Differential Equations and Exact Solutions in the Moving Sofa Problem

ABSTRACT

The moving sofa problem, posed by Moser in 1966, asks for the planar shape of maximal area that can move around a right-angled corner in a hallway of unit width and is conjectured to have as its solution a complicated shape derived by Gerver in 1992. We extend Gerver's techniques by deriving a family of six differential equations arising from the area-maximization property. We then use this result to derive a new shape that we propose as a possible solution to the ambidextrous moving sofa problem, a variant of the problem previously studied by Conway and others in which the shape is required to be able to negotiate a right-angle turn both to the left and to the right. Unlike Gerver's construction, our new shape can be expressed in closed form, and its boundary is a piecewise algebraic curve. Its area is equal to , where X and Y are solutions to the cubic equations x²(x + 3) = 8 and x(4x² + 3) = 1, respectively.

KEYWORDS:

• algebraic number
• geometric optimization
• moving sofa problem
• ordinary differential equation

2000 AMS Subject Classification:

• 49K15
• 49Q10

Acknowledgments

The author wishes to thank Greg Kuperberg, Alexander Coward, Alexander Holroyd, James Martin, and Anastasiia Tsvietkova for helpful conversations and suggestions during the work described in this article.

Funding

The author was supported by the National Science Foundation under grant DMS-0955584.

Notes

1 Throughout the article, we denote vectors in with boldface letters and consider them as column vectors.