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Inscribed rectangle coincidences
Abstract
We prove an integral formula for continuous paths of rectangles inscribed in a piecewise smooth loop. We use this integral formula to prove the inequality M ( γ ) ≥ Δ ( γ )/2 – 1, where M ( γ ) denotes the total multiplicity of rectangle coincidences, i.e. pairs, triples, etc. of isometric rectangles inscribed in γ , and Δ ( γ ) denotes the number of stable diameters of γ , i.e. critical points of the distance function on γ .
... Compactified configuration spaces have also been used by S.T. Vrećica and R.T.Živaljević, T. Rade [50] in their paper looking at the polygonal peg problem (inscribed affine regular hexagons in smooth Jordan curves, and inscribed parallelograms in smooth simple closed curves in R 3 ). There have also been many papers [1,3,14,15,21,22,26,31,41,40] examining quadrilaterals inscribed in curves and, more recently, making progress towards solving the rectangular-peg problem (finding rectangles of any aspect ratio inscribed in Jordan curves). ...
We prove a transversality "lifting property" for compactified configuration spaces as an application of the multijet transversality theorem: given a submanifold of configurations of points on an embedding of a compact manifold M in Euclidean space, we can find a dense set of smooth embeddings of M for which the corresponding configuration space of points is transverse to any submanifold of the configuration space of points in Euclidean space, as long as the two submanifolds of compactified configuration space are boundary-disjoint. We use this setup to provide an attractive proof of the square-peg problem: there is a dense family of smoothly embedded circles in the plane where each simple closed curve has an odd number of inscribed squares, and there is a dense family of smoothly embedded circles in where each simple closed curve has an odd number of inscribed square-like quadrilaterals.
We use Batson's lower bound on the nonorientable slice genus of -torus knots to prove that for any , every smooth Jordan curve has an inscribed rectangle of of aspect ratio for some . Setting , we have that every smooth Jordan curve has an inscribed rectangle of aspect ratio .
We classify the set of quadrilaterals that can be inscribed in convex Jordan curves, in the continuous as well as in the smooth case.
We prove that any cyclic quadrilateral can be inscribed in any closed convex
-curve. The smoothness condition is not required if the quadrilateral is a rectangle.
The ‘square peg problem’ or ‘inscribed square problem’ of Toeplitz asks if every simple closed curve in the plane inscribes a (nondegenerate) square, in the sense that all four vertices of that square lie on the curve. By a variety of arguments of a ‘homological’ nature, it is known that the answer to this question is positive if the curve is sufficiently regular. The regularity hypotheses are needed to rule out the possibility of arbitrarily small squares that are inscribed or almost inscribed on the curve; because of this, these arguments do not appear to be robust enough to handle arbitrarily rough curves. In this paper, we augment the homological approach by introducing certain integrals associated to the curve. This approach is able to give positive answers to the square peg problem in some new cases, for instance if the curve is the union of two Lipschitz graphs
f
,
that agree at the endpoints, and whose Lipschitz constants are strictly less than one. We also present some simpler variants of the square problem which seem particularly amenable to this integration approach, including a periodic version of the problem that is not subject to the problem of arbitrarily small squares (and remains open even for regular curves), as well as an almost purely combinatorial conjecture regarding the sign patterns of sums
for
ranging in finite sets of real numbers.
This paper presents some configuration theorems concerning rectangles inscribed in four lines.
This is a short survey article on the 102 years old Square Peg Problem of Toeplitz, which is also called the Inscribed Square Problem. It asks whether every continuous simple closed curve in the plane contains the four vertices of a square. This article rst appeared in a more compact form in Notices Amer. Math. Soc. 61(4), 2014.
Let ABCDE be a pentagon inscribed in a circle. It is proved that if
O\mathcal{O}
is a C4-generic smooth convex planar oval with four vertices (stationary points of curvature), then there are two similarities φ
such that the quadrangle φ(ABCD) is inscribed in
O\mathcal{O}
and the point φ(E)lies inside
O\mathcal{O}
, as well as two similarities ψ such that the quadrangle ψ(ABCD) is inscribed in
O\mathcal{O}
and ψ(E)lies outside
O\mathcal{O}
. Itisalsoprovedthatif n is odd, then any smoothly embedded circle γ ↪ ℝn contains the vertices of an equilateral (n + 1)-link polygonal line lying in a hyperplane of ℝn. Bibliography: 7 titles.
LetQ be any rectangle and letK
d
(d 2) be a continuum which is either symmetric across a hyperplane or symmetric through a pointz K. We show thatK contains the vertices of a rectangle similar toQ which exhibits the same symmetry as doesK.
Rectangles and simple closed curves. Lecture, Univ. of Illinois at Urbana-Champaign. (See page 71 in M. D. Myerson, Balancing acts
- H Vaughan
Splitting Loops and necklaces: Variants of the Square Peg Problem
- J Aslam
- S Chen
- F Frick
- S Saloff-Coste
- L Setiabrate
- H Thomas