اشترك بالحزمة الذهبية واحصل على وصول غير محدود شمرا أكاديميا
تسجيل مستخدم جديدHinged Dissections Exist
30
0
0.0
(
0
)
اسأل ChatGPT حول البحث
ﻻ يوجد ملخص باللغة العربية
We prove that any finite collection of polygons of equal area has a common hinged dissection. That is, for any such collection of polygons there exists a chain of polygons hinged at vertices that can be folded in the plane continuously without self-intersection to form any polygon in the collection. This result settles the open problem about the existence of hinged dissections between pairs of polygons that goes back implicitly to 1864 and has been studied extensively in the past ten years. Our result generalizes and indeed builds upon the result from 1814 that polygons have common dissections (without hinges). We also extend our common dissection result to edge-hinged dissections of solid 3D polyhedra that have a common (unhinged) dissection, as determined by Dehns 1900 solution to Hilberts Third Problem. Our proofs are constructive, giving explicit algorithms in all cases. For a constant number of planar polygons, both the number of pieces and running time required by our construction are pseudopolynomial. This bound is the best possible, even for unhinged dissections. Hinged dissections have possible applications to reconfigurable robotics, programmable matter, and nanomanufacturing.
قيم البحث
اقرأ أيضاً
Let $P$ be a set of $2n$ points in convex position, such that $n$ points are colored red and $n$ points are colored blue. A non-crossing alternating path on $P$ of length $ell$ is a sequence $p_1, dots, p_ell$ of $ell$ points from $P$ so that (i) all
points are pairwise distinct; (ii) any two consecutive points $p_i$, $p_{i+1}$ have different colors; and (iii) any two segments $p_i p_{i+1}$ and $p_j p_{j+1}$ have disjoint relative interiors, for $i eq j$. We show that there is an absolute constant $varepsilon > 0$, independent of $n$ and of the coloring, such that $P$ always admits a non-crossing alternating path of length at least $(1 + varepsilon)n$. The result is obtained through a slightly stronger statement: there always exists a non-crossing bichromatic separated matching on at least $(1 + varepsilon)n$ points of $P$. This is a properly colored matching whose segments are pairwise disjoint and intersected by common line. For bo
Monskys celebrated equidissection theorem follows from his more general proof of the existence of a polynomial relation $f$ among the areas of the triangles in a dissection of the unit square. More recently, the authors studied a different polynomial
$p$, also a relation among the areas of the triangles in such a dissection, that is invariant under certain deformations of the dissection. In this paper we study the relationship between these two polynomials. We first generalize the notion of dissection, allowing triangles whose orientation differs from that of the plane. We define a deformation space of these generalized dissections and we show that this space is an irreducible algebraic variety. We then extend the theorem of Monsky to the context of generalized dissections, showing that Monskys polynomial $f$ can be chosen to be invariant under deformation. Although $f$ is not uniquely defined, the interplay between $p$ and $f$ then allows us to identify a canonical pair of choices for the polynomial $f$. In many cases, all of the coefficients of the canonical $f$ polynomials are positive. We also use the deformation-invariance of $f$ to prove that the polynomial $p$ is congruent modulo 2 to a power of the sum of its variables.
Existing studies for environment interaction with an aerial robot have been focused on interaction with static surroundings. However, to fully explore the concept of an aerial manipulation, interaction with moving structures should also be considered
. In this paper, a multirotor-based aerial manipulator opening a daily-life moving structure, a hinged door, is presented. In order to address the constrained motion of the structure and to avoid collisions during operation, model predictive control (MPC) is applied to the derived coupled system dynamics between the aerial manipulator and the door involving state constraints. By implementing a constrained version of differential dynamic programming (DDP), MPC can generate position setpoints to the disturbance observer (DOB)-based robust controller in real-time, which is validated by our experimental results.
We study a $2 times 2$ matrix equation arising naturally in the theory of Coxeter frieze patterns. It is formulated in terms of the generators of the group $mathrm{PSL}(2,mathbb{Z})$ and is closely related to continued fractions. It appears in a numb
er of different areas, for example, toric varieties. We count its positive solutions, obtaining a series of integer sequences, some known and some new. This extends classical work of Conway and Coxeter proving that the first of these sequences is the Catalan numbers.
Does regulation in the genome use collective behavior, similar to the way the brain or deep neural networks operate? Here I make the case for why having a genomic network capable of a high level of computation would be strongly selected for, and sugg
est how it might arise from biochemical processes that succeed in regulating in a collective manner, very different than the usual way we think about genetic regulation.
سجل دخول لتتمكن من نشر تعليقات
التعليقات
جاري جلب التعليقات
سجل دخول لتتمكن من متابعة معايير البحث التي قمت باختيارها
