Trying to manoeuvre a big sofa around a corner is a challenge. When faced with the challenge, you not only need muscles – but good navigational skills too.
Even mathematicians are scrambling to find a solution for the perfect sofa shape – even decades after raising the issue. It is known as the ‘moving sofa problem’.
A quarter of a century ago, in 1996, The mathematician Leo Moser first asked the question: ‘What is the shape of the largest area in the plane that can be moved around a right-angled corner in a two-dimensional hallway of width 1?’.
Despite little knowledge of geometry, it can be easy to find different shapes that will fit around the corner, but even for those with a whole host of knowledge, it’s hard to come up with large shapes that will still fit.
Professor of mathematics at The University of California, Dan Romik, talks, in more detail, about the problem in a YouTube video by Numberphile.
Romik explains how it’s not about the longest, nor the heaviest – it’s all about the area of the sofa.
A simple semi-circular sofa shape would need the width of at least one unit and the semi circle would have a radius of one. The formula used would be:
π over two, which = 1.57.
John Hammersley, a mathematician, noticed that if the semicircle was cut into two quarter-circles and the gap between them was filled with a rectangular block, there would be a larger sofa shape that could be moved around the corner.
In his blog, Romik explained that “Hammersley’s idea would work for every value between 0 and 1 of the radius of the semi-circular hole at the bottom.
“The shape of maximal area in this family is obtained when the radius is chosen to be 2/ᴨ (approximately 0.637), which gives an area of 2/ᴨ+ᴨ/2, or approximately 2.2074.
“This is much better than the area of our ‘idiot’s sofa,’ the unit square. Hammersley thought his construction may be optimal, but this turned out to be false.”