Moving sofa problem

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The Hammersley sofa has area 2.2074... but is not the largest solution
The Hammersley sofa has area 2.2074... but is not the largest solution

The moving sofa problem was formulated by the Austrian-Canadian mathematician Leo Moser in 1966. The problem is a two-dimensional idealisation of real-life furniture moving problems, and asks for the rigid two-dimensional shape of largest area A that can be maneuvered through an L-shaped planar region with legs of unit width. The area A thus obtained is referred to as the 'sofa constant'.

As a semi-circular disk of unit radius can pass through the corner, a lower bound for the sofa constant \scriptstyle A = \pi/2 or 1.570796327 is readily obtained. Hammersley derived a considerably higher lower bound \scriptstyle A\, =\, \pi/2 + 2/\pi or 2.207416099 based on a handset-type shape consisting of two quarter-circles on either side of a 1 by 4/π rectangle from which a semicircle of radius \scriptstyle 2/\pi\, has been removed.[1][2]

Gerver found a sofa that further increased the lower bound for the 'sofa constant' to 2.219531669.[3][4] The exact value of the sofa constant constitutes an unsolved problem in mathematics.

[edit] References

  1. ^ H.T. Croft, K.J. Falconer, and R.K. Guy, Unsolved Problems in Geometry, Springer-Verlag, 1994
  2. ^ Moving sofa problem on Mathsoft includes a diagram of Gerver's sofa
  3. ^ J.L. Gerver, "On Moving a Sofa Around a Corner" Geometriae Dedicata 42, 267-283, 1992.
  4. ^ Moving sofa problem on MathWorld
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