{"id":1245,"date":"2015-01-15T01:00:32","date_gmt":"2015-01-15T01:00:32","guid":{"rendered":"http:\/\/blogs.ams.org\/visualinsight\/?p=1245"},"modified":"2016-12-25T20:26:50","modified_gmt":"2016-12-25T20:26:50","slug":"hammersley-sofa","status":"publish","type":"post","link":"https:\/\/blogs.ams.org\/visualinsight\/2015\/01\/15\/hammersley-sofa\/","title":{"rendered":"Hammersley Sofa"},"content":{"rendered":"
\n
\"Hammersley<\/a>

Hammersley Sofa – Claudio Rocchini<\/p><\/div>\n<\/div>\n

You’ve probably tried to move a sofa around a bend in a hallway. It’s annoying. But it leads to some interesting math puzzles. Let’s keep things simple and work in 2 dimensions. Then the moving sofa problem<\/b> asks:<\/p>\n

\nWhat’s the largest possible area of a 2-dimensional region that can be maneuvered using rigid motions through an L-shaped hallway of width 1?\n<\/p><\/blockquote>\n

This question was first published by Leo Moser in 1966. The animation above, made by Claudio Rocchini, shows one attempt to solve this problem. It’s called the Hammersley sofa<\/b>, since it was discovered by John Michael Hammersley in 1968. Its area has this charming value:<\/p>\n

$$ \\frac{\\pi}{2} + \\frac{2}{\\pi} = 2.2074 \\dots $$<\/p>\n

But it’s not the best known solution! In 1992, Joseph Gerver found a shape of area <\/p>\n

$$ 2.219531668871 \\dots $$<\/p>\n

that fits around the bend in the hallway. Basically Gerver rounded off some of the corners of Hammersley’s sofa. The resulting shape has a boundary consisting of 3 straight line segments and 15 curved segments, each described by an analytic expression:<\/p>\n

\n
\"Gerver<\/a>

Gerver Sofa – Dan Romik<\/p><\/div>\n<\/div>\n

On the other hand, Hammersley showed that any solution has to have area at most<\/p>\n

$$ 2 \\sqrt{2} = 2.8282 \\dots $$<\/p>\n

So, the moving sofa problem remains unsolved. However, Philip Gibbs recently used a numerical method to solve a discretized version of this problem, and he obtained a sofa whose area agrees with Gerver’s to about 8 significant figures:<\/p>\n

• Philip Gibbs, A computational study of sofas and cars<\/a>, 13 November 2014.<\/p>\n

So, it seems reasonable to conjecture that Gerver’s solution is optimal!<\/p>\n

For more, see:<\/p>\n

• Moving sofa problem<\/a>, Wikipedia.<\/p>\n

• Neal R. Wagner, The sofa problem<\/a>, American Mathematical Monthly<\/i> 83<\/b> (1976), 188–189.<\/p>\n

• Moving sofa problem<\/a>, Wolfram Mathworld.<\/p>\n

• Steven Finch Moving sofa constant<\/a>, MathSoft Constants.<\/p>\n

The last two pages explain Gerver’s sofa and compute its area. Gerver’s original paper is here:<\/p>\n

• Joseph L. Gerver, On moving a sofa around a corner, Geometriae Dedicata<\/i> 42<\/b> (1992), 267–283.<\/p>\n

while Hammersley’s proposed solution appeared in this tendentiously titled article:<\/p>\n

• John Michael Hammersley, On the enfeeblement of mathematical skills by modern mathematics and by similar soft intellectual trash in schools and universities, Bull. Inst. Math. App.<\/i> 4<\/b> (1968), 66–85.<\/p>\n

Dan Romik has an interesting paper on the sofa problem, and a variant that demands the sofa be symmetrical and optimal for negotiating both right and left turns in a hallway:<\/p>\n

• Dan Romik, Differential equations and exact solutions in the sofa problem<\/a>, 2016.<\/p>\n

He also has a nice webpage featuring movies of several sofas, including the Gerver sofa shown above:<\/p>\n

• Dan Romik, The moving sofa problem<\/a>.<\/p>\n

Claudio Rocchini, who created the animated gif above, put it on Wikimedia Commons<\/a> with a Creative Commons Attribution-Share Alike 3.0 Unported<\/a> license.<\/p>\n

\n

Visual Insight<\/i> is a place to share striking images that help explain advanced topics in mathematics. I\u2019m always looking for truly beautiful images, so if you know about one, please drop a comment here<\/a> and let me know!<\/p>\n

<\/div>","protected":false},"excerpt":{"rendered":"

The moving sofa problem<\/b> asks: what is the shape of largest area that can be maneuvered through an L-shaped hallway of width 1? This animated image made by Claudio Rocchini shows one attempt to solve this problem. <\/p>\n

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