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 which can be moved around a right-angled hallway of unit width? Hammersley (Croft et al. 1994) showed that

$\begin{displaymath} S\geq {\pi\over 2}+{2\over\pi}=2.2074\ldots. \end{displaymath}$

(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

, $\phi$ , and $\theta$ by solving

$\begin{displaymath} A(\cos\theta-\cos\phi)-2B\sin\phi+(\theta-\phi-1)\cos\theta-\sin\theta+\cos\phi+\sin\phi=0 \end{displaymath}$

(2)

$\begin{displaymath} A(3\sin\theta+\sin\phi)-2B\cos\phi+3(\theta-\phi-1)\sin\theta+3\cos\theta-\sin\phi+\cos\phi=0 \end{displaymath}$

(3)

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

(4)

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

(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

$\begin{displaymath} r(\alpha)\equiv\left\{\begin{array}{ll} {\textstyle{1\over ... ...a\leq\alpha<{\textstyle{1\over 2}}\pi-\phi, \end{array}\right. \end{displaymath}$

(10)

where

$\displaystyle s(\alpha)$	$\textstyle \equiv$	$\displaystyle 1-r(\alpha)$	(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.$
			(12)
$\displaystyle D_u(\alpha)$	$\textstyle =$	$\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.$
			(13)

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)

(Finch).

See also Piano Mover's Problem

References

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

Finch, S. ``Favorite Mathematical Constants.'' http://www.mathsoft.com/asolve/constant/sofa/sofa.html

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.

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