Zobrazeno 1 - 10
of 953
pro vyhledávání: '"Schwartz, Richard P."'
Autor:
Schwartz, Richard Evan
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 re
Externí odkaz:
http://arxiv.org/abs/2412.02417
Autor:
Schwartz, Richard Evan
Let $\epsilon<1/384$ and let $\Omega$ be a smooth embedded paper Moebius band of aspect ratio less than $\sqrt 3 + \epsilon$. We prove that $\Omega$ is within Hausdorff distance $18 \sqrt \epsilon$ of an equilateral triangle of perimeter $2 \sqrt 3$.
Externí odkaz:
http://arxiv.org/abs/2412.00572
Autor:
Schwartz, Richard Evan
We study the $(k+1,k)$ diagonal map for $k=2,3,4,...$. We call this map $\Delta_k$. The map $\Delta_1$ is the pentagram map and $\Delta_k$ is a generalization. $\Delta_k$ does not preserve convexity, but we prove that $\Delta_k$ preserves a subset $B
Externí odkaz:
http://arxiv.org/abs/2403.05735
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
Externí odkaz:
http://arxiv.org/abs/2310.10000
Autor:
Schwartz, Richard Evan
A smooth twisted paper cylinder of aspect ratio $\lambda$ is an isometric embedding of a $1 \times \lambda$ cylinder into $\pmb{R}^3$ such that the images of the boundary components are linked. We prove that for such an object to exist we must have $
Externí odkaz:
http://arxiv.org/abs/2309.14033
Autor:
Schwartz, Richard Evan
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 isom
Externí odkaz:
http://arxiv.org/abs/2308.12641
Autor:
Schwartz, Richard Evan
The purpose of this paper is to 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
Externí odkaz:
http://arxiv.org/abs/2307.12259
Autor:
Schwartz, Richard Evan
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 ener
Externí odkaz:
http://arxiv.org/abs/2301.05090
Autor:
Schwartz, Richard Evan
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, wh
Externí odkaz:
http://arxiv.org/abs/2208.05254
Autor:
Schwartz, Richard Evan
In this paper we will give a short and direct proof that Wolfgang Kuehnel's 9-vertex triangulation of the complex projective plane really is the complex projective plane. The idea of our proof is to recall the trisection of the complex projective pla
Externí odkaz:
http://arxiv.org/abs/2205.00595