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Moving Sofa Constant

N.B. A detailed on-line essay by S. Finch was the starting point for this entry.

What is the sofa of greatest Area $S$ which can be moved around a right-angled hallway of unit width? Hammersley (Croft et al. 1994) showed that

S\geq {\pi\over 2}+{2\over\pi}=2.2074\ldots.
\end{displaymath} (1)

Gerver (1992) found a sofa with larger Area and provided arguments indicating that it is either optimal or close to it. The boundary of Gerver's sofa is a complicated shape composed of 18 Arcs. Its Area can be given by defining the constants $A$, $B$, $\phi$, and $\theta$ by solving

\end{displaymath} (2)

\end{displaymath} (3)

A\cos\phi-(\sin\phi+{\textstyle{1\over 2}}-{\textstyle{1\over 2}}\cos\phi+B\sin\phi)=0
\end{displaymath} (4)

(A+{\textstyle{1\over 2}}\pi-\phi-\theta)-[B-{\textstyle{1\o...
...}}(\theta-\phi)(1+A)-{\textstyle{1\over 4}}(\theta-\phi)^2]=0.
\end{displaymath} (5)

This gives
$\displaystyle A$ $\textstyle =$ $\displaystyle 0.094426560843653\ldots$ (6)
$\displaystyle B$ $\textstyle =$ $\displaystyle 1.399203727333547\ldots$ (7)
$\displaystyle \phi$ $\textstyle =$ $\displaystyle 0.039177364790084\ldots$ (8)
$\displaystyle \theta$ $\textstyle =$ $\displaystyle 0.681301509382725\ldots.$ (9)

Now define

{\textstyle{1\over ...
...a\leq\alpha<{\textstyle{1\over 2}}\pi-\phi,
\end{displaymath} (10)

$\displaystyle s(\alpha)$ $\textstyle \equiv$ $\displaystyle 1-r(\alpha)$ (11)
$\displaystyle u(\alpha)$ $\textstyle \equiv$ $\displaystyle \left\{\begin{array}{ll} B-{\textstyle{1\over 2}}(\alpha-\phi)(1+...
...pha & \mbox{for $\theta\leq\alpha<{\textstyle{1\over 4}}\pi$}\end{array}\right.$  
$\displaystyle D_u(\alpha)$ $\textstyle =$ $\displaystyle {du\over d\alpha}=\left\{\begin{array}{ll} -{\textstyle{1\over 2}...
... -1 & \mbox{if $\theta\leq\alpha<{\textstyle{1\over 4}}\pi$.}\end{array}\right.$  

Finally, define the functions
$\displaystyle y_1(\alpha)$ $\textstyle \equiv$ $\displaystyle 1-\int_0^\alpha r(t)\sin t\,dt$ (14)
$\displaystyle y_2(\alpha)$ $\textstyle \equiv$ $\displaystyle 1-\int_0^\alpha s(t)\sin t\,dt$ (15)
$\displaystyle y_3(\alpha)$ $\textstyle \equiv$ $\displaystyle 1-\int_0^\alpha s(t)\sin t\,dt-u(\alpha)\sin\alpha.$ (16)

The Area of the optimal sofa is given by

$A=2\int_0^{\pi/2-\phi} y_1(\alpha)r(\alpha)\cos\alpha\,d\alpha+2\int_0^\theta y_2(\alpha)s(\alpha)\cos\alpha\,d\alpha$
$ +2\int_\phi^{\pi/4} y_3(\alpha)[u(\alpha)\sin\alpha-D_u(\alpha)\cos\alpha-s(\alpha)\cos\alpha]\,d\alpha=2.21953166887197\ldots\quad$ (17)

See also Piano Mover's Problem


Croft, H. T.; Falconer, K. J.; and Guy, R. K. Unsolved Problems in Geometry. New York: Springer-Verlag, 1994.

Finch, S. ``Favorite Mathematical Constants.''

Gerver, J. L. ``On Moving a Sofa Around a Corner.'' Geometriae Dedicata 42, 267-283, 1992.

Stewart, I. Another Fine Math You've Got Me Into.... New York: W. H. Freeman, 1992.

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© 1996-9 Eric W. Weisstein