Inscribed square problem
Encyclopedia
The inscribed square problem is an unsolved question in geometry
: Does every plane simple curve contain all four vertices of some square
? This is known to be true if the curve is convex
or piecewise smooth and in other special cases. The problem was proposed by Otto Toeplitz
in 1911. Some early positive results were obtained by Arnold Emch and Lev Schnirelmann
. As of 2007, the general case remains open.
P is inscribed in C if all vertices of P belong to C. The inscribed square problem asks:
It is not required that the vertices of the square appear along the curve in any particular order.
Some figures, such circle
s and square
s, admit infinitely many inscribed squares. If C is an obtuse triangle then it admits exactly one inscribed square.
The most encompassing result to date is due to Stromquist, who proved that every local monotone plane simple curve admits an inscribed square. The condition is that for any point p, the curve C can be locally represented as a graph of a function y = f(x). More precisely, for any point p on C there is a neighborhood U(p) such that no chord of C in this neighborhood is parallel to a fixed direction n(p) (the direction of the "y-axis"). Locally monotone curves includes all closed convex curves and all piecewise-C1 curves without cusps.
The affirmative answer is also known for centrally symmetric curves.
in C. In particular, there is always an inscribed equilateral triangle. It is also known that any Jordan curve admits an inscribed rectangle
.
Some generalizations of the inscribed square problem consider inscribed polygons for curves and even more general continua
in higher dimensional Euclidean space
s. For example, Stromquist proved that every continuous closed curve C in Rn satisfying "Condition A" that no two chords of C in a suitable neighborhood of any point are perpendicular admits an inscribed quadrilateral with equal sides and equal diagonals. This class of curves includes all C2 curves. Nielsen and Wright proved that any symmetric continuum K in Rn contains many inscribed rectangles. H.W. Guggenheimer proved that every hypersurface C3-diffeomorphic to the sphere Sn−1 contains 2n vertices of a regular Euclidean n-cube.
Geometry
Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....
: Does every plane simple curve contain all four vertices of some square
Square (geometry)
In geometry, a square is a regular quadrilateral. This means that it has four equal sides and four equal angles...
? This is known to be true if the curve is convex
Convex set
In Euclidean space, an object is convex if for every pair of points within the object, every point on the straight line segment that joins them is also within the object...
or piecewise smooth and in other special cases. The problem was proposed by Otto Toeplitz
Otto Toeplitz
Otto Toeplitz was a German Jewish mathematician working in functional analysis.- Life and work :...
in 1911. Some early positive results were obtained by Arnold Emch and Lev Schnirelmann
Lev Schnirelmann
Lev Genrikhovich Schnirelmann , also Shnirelman, Shnirel'man was a Soviet mathematician who sought to prove Goldbach's conjecture...
. As of 2007, the general case remains open.
Overview
Let C be a Jordan curve. A polygonPolygon
In geometry a polygon is a flat shape consisting of straight lines that are joined to form a closed chain orcircuit.A polygon is traditionally a plane figure that is bounded by a closed path, composed of a finite sequence of straight line segments...
P is inscribed in C if all vertices of P belong to C. The inscribed square problem asks:
- Does every Jordan curve admit an inscribed square?
It is not required that the vertices of the square appear along the curve in any particular order.
Some figures, such circle
Circle
A circle is a simple shape of Euclidean geometry consisting of those points in a plane that are a given distance from a given point, the centre. The distance between any of the points and the centre is called the radius....
s and square
Square (geometry)
In geometry, a square is a regular quadrilateral. This means that it has four equal sides and four equal angles...
s, admit infinitely many inscribed squares. If C is an obtuse triangle then it admits exactly one inscribed square.
The most encompassing result to date is due to Stromquist, who proved that every local monotone plane simple curve admits an inscribed square. The condition is that for any point p, the curve C can be locally represented as a graph of a function y = f(x). More precisely, for any point p on C there is a neighborhood U(p) such that no chord of C in this neighborhood is parallel to a fixed direction n(p) (the direction of the "y-axis"). Locally monotone curves includes all closed convex curves and all piecewise-C1 curves without cusps.
The affirmative answer is also known for centrally symmetric curves.
Variants and generalizations
One may ask whether other shapes can be inscribed into an arbitrary Jordan curve. It is known that for any triangle T and Jordan curve C, there is a triangle similar to T and inscribed in C. Moreover, the set of the vertices of such triangles is denseDense set
In topology and related areas of mathematics, a subset A of a topological space X is called dense if any point x in X belongs to A or is a limit point of A...
in C. In particular, there is always an inscribed equilateral triangle. It is also known that any Jordan curve admits an inscribed rectangle
Rectangle
In Euclidean plane geometry, a rectangle is any quadrilateral with four right angles. The term "oblong" is occasionally used to refer to a non-square rectangle...
.
Some generalizations of the inscribed square problem consider inscribed polygons for curves and even more general continua
Continuum (topology)
In the mathematical field of point-set topology, a continuum is a nonempty compact connected metric space, or less frequently, a compact connected Hausdorff topological space...
in higher dimensional Euclidean space
Euclidean space
In mathematics, Euclidean space is the Euclidean plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions...
s. For example, Stromquist proved that every continuous closed curve C in Rn satisfying "Condition A" that no two chords of C in a suitable neighborhood of any point are perpendicular admits an inscribed quadrilateral with equal sides and equal diagonals. This class of curves includes all C2 curves. Nielsen and Wright proved that any symmetric continuum K in Rn contains many inscribed rectangles. H.W. Guggenheimer proved that every hypersurface C3-diffeomorphic to the sphere Sn−1 contains 2n vertices of a regular Euclidean n-cube.
Additional reading
- Victor KleeVictor KleeVictor L. Klee, Jr. was a mathematician specialising in convex sets, functional analysis, analysis of algorithms, optimization, and combinatorics. He spent almost his entire career at the University of Washington in Seattle.Born in San Francisco, Vic Klee earned his B.A...
and Stan Wagon, Old and New Unsolved Problems in Plane Geometry and Number Theory, The Dolciani Mathematical Expositions, Number 11, Mathematical Association of AmericaMathematical Association of AmericaThe Mathematical Association of America is a professional society that focuses on mathematics accessible at the undergraduate level. Members include university, college, and high school teachers; graduate and undergraduate students; pure and applied mathematicians; computer scientists;...
, 1991
External links
- Mark J. Nielsen, Figures Inscribed in Curves. A short tour of an old problem
- Inscribed squares: Denne speaks at Jordan Ellenberg's blog