|
|
| Line 1: |
Line 1: |
| [[Image:Hammersley sofa animated.gif|right|280px|thumb|The Hammersley sofa has area 2.2074... but is not the largest solution]]
| | Alyson is the title people use to contact me and I believe it sounds fairly great when you say it. Doing ballet is something she would never give up. Some time in the past she selected to reside in Alaska and her parents live close by. I am an invoicing officer and I'll be promoted soon.<br><br>Look into my weblog: [http://Myoceancounty.net/groups/apply-these-guidelines-when-gardening-and-grow/ online psychic readings] |
| | |
| The '''moving sofa problem''' was formulated by the Austrian-Canadian mathematician [[Leo Moser]] in 1966. The problem is a two-dimensional idealisation of real-life furniture moving problems, and asks for the rigid two-dimensional shape of largest area ''A'' that can be maneuvered through an L-shaped planar region with legs of unit width. The area ''A'' thus obtained is referred to as the ''sofa constant''. The exact value of the sofa constant is an [[Unsolved problems in mathematics|open problem]].
| |
| | |
| ==Lower and upper bounds==
| |
| As a semicircular disk of unit radius can pass through the corner, a lower bound for the sofa constant <math>\scriptstyle A\,\geq\,\pi/2\,\approx\, 1.570796327</math> is readily obtained.
| |
| | |
| [[John Hammersley]] derived a considerably higher lower bound <math>\scriptstyle A\,\geq\,\pi/2 + 2/\pi\,\approx\,2.207416099</math> based on a [[handset]]-type shape consisting of two quarter-circles on either side of a 1 by 4/π rectangle from which a semicircle of radius <math>\scriptstyle 2/\pi\,</math> has been removed.<ref>{{cite book |last1=Croft |first1=Hallard T. |last2=Falconer |first2=Kenneth J. |last3=Guy |first3=Richard K. |authorlink3=Richard K. Guy |title=Unsolved Problems in Geometry |series=Problem Books in Mathematics; Unsolved Problems in Intuitive Mathematics |volume=II |editor-last=Hamos |editor-first=Paul R. |publisher=Springer-Verlag |year=1994 |isbn=978-0-387-97506-1 |url=http://www.springer.com/mathematics/geometry/book/978-0-387-97506-1 |accessdate=24 April 2013}}</ref><ref>[http://web.archive.org/web/20080107101427/http://mathcad.com/library/constants/sofa.htm Moving Sofa Constant] by Steven Finch at MathSoft, includes a diagram of Gerver's sofa</ref>
| |
| | |
| Gerver found a sofa that further increased the lower bound for the sofa constant to 2.219531669.<ref>{{cite journal |last=Gerver |first=Joseph L. |title=On Moving a Sofa Around a Corner |journal=Geometriae Dedicata |issn=0046-5755 |volume=42 |issue=3 |pages=267–283 |year=1992 |doi=10.1007/BF02414066}}</ref><ref>{{MathWorld|urlname=MovingSofaProblem|title=Moving sofa problem}}</ref>
| |
| | |
| In a different direction, an easy argument by Hammersley shows that the sofa constant is at most <math>\scriptstyle 2\sqrt{2}\,\approx\, 2.8284</math>.<ref>{{cite journal |last=Wagner |first=Neal R. |title=The Sofa Problem |journal=The American Mathematical Monthly |volume=83 |issue=3 |year=1976 |pages=188–189 |doi=10.2307/2977022 |url=http://www.cs.utsa.edu/~wagner/pubs/corner/corner_final.pdf |jstor=2977022}}</ref><ref>{{cite book |last=Stewart |first=Ian |authorlink=Ian Stewart (mathematician) |title=Another Fine Math You've Got Me Into... |date=January 2004 |publisher=Dover Publications |location=Mineola, N.Y. |isbn=0486431819 |url=http://store.doverpublications.com/0486431819.html |accessdate=24 April 2013}}</ref>
| |
| | |
| ==See also==
| |
| *[[Mountain climbing problem]]
| |
| *[[Moser's worm problem]]
| |
| | |
| ==References==
| |
| {{reflist}}
| |
| | |
| [[Category:Discrete geometry]]
| |
| [[Category:Unsolved problems in mathematics]]
| |
| [[Category:Recreational mathematics]]
| |
Alyson is the title people use to contact me and I believe it sounds fairly great when you say it. Doing ballet is something she would never give up. Some time in the past she selected to reside in Alaska and her parents live close by. I am an invoicing officer and I'll be promoted soon.
Look into my weblog: online psychic readings