Optimality of Gerver's Sofa | Hacker News Reader

Optimality of Gerver's Sofa

arxiv.org

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Here's a fun render of how a real life Gerver's Sofa could look:

https://www.mdpi.com/symmetry/symmetry-14-01409/article_depl...

I kinda want one...

Source: https://www.mdpi.com/2073-8994/14/7/1409

It would have to be custom made because the sofa size depends on your hallway width.

A Gerber sofa would work for all hallways that are X amount wide or wider, wouldn't it? Given how it slides over a corner.

Of course, but it's only the biggest possible sofa for a certain size hallway

A real self respecting department would (a) order the sofa, and then (b) get Buildings&Grounds to adjust at least one of their lunch room-adjacent hallway corners to be the minimum size it fits through?

Also a Gerber sofa is based on a flat 2D world. Tilting the sofa to get around a 3D corner seems a reasonable solution.

Presumably there's another Sofa design for the maximum that can get around a 3D corridor corner (and what height do you choose for the corridor given humans tend to choose corridors that are taller than they are wide?)

Should be mandatory in the lunch room of any self respecting math department

For those that don't understand - https://en.wikipedia.org/wiki/Moving_sofa_problem

And perhaps expanded upon into 3D space in Dirk Gently’s Holistic Detective Agency (in which the sofa becomes impossibly stuck, at least in human understood physics) -

https://en.m.wikipedia.org/wiki/Dirk_Gently%27s_Holistic_Det...

In a similar incident that occurred while Douglas Adams attended St John's College, Cambridge, furniture was placed in the rooms overlooking the river in Third Court while the staircases were being refurbished. When the staircases were completed, it was discovered that the sofas could no longer be removed from the rooms, and the sofas remained in those rooms for several decades.

lol

Couldn't take them out the windows?

Old buildings tend to have smaller windows,so it would be unlikely. (It's a 17th century building)

Also could probably partially disassemble them.

Looks like the author researched this during their PhD which ended this year, obtained a postdoc and then finished it off. Good on them!

I wonder how the result varies if one of the corridors (the second one for simplicity) is given a variable width. And if the angle of turn is variable.

I am not well versed with mathematics publishing, but has this proof been already been peer reviewed by other mathematicians, or is it still awaiting confirmation/proof replication?

This is just a preprint, so no peer review or anything yet. Putting out a preprint on the Arxiv like this is generally the first step of the author sharing their work for other mathematicians to take a look at, before it gets submitted for peer review and publication

Just a quick well, actually. It's common at least in my corner of mathematics to put something on arxiv simultaneously with submission to a journal or conference. If I wanted feedback on a paper before submitting I would definitely do that privately. But everyone is different.

You'd think most people who put stuff on arXiv would already be reasonably confident it is correct, at least through correspondence with their peers. However...

That said, I've no reason to doubt this proof (it is not within my wheelhouse).

I was trying to search in the paper for a definition of the shape, to, for example, draw it in SVG.

It seems there is no closed-form solution. I saw this paper is maybe easier to follow for the definition:

https://www.math.ucdavis.edu/~romik/data/uploads/papers/sofa...

Quote:

It is worth noting that Gerver’s description of his shape is not fully explicit, in the sense that the analytic formulas for the curved pieces of the shape are given in terms of four numerical constants A, B, φ and θ (where 0 < φ < θ < π/4 are angles with a certain geometric meaning), which are defined only implicitly as solutions of the nonlinear system of four equations

The downloadable extras at the bottom of https://www.math.ucdavis.edu/~romik/movingsofa/ may be useful – especially the Mathematica package, if you have Mathematica.

Speaking of SVGs, you reminded me of the beautiful SVG documents in David Ellsworth's "Squares in Squares" page [0]. If you view the source of any of them, you'll find the full Mathematica code to generate any constants needed (which is usually simple enough to implement in other CAS software, if you know how to read it), as well as their values listed to dozens of digits within an inline DTD, to be referenced from the actual elements.

[0] https://kingbird.myphotos.cc/packing/squares_in_squares.html

Interesting, but not practical. All real furniture movers would make use of the third dimension.

The difficulty of extending the definition to 3 dimensions is that the restriction to 2 separates two classes of constraint: being able to move the sofa round the corner, and the shape of the sofa being comfortable to sit on.

I introduced my kids (13 and 11) to the sofa problem last week as we installed this cabinet organizer from IKEA: https://www.ikea.com/us/en/p/utrusta-corner-base-cab-pull-ou...

After showing them a youtube video about the problem they saw clearly how the organizer is a sofa and even made a joke about it a few days later.

Relatable math is pretty great. Also really cool is showing how academia translates to enriching our lives in benign ways.

Wow. This solves a problem that's been open for at least 58 years.

This might solve the problem - gotta stand up to scrutiny first which takes some time

An interesting thing about this proof is that it looks as though an earlier draft relied on computer assistance – see the author’s code repository at https://github.com/jcpaik/sofa-designer – whereas this preprint contains a proof that “does not require computer assistance, except for numerical computations that can be done on a scientific calculator.”

Numberphile had a great video on this a few years ago: https://youtu.be/rXfKWIZQIo4

Big if true! But has it been reviewed by experts?

This paper makes use of Mamikons theorem. This theorem is not widely known, but it should be: https://en.m.wikipedia.org/wiki/Visual_calculus

So say you have a hallway shaped like a 5, what is the maximum volume 3d-gerver's that can make it through (by being possible to rotate to swap the turning direction?

Seems like he could have saved himself a whole lot of trouble by just getting a sectional.

Pivot!

My favourite mathematical problem looks to have been solved.

Dies Ikea offer one of these?

There are more interesting investigations and results This one is ok but not remarkable

What is the practical application of this?

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