Searcharxiv⌕ Search

arXiv subjects

Richard Evan Schwartz

Publications and source records attributed to Richard Evan Schwartz.

At least 19 recordsLinked to original sources
arXiv2026

Divide and Conquer: A Distributed Approach to Five Point Energy Minimization

This work rigorously verifies the phase transition in 5-point energy minimization first observed by Melnyk-Knop-Smith in 1977. More precisely, we prove that there is a constant S = [15+24/512,15+25/512] such that the triangular bi-pyramid is the energy minimizer with respect to the s-power law potential for all s in (0,S) and some pyramid with square base is the unique minimizer for all s in (S,15+512/25]. Taking s=1 gives another solution to Thomson's 5 electron problem from 1904.

math.MG↗
arXiv2026

The Four Color Theorem meets Shapes of Polyhedra

We consider solutions to the $4$-color problem for the vertices of sphere triangulations with degree sequence $6,...,6,4,4,4,4,4,4$. We sort these solutions into combinatorial types and show that each generic type $τ$ is parametrized by the set of integer lattice points inside a rational polyhedral convex cone ${\cal C\/}_τ$ of dimension at least 4. There is an integral quadratic form $Q_τ$ on ${\cal C\/}_τ$ whose diagonal part, evaluated on a lattice point, is $3$ times the number of triangles in the corresponding triangulation. We relate this structure to the octahedral stratum of Thurston's moduli space of flat cone structures on the sphere.

math.MG↗
arXiv2026

On Pappus and Anosov Representations of the Modular Group

Let $X=SL_3(\R)/SO(3)$. Let $\cal DFR$ be the space of discrete faithful representations of the modular group into ${\rm Isom\/}(X)$ which map the order $2$ generator to an isometry with a unique fixed point. In this paper, we prove that $\cal DFR$ has a component $\cal B$, the so-called Barbot component, that is homeomorphic to $\R^2 \times [0,\infty)$. The boundary of $\cal B$ parametrizes the Pappus representations and the interior consists of Anosov representations.

math.GT↗
arXiv2026

Patterns of Geodesics, Shearing, and Anosov Representations of the Modular Group

Let $X=SL_3(\R)/SO(3)$. Let $\cal DFR$ be the space of discrete faithful representations of the modular group into ${\rm Isom\/}(X)$ which map the order $2$ generator to an isometry with a unique fixed point. I prove many things about the component $\cal B$ of $\cal DFR$ known as the Barbot component: It is homeomorphic to $\R^2 \times [0,\infty)$. The boundary parametrizes the Pappus representations from [{\bf S0\/}]. The interior parametrizes the complete extension of the family of Anosov representations from [{\bf BLV\/}]. The members of $\cal B$ are isometry groups of embedded patterns of geodesics in $X$ which have asymptotic properties like the edges of the Farey triangulation or shears thereof. The Anosov representations are obtained from the Pappus representations by either of two shearing operations in $X$. The shearing structure is encoded by two proper foliations of $\cal B$ into rays.

math.GT↗
arXiv2026

Mostow Rigidity Made Easier

This article gives a self-contained proof of Mostow Rigidity, at least modulo undergrad real analysis. The proof should be accessible to grad students interested in geometry and topology. It has no new research, but I think that this is an unusually clean and analytically light proof of this famous result. I am posting this because I think it will be useful to geometry/topology students.

math.GT↗
arXiv2026

Vertex-Minimal Paper Tori

A paper torus is an embedded polyhedral torus that is isometric to a flat torus in the intrinsic sense. We prove that there does not exist a paper torus with $7$ vertices, and that there does exist a paper torus with $8$ vertices. This settles the question of the minimum number of vertices needed for a paper torus.

math.MG↗
arXiv2026

Build Boy's Surface

This article describes Boy's surface in a nice way that does not make many demands on three-dimensional visualization. The article includes a kit that you can print out onto card stock and assemble with scissors and tape.

math.HO↗
arXiv2026

Continued Fractions and the 4-Color Theorem

We study the geometry of some proper 4-colorings of the vertices of sphere triangulations with degree sequence 6,...,6,2,2,2. Such triangulations are the simplest examples which have non-negative combinatorial curvature. The examples we construct, which are roughly extremal in some sense, are based on a novel geometric interpretation of continued fractions. We also present a conjectural sharp "isoperimetric inequality" for colorings of this kind of triangulation.

math.CO↗
arXiv2025

Collapsibility and Near Universality for Vertex Minimal Paper Tori

A paper torus is a piecewise linear isometric embedding of a flat torus into $\R^3$. Following up on the $8$-vertex paper tori discovered by the second author, we prove universality and collapsibility results about these objects. One corollary is that any flat torus without reflection symmetry is realized as an $8$-vertex paper torus. Another corollary is that, for any $ε>0$, there is an $8$-vertex paper torus within $ε$ of a unit equilateral triangle in the Hausdorff metric.

math.MG↗
arXiv2025

The Optimal Twisted Paper Cylinder

An embedded twisted paper cylinder of aspect ratio $λ$ is a smooth isometric embedding of a flat $λ\times 1$ cylinder into $\R^3$ such that the images of the boundary components are linked. We prove that for such an object to exist we must have $λ>2$ and that this bound is sharp. We also show that any sequence of examples having aspect ratio converging to $2$ must converge to a (non-smooth) $4$-fold wrapping of a right-angled isosceles triangle.

math.MG↗
arXiv2025

Le Retour de Pappus

In my 1993 paper, "Pappus's Theorem and the Modular Group", I explained how the iteration of Pappus's Theorem gives rise to a $2$-parameter family of representations of the modular group into the group of projective automorphisms. In this paper we realize these representations as isometry groups of patterns of geodesics in the symmetric space $X=SL_3(\R)/SO(3)$. The patterns have the same asymptotic structure as the geodesics in the Farey triangulation, so our construction gives a $2$ parameter family of deformations of the Farey triangulation inside $X$. We also describe a bending phenomenon associated to these patterns.

math.GT↗
arXiv2025

Symplectic Tiling Billiards, Planar Linkages, and Hyperbolic Geometry

In this paper I will unite two games, symplectic billiards and tiling billiards. The new game is called symplectic tiling billiards. I will prove a result about periodic orbits of symplectic tiling billiards in a very special case and then show how this result combines with the construction in Thurston's paper {\it Shapes of Polyhedra\/} to give hyperbolic structures on moduli spaces of planar equilateral polygons. One corollary is that the configuration space of the hexagonal planar linkage with unit-length rods (modulo isometry) has an algebraically defined hyperbolic structure in which it is a $10$-cusped hyperbolic $3$-manifold that is tiled by $15$ regular ideal octahedra. The $10$ cusps correspond to the $10$ maximally degenerate configurations.

math.DS↗
arXiv2025

The Flapping Birds in the Pentagram Zoo

We study the $(k+1,k)$ diagonal map for $k=2,3,4,...$. We call this map $Δ_k$. The map $Δ_1$ is the pentagram map and $Δ_k$ is a generalization. $Δ_k$ does not preserve convexity, but we prove that $Δ_k$ preserves a subset $B_k$ of certain star-shaped polygons which we call $k$-birds. The action of $Δ_k$ on $B_k$ seems similar to the action of $Δ_1$ on the space of convex polygons. We show that some classic geometric results about $Δ_1$ generalize to this setting.

math.DS↗
arXiv2024

On Nearly Optimal Paper Moebius Bands

Let $ε<1/384$ and let $Ω$ be a smooth embedded paper Moebius band of aspect ratio less than $\sqrt 3 + ε$. We prove that $Ω$ is within Hausdorff distance $18 \sqrt ε$ of an equilateral triangle of perimeter $2 \sqrt 3$. This is an effective and fairly sharp version of our recent theorems in [{\bf S0\/}] about the optimal paper Moebius band.

math.MG↗
arXiv2024

The Optimal Paper Moebius Band

In this paper we prove that a smooth embedded paper Moebius band must have aspect ratio greater than $\sqrt 3$. We also prove that any sequence of smooth embedded paper Moebius bands whose aspect ratio converges to $\sqrt 3$ must converge, up to isometry, to the famous triangular Moebius band. These results answer the minimum aspect ratio question discussed by W. Wunderlich in 1962 and prove the more specific conjecture of B Halpern and C. Weaver from 1977.

math.MG↗
arXiv2024

Paper Moebius bands with T Patterns

This paper gives another proof of the key lemma in my recent paper which solves the optimal paper Moebius band conjecture of Halpern and Weaver, namely Lemma T. The proof here is longer but it offers more geometric intuition about what is going on.

math.MG↗
arXiv2024

Pentagram Rigidity for Centrally Symmetric Octagons

In this paper I will establish a special case of a conjecture that intertwines the deep diagonal pentagram maps and Poncelet polygons. The special case is that of the 3-diagonal map acting on affine equivalence classes of centrally symmetric octagons. This is the simplest case that goes beyond an analysis of elliptic curves. The proof involves establishing that the map is Arnold-Liouville integrable in this case, and then exploring the Lagrangian surface foliation in detail.

math.SG↗
arXiv2023

The Crisscross and the Cup: Two Short 3-Twist Paper Moebius Bands

We introduce the crisscross and the cup, both of which are immersed $3$-twist polygonal paper Moebius band of aspect ratio $3$. We explain why these two objects are limits of smooth embedded paper Moebius bands having knotted boundary. We conjecture that any smooth embedded paper Moebius band with knotted boundary has aspect ratio greater than $3$. The crisscross is planar but the cup is not.

math.MG↗