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Sofas and ladders round corners

Sofas and ladders round corners

Two corridors of widths a and b intersect at right angles.

Q: What is the longest ladder (line segment) you can take round the corner?

A: (a2/3+b2/3)3/2. Dropping the restriction that the ladder be straight means it can be infinite - use a snowflake curve.

Q: What is the largest (wrt area) rectangular sofa you can take round the corner?

A: Length = sqrt(a2 + b2), width = ab / length. Note that this sofa has the same area (viz ab) as the largest one you can get round the corner without rotating it.

Q: ??? What about a non-rectangular sofa? What if it must be convex?

A: Hereafter restrict to a = b = 1.

"I posted that the best known area is pi/2 + 2/pi [ = ~2.207416099]. The shape that gives this is [a rectangle minus a semicircle and two quarter-circles].

Terrible ascii drawing follows:

          /                \
         /        __        \
        |       /    \       |
        |______+      +______|

As the shape enters the corner, it makes contact at 5 points: one on each quarter circular arc, the two inside corners (+ above), and one on the semicircular arc.

I have no proof that this shape gives the maximal area, but as far as I know, no one has come up with a larger shape that will make the corner."
[Dave Dodson , rec.puzzles, 21 May 1998]

"Ian Stewart's book "Another Fine Math You've Got Me Into" discusses this problem (the puzzle is there called "Conway's Sofa"). The shape described above is the "Hammersley sofa". It can be improved slightly by trimming small pieces from the inside corners, allowing the outside curves to be made fatter.

The best known shape was discovered independently by Ben Logan and Joseph Gerver. It is very similar to the above shape, but more subtle. Unfortunately, the proof that it is maximal has a few gaps. See the book for more details."
[pfoster.No.SpaM@pcug.org.au (Peter Foster), rec.puzzles, 27 May 1998]

Another rec.puzzles suggestion is to consider two disks of diameter 1 a fixed distance apart, and take the largest sofa containing them - at the ASCII-art level it looks the same as the drawing above. For maximal area the disks should be 0.947106134 apart, but even then the area is only 2.097365326.

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