxmin-angle polygonal curve in 3d, which we show to be $\mathsf{NP}$-hard to find if repetitions are forbidden but can be found in near-cubic time if repeated vertices or line segments are allowed, by reducing the problem to finding a bottleneck cycle in a graph. We formalize …

Billiards and Hofer's Geometry
Mark Berezovik, Konstantin Kliakhandler, Yaron Ostrover, Leonid Polterovich
https://arxiv.org/abs/2507.04767 https://

Billiards and Hofer's Geometry
We present a link between billiards in convex plane domains and Hofer's geometry, an area of symplectic topology. For smooth strictly convex billiard tables, we prove that the Hofer distance between the corresponding billiard ball maps admits an upper bound in terms of a simple geometric distance between the tables. We use this result to show that the billiard ball map of a convex polygon lies in the completion, with respect to Hofer's metric, of the group of smooth area-preserving maps of the …

Polygon co-founder Sandeep Nailwal is named CEO of the Polygon Foundation after the departure of Mihailo Bjelic, and unveils changes to streamline the nonprofit (Daniel Kuhn/The Block)
https://www.theblock.co/post/35…

🇺🇦 #NowPlaying on #BBC6Music's #ThePeoplesParty with #AFRODEUTSCHE
Polygon Window:
🎵 Bike Pump Meets Bucket
#PolygonWindow
https://aphextwin.bandcamp.com/track/bike-pump-meets-bucket
https://open.spotify.com/track/2rqDZIpxVFhkriNBKf4EAS

Soft-Clamped Perimeter Modes of Polygon Resonators
Zhihao Niu, Yuanyuan Zhao
https://arxiv.org/abs/2506.18109 https://arxiv.org/pdf/2…

Soft-Clamped Perimeter Modes of Polygon Resonators
Polygon resonators are known for their small size and high force sensitivity. We propose an analytical model to predict the mechanical quality factors (Q) of polygon perimeter modes by incorporating a stress term in the Timoshenko-Gere equation. We discuss the origin of losses and their relationship with polygon structure dimensions. Notably, we demonstrate that the torsion loss can be suppressed through the torsion angle of supporting tethers. These results offer new insights into optimizing h…

Conductive homogeneity of locally symmetric polygon-based self-similar sets
J. Kigami amd Y. Ota
https://arxiv.org/abs/2505.23057 https://

Conductive homogeneity of locally symmetric polygon-based self-similar sets
We provide a rich family of self-similar sets, called locally symmetric polygon-based self-similar sets, as examples of metric spaces having conductive homogeneity, which was introduced as a sufficient condition for the construction of counterparts of ``Sobolev spaces'' on compact metric spaces. In particular, our results imply the existence of ``Brownian motions'' on our family of self-similar sets at the same time. Unlike the known examples like the Sierpinski carpet by Barlow-Bass, unconstra…

This https://arxiv.org/abs/2407.15625 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_csCY_…

Dressed to Gamble: How Poker Drives the Dynamics of Wearables and Visits on Decentraland's Social Virtual World
Decentraland is a blockchain-based social virtual world where users can publish and sell wearables to customize avatars. In it, the third-party Decentral Games (DG) allows players of its flagship game ICE Poker to earn cryptocurrency only if they possess certain wearables. Herein, we present a comprehensive study on how DG and its game influence the dynamics of wearables and in-world visits in Decentraland. To this end, we analyzed 5.9 million wearable transfers made on the Polygon blockchain (…

„Donkey Kong Bananza is Nintendo going full galaxy brain.“
Can’t wait! 🫨
https://www.polygon.com/news/613165/switch-2-donkey-kong-bananza-creation-secrets-development

Nintendo’s Donkey Kong team sounds increasingly bananas
Many of the core ideas for the new Switch 2 game came from unexpected sources, Nintendo reveals

Multipoint conformal integrals in $D$ dimensions. Part II: Polygons and basis functions
K. B. Alkalaev, Semyon Mandrygin
https://arxiv.org/abs/2507.01904 h…

Multipoint conformal integrals in $D$ dimensions. Part II: Polygons and basis functions
We explicitly construct a class of multivariate generalized hypergeometric series which is conjectured in our previous paper [Alkalaev & Mandrygin 2025] to calculate multipoint one-loop parametric conformal integrals in $D$ dimensions. Our approach is based on a simple diagrammatic algorithm which systematically builds both arguments and series coefficients in terms of a convex polygon which is part of the Baxter lattice. The examples of the box, pentagon, and hexagon integrals are considered i…

Uniqueness in determining a convex polygonal source of an elastic body
Jianli Xiang
https://arxiv.org/abs/2506.02870 https://arxiv.or…

Uniqueness in determining a convex polygonal source of an elastic body
In this work, we consider the time-harmonic inverse elastic source problem of a fixed frequency for the Navier equation in two dimensions. We show that a convex polygon can be uniquely determined by a single far field measurement. Our approach relies on the corner singularity analysis of solutions to the inhomogeneous Navier equation with a source term in a sector. This paper also contributes to corner scattering theory for the Navier equation in an non-convex domain.

Horofunctions of infinite Sierpinski polygon graphs
Daniele D'Angeli, Francesco Matucci, Davide Perego, Emanuele Rodaro
https://arxiv.org/abs/2507.23681 https://

Horofunctions of infinite Sierpinski polygon graphs
Generalizing works of D'Angeli and Donno, we describe, starting from an infinite sequence over $r$ letters with $r \neq 4i$ and $i \in \mathbb{N}$, a sequence of pointed finite graphs. We study the pointed Gromov-Hausdorff limit graphs giving a description of isomorphim classes in terms of dihedral groups and providing insights on the horofunction boundaries in terms of Busemann and non-Busemann points.

Ms. Marvel is #1. 🤷
Sure, I liked the show... but is it better than Loki, Wandavision, Moon Knight, Hawkeye... even She-Hulk?
No. Just... no. I gotta wonder if this is an #MCU fan troll.
✅ Every #Marvel TV show in the Cinematic Universe, ranked | Polygon

Every Marvel Cinematic Universe TV show, ranked
From Ironheart to Loki, from WandaVision to Jessica Jones, we rank the best MCU TV shows and the worst, and cover which Marvel shows are canon.

Can we "seamlessly" divide a polygon?
Byungchang So
https://arxiv.org/abs/2506.11742 https://arxiv.org/pdf/2506.11742

Can we "seamlessly" divide a polygon?
While the contents of Euclid's Elements are well-known these days, some characters of the original text have been overlooked due to interpretation by modern mathematical languages. The lens of modern mathematics once anachronistically misled researchers of the history of mathematics, and this shows that the classic text itself contains ideas that have not been completely accurately translated. This article concentrates on the division of geometric figures(line, polygon, polyhedron, etc.). In mo…

This https://arxiv.org/abs/2405.12949 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_csCG_…

Exact predicates, exact constructions and combinatorics for mesh CSG
This article introduces a general mesh intersection algorithm that exactly computes the so-called Weiler model (also called an arrangement) and that uses it to implement boolean operations with arbitrary multi-operand expressions, CSG (constructive solid geometry) and some mesh repair operations. From an input polygon soup, the algorithm first computes the co-refinement, with an exact representation of the intersection points. Then, the decomposition of 3D space into volumetric regions (Weiler …

Cannonball Polygons with Multiplicities
Anji Dong, Katerina Saettone, Kendra Song, Alexandru Zaharescu
https://arxiv.org/abs/2507.18057 https://arxiv.org/p…

Cannonball Polygons with Multiplicities
We generalize the Cannonball Problem by introducing integer-valued and non-increasing arithmetic functions $w$. We associate these functions $w$ with certain polygons, which we call cannonball polygons. Through this correspondence, we show that for any $Z\in\mathbb{N}$, there exists a cannonball polygon with multiplicity 8 and largest side of length $Z$. Moreover, for any multiplicity $s$ greater than 8, we provide an asymptotic formula for the number of distinct classes of cannonball polygons …

This https://arxiv.org/abs/2307.07774 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

Heptagon relations from a simplicial 3-cocycle, and their cohomology
We introduce new algebraic structures associated with heptagon relations -- higher analogue of the well-known pentagon. The main points we deal with are: (i) polygon relations as algebraic imitations of Pachner moves, on the example of heptagon, (ii) parameterization of heptagon relations by simplicial 3-cocycles, (iii) applications to invariants of pairs "piecewise linear 5-manifold, a 3rd cohomology class on it".

Shape-determined kinetic pathways in two-dimensional solid-solid phase transitions
Ruijian Zhu, Yi Peng, Yanting Wang
https://arxiv.org/abs/2506.20423 http…

Shape-determined kinetic pathways in two-dimensional solid-solid phase transitions
Solid-solid phase transitions are ubiquitous in nature, but the kinetic pathway of anisotropic particle systems remains elusive, where the coupling between translational and rotational motions plays a critical role in various kinetic processes. Here we investigate this problem by molecular dynamics simulation for two-dimensional ball-stick polygon systems, where pentagon, hexagon, and octagon systems all undergo an isostructural solid-solid phase transition. During heating, the translational mo…

The number of maximal matchings in polygon rings
Chengqi Li, Jinhui Yin, Lingjuan Shi
https://arxiv.org/abs/2506.08687 https://arxiv.…

The number of maximal matchings in polygon rings
A matching of graph $G$ is maximal if it cannot be expanded by adding any edge to create a larger matching. In this paper, for a hexagonal ring $H$ with $n$ hexagons, we show that the number of maximal matchings of $H$ equals to the trace of the product of $n$ matrices, each of which is $S$, $L$, or $R$ according to the type of the connection mode of $H$. Finally, we extend this conclusion to arbitrary polygon rings and provide an algorithm to determine the transition matrices of polygon chains…

Computing Diverse and Nice Triangulations
Waldo G\'alvez, Mayank Goswami, Arturo Merino, GiBeom Park, Meng-Tsung Tsai
https://arxiv.org/abs/2506.01323 …

Computing Diverse and Nice Triangulations
We initiate the study of computing diverse triangulations to a given polygon. Given a simple $n$-gon $P$, an integer $ k \geq 2 $, a quality measure $σ$ on the set of triangulations of $P$ and a factor $ α\geq 1 $, we formulate the Diverse and Nice Triangulations (DNT) problem that asks to compute $k$ \emph{distinct} triangulations $T_1,\dots,T_k$ of $P$ such that a) their diversity, $\sum_{i < j} d(T_i,T_j) $, is as large as possible \emph{and} b) they are nice, i.e., $σ(T_i) \leq ασ^* $ …

Model Simplification through refinement
Dmitry Brodsky, Benjamin Watson
https://arxiv.org/abs/2507.15186 https://arxiv.org/pdf/2507.1…

Model Simplification through refinement
As modeling and visualization applications proliferate, there arises a need to simplify large polygonal models at interactive rates. Unfortunately existing polygon mesh simplification algorithms are not well suited for this task because they are either too slow (requiring the simplified model to be pre-computed) or produce models that are too poor in quality. These shortcomings become particularly acute when models are extremely large. We present an algorithm suitable for simplification of larg…

On the Performance of Cloud-based ARM SVE for Zero-Knowledge Proving Systems
Dumitrel Loghin, Shuang Liang, Shengwei Liu, Xiong Liu, Pingcheng Ruan, Zhigang Ye
https://arxiv.org/abs/2506.09505

On the Performance of Cloud-based ARM SVE for Zero-Knowledge Proving Systems
Zero-knowledge proofs (ZKP) are becoming a gold standard in scaling blockchains and bringing Web3 to life. At the same time, ZKP for transactions running on the Ethereum Virtual Machine require powerful servers with hundreds of CPU cores. The current zkProver implementation from Polygon is optimized for x86-64 CPUs by vectorizing key operations, such as Merkle tree building with Poseidon hashes over the Goldilocks field, with Advanced Vector Extensions (AVX and AVX512). With these optimizations…

Spectral asymptotics of pseudodifferential operators with discontinuous symbols
Alexey Derkach, Alexander V. Sobolev
https://arxiv.org/abs/2506.17426 https…

Spectral asymptotics of pseudodifferential operators with discontinuous symbols
We study discrete spectrum of self-adjoint Weyl pseudodifferential operators with discontinuous symbols of the form $1_Ωϕ$ where $1_Ω$ is the indicator of a domain in $Ω\subset\mathbb R^2$, and $ϕ\in C^\infty_0(\mathbb R^2)$ is a real-valued function. It was known that in general, the singular values $s_k$ of such an operator satisfy the bound $s_k = O(k^{-3/4})$, $k = 1, 2, \dots$. We show that if $Ω$ is a polygon, the singular values decrease as $O(k^{-1}\log k)$. In the case where $Ω$…

Finding Some Impossibility of Flat-Folding of Given Origami Crease Pattern by Graphical Representation
Chihiro Nakajima
https://arxiv.org/abs/2506.12659 ht…

Finding Some Impossibility of Flat-Folding of Given Origami Crease Pattern by Graphical Representation
Flat-foldability problem of origami is the problem to determine whether a given crease pattern drawn on a piece of paper is possible to fold without any penetration or intrusion of a polygon into any connections among them. It is known from the results of Bern and Hayes and following studies that determining whether an origami diagram which constitute of polygons in general shapes can be flat-folded is an NP-hard problem. In this manuscript, on determining the flat foldability of unsigned creas…

Collapsing in polygonal dynamics
Stiegler Jean-Baptiste
https://arxiv.org/abs/2507.16432 https://arxiv.org/pdf/2507.16432

Collapsing in polygonal dynamics
Polygonal dynamics is a family of dynamical systems containing many studied systems, like the famous pentagram map. Similar collapsing phenomena seem to occur in most of these systems. We give a unifying, general definition of polygonal dynamics, and conjecture that a generic orbit collapses towards a predictable point. For the special case of ``closed polygons'', we show that the collapse point depends algebraicly on the vertices of the starting polygon, using tools called scaling symmetry and…

Bicriteria Polygon Aggregation with Arbitrary Shapes
Lotte Blank, David Eppstein, Jan-Henrik Haunert, Herman Haverkort, Benedikt Kolbe, Philip Mayer, Petra Mutzel, Alexander Naumann, Jonas Sauer
https://arxiv.org/abs/2507.11212

Bicriteria Polygon Aggregation with Arbitrary Shapes
We study the problem of aggregating polygons by covering them with disjoint representative regions, thereby inducing a clustering of the polygons. Our objective is to minimize a weighted sum of the total area and the total perimeter of the regions. This problem has applications in cartographic map generalization and urban analytics. Here, the polygons represent building footprints and the clusters may represent urban areas. Previous approaches forced the boundaries of the regions to come from a…

Shape and class of Bruhat Intervals
Gaston Burrull, Nicolas Libedinsky, Rodrigo Villegas
https://arxiv.org/abs/2507.14033 https://arx…

Shape and class of Bruhat Intervals
We study Bruhat intervals in affine Weyl groups by viewing them as regions of alcoves. In type $\widetilde{A}_2$ we show that each interval coincides with a generalized permutohedron minus a star-shaped polygon, and we prove a subtler version inside the dominant chamber of type $\widetilde{A}_n$. Motivated by this geometry, we conjecture that whenever two Bruhat intervals are isomorphic, there exists an isomorphism realized by a piecewise isometry. We prove this when both endpoints are dominant…

Line Aspect Ratio
Arash Vaezi
https://arxiv.org/abs/2506.20508 https://arxiv.org/pdf/2506.20508…

Line Aspect Ratio
We address the problem of covering a target segment $\overline{uv}$ using a finite set of guards $\mathcal{S}$ placed on a source segment $\overline{xy}$ within a simple polygon $\mathcal{P}$, assuming weak visibility between the target and source. Without geometric constraints, $\mathcal{S}$ may be infinite, as shown by prior hardness results. To overcome this, we introduce the {\it line aspect ratio} (AR), defined as the ratio of the \emph{long width} (LW) to the \emph{short width} (SW) of $\…

Hi-d maps: An interactive visualization technique for multi-dimensional categorical data
Radi Muhammad Reza, Benjamin A Watson
https://arxiv.org/abs/2507.07890

Hi-d maps: An interactive visualization technique for multi-dimensional categorical data
In this paper, we present Hi-D maps, a novel method for the visualization of multi-dimensional categorical data. Our work addresses the scarcity of techniques for visualizing a large number of data-dimensions in an effective and space-efficient manner. We have mapped the full data-space onto a 2D regular polygonal region. The polygon is cut hierarchically with lines parallel to a user-controlled, ordered sequence of sides, each representing a dimension. We have used multiple visual cues such as…

A Discrete Analog of Tutte's Barycentric Embeddings on Surfaces
\'Eric Colin de Verdi\`ere, Vincent Despr\'e, Lo\"ic Dubois
https://arxiv.org/abs/2507.13096

A Discrete Analog of Tutte's Barycentric Embeddings on Surfaces
Tutte's celebrated barycentric embedding theorem describes a natural way to build straight-line embeddings (crossing-free drawings) of a (3-connected) planar graph: map the vertices of the outer face to the vertices of a convex polygon, and ensure that each remaining vertex is in convex position, namely, a barycenter with positive coefficients of its neighbors. Actually computing an embedding then boils down to solving a system of linear equations. A particularly appealing feature of this metho…

On geodesic disks enclosing many points
Prosenjit Bose, Guillermo Esteban, David Orden, Rodrigo Silveira, Tyler Tuttle
https://arxiv.org/abs/2506.06477 htt…

On geodesic disks enclosing many points
Let $ Π(n) $ be the largest number such that for every set $ S $ of $ n $ points in a polygon~$ P $, there always exist two points $ x, y \in S $, where every geodesic disk containing $ x $ and $ y $ contains $ Π(n) $ points of~$ S $. We establish upper and lower bounds for $ Π(n)$, and show that $ \left\lceil \frac{n}{5}\right\rceil+1 \leq Π(n) \leq \left\lceil \frac{n}{4} \right\rceil +1 $. We also show that there always exist two points $x, y\in S$ such that every geodesic disk with $x$ …