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Moving Sofa Problem

المؤلف:  Gerver, J. L.

المصدر:  "On Moving a Sofa Around a Corner." Geometriae Dedicata 42

الجزء والصفحة:  ...

11-2-2020

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Moving Sofa Problem

 

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

(1)

(OEIS A086118). 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 , , , and  by solving

	(2)
	(3)
	(4)
	(5)

This gives

			(6)
			(7)
			(8)
			(9)

Now define

{1/2 for 0<=alpha<phi; 1/2(1+A+alpha-phi) for phi<=alpha<theta; A+alpha-phi for theta<=alpha<1/2pi-theta; B-1/2(1/2pi-alpha-phi)(1+A) for 1/2pi-theta<=alpha<1/2pi-phi,; -1/4(1/2pi-alpha-phi)^2 " src="http://mathworld.wolfram.com/images/equations/MovingSofaProblem/NumberedEquation2.gif" style="height:146px; width:364px" />

(10)

where

			(11)
		{B-1/2(alpha-phi)(1+A) for phi<=alpha<theta-1/4(alpha-phi)^2; A+1/2pi-phi-alpha for theta<=alpha<1/4pi" src="http://mathworld.wolfram.com/images/equations/MovingSofaProblem/Inline27.gif" style="height:60px; width:282px" />	(12)
		{-1/2(1+A)-1/2(alpha-phi) for phi<=alpha<=theta; -1 if theta<=alpha<1/4pi." src="http://mathworld.wolfram.com/images/equations/MovingSofaProblem/Inline30.gif" style="height:58px; width:270px" />	(13)

Finally, define the functions

			(14)
			(15)
			(16)

The area of the optimal sofa is then given by

				(17)
				(18)

(Finch 2003).

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REFERENCES:

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

Finch, S. R. "Moving Sofa Constant." §8.12 in Mathematical Constants. Cambridge, England: Cambridge University Press, pp. 519-523, 2003.

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

Sloane, N. J. A. Sequence A086118 in "The On-Line Encyclopedia of Integer Sequences."

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

Trott, M. The Mathematica GuideBook for Programming. New York: Springer-Verlag, p. 104, 2004. http://www.mathematicaguidebooks.org/.