Math 2.0

Geometry Conjecture

Inscribed square problem

Builds on Closed curve

Statement

Let C⊂ℝ2C \subset \R^2 be a Jordan curve, that is, a closed curve in the plane that does not meet itself. Then CC contains four points that are the vertices of a square.

Remarks

The conjecture is open in general. It is known for many classes of curves, among them convex curves, polygons, and curves that are smooth enough. For smooth Jordan curves, Joshua Greene and Andrew Lobb proved more: such a curve contains the vertices of a rectangle of every prescribed aspect ratio. The difficulty for general continuous curves is that limits of inscribed squares of approximating smooth curves may shrink to a single point.

Reading path

To understand this page from the ground up, read these 6 pages in order. The foundations come first.

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Euclidean space, object, no evidence levelEuclidean spaceobjectFunction, concept, no evidence levelFunctionconceptVector-valued function, concept, no evidence levelVector-valued functionconceptParametrised curve, concept, no evidence levelParametrised curveconceptClosed curve, concept, no evidence levelClosed curveconceptInscribed square problem, conjecture, no evidence levelInscribed square prob…conjecture
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Euclidean space object
  2. Function concept
  3. Vector-valued function concept
  4. Parametrised curve concept
  5. Closed curve concept
  6. Inscribed square problem conjecture

Further reading

  1. Web https://en.wikipedia.org/wiki/Inscribed_square_problem
About this page
TagJ72DEZHNAW
URIm2://J72DEZHNAW
Kindconjecture
WritingW0 stub
MSC51M04, 52A10
StatusThis copy of the site has no build data. Run m2 status J72DEZHNAW.
AboutClosed curve
UsesClosed curve
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