## 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

 (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 , , , and by solving

 (2)

 (3)

 (4)

 (5)

This gives
 (6) (7) (8) (9)

Now define

 (10)

where
 (11) (12) (13)

Finally, define the functions
 (14) (15) (16)

The Area of the optimal sofa is given by

 (17)
(Finch).

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.