The fix-the-wobbly-table theorem

by birtanpublished on August 15, 2020

welcome to another muffuletta video today's video is about the absolutely wonderful wobbly table theorem some of you may already be familiar with a very pretty special case of this theorem which became quite well known a few

Years ago when numberphile dedicated a nice video to it this special case of the theorem runs as follows take a square table like the one over there at the moment the tables hovering above the floor with unusual behavior for a table

But perhaps it's due to the ghostly presence at our sales anyway the feet of our spooky table are blue and the points directly below the four feet are marked in green the ghost and disappears in the table plunks onto the surface chances

Are that it will topple right not because the ghost is still playing games but just because the floor is uneven how do we stabilize a table well usually people will wedge a napkin or whatever under one of the wobbling feet like this

However the wobbly table theorem says there's another method the theorem says that if the floor is not too crazily uneven then you can stabilize the square table simply by rotating the table on the spot like in this example so just

Rotate it and eventually all four feet will be touching the ground super neat isn't it and this actually works very well in practice try it on your next cafe trip for example and this is only a small piece of some nice and mostly

Unknown table turning mats which also applies to non square tables well let's see now before we get going I have to make two very important disclaimers first when I'm talking about a square table I always mean that the feet of the

Table form a square of course that's not always true even if the table surface is square often enough real-life wobbling is not caused by the uneven floor but by the table having feet of unequal lengths for example over there we've got a table

On a nice flat surface but one of the legs has been gnawed at by a beaver or a ghost or whatever anyway one leg is shorter so even though the table is standing on a flat ground obviously there's no way of shifting the

Table around to make it stable well we could turn it over with its legs in the air but that would be cheating so again by a square table we mean the feet of the table form a square second disclaimer by having stabilized a table

We do not mean that the tabletop is necessarily horizontal usually the stabilized table will still slope in some manner consistent with the uneven floor of course such a slope can also be annoying but definitely not as annoying

As a wobble and typically in real life a slight unevenness in the ground will translate into a slide and not too annoying slope of the tabletop with those two disclaimer stored away it's time to have some serious fun turning

The tables now to do that we'll start with Homer Simpson of course it's never the wrong time to introduce some Simpsons into the discussion recall that my last video was all about Furious series and how to draw complicated

Pictures like this Homer drawing with epicycles I'll announce the winner of the epicycle competition at the end of this video for now let me recycle cycle Homer as a warm-up exercise for our table turning let's begin by Framing

Homer in a rectangle like this I call this sort of rectangle a hugging rectangle because Homer touches all four sides of the rectangle of course they're infinitely many hugging rectangles one for each possible orientation here a few

More there's one and there's another one and one more now the aspect ratios of homers hugging rectangles will vary but it turns out that one of them is special one of the hugging rectangles is a perfect square and that was not luck it

Turns out that any picture will have a hugging square not obvious at all but it's true we'll give the idea of a proof a minute but first here a couple more examples to illustrate Apple Apple Apple and here is its square Linux penguin and

Here's its square fish square just a few random points square now it can happen that the shape has more than one having square for example all hugging rectangles of circles are squares question for you are there any other

Shapes all of whose hugging rectangles are squares leave your answers in the comments here's a hint take a close look at the mythology homepage okay here's the gist of a proof that every picture has a

Hugging square we start with any hugging rectangle as in the example over there and color pairs of opposite sides red and blue now let's rotate this rectangle through an angle of 90 degrees so that at every stage we have a hugging

Rectangle all right the key observation is that the dimensions of the hugging rectangle will change slowly as we rotate with no sudden jumps using maths jargon we would say that the dimensions change continuously as we rotate now

Having rotated through 90 degrees we're back where we started with the same hugging rectangle let's double check mmm yep exactly the same right well not quite have a closer look what has happened is that the red and blue sides

Have swapped places that may not seem important but it's the key to capturing our hugging square okay back to the beginning zero degrees let's record a difference between the red and blue lengths as we rotate to begin red is

Longer than blue so the starting difference will be what positive negative or zero well positive of course however because of the swapping at the end of the 90 degree turn red will be shorter than blue and so the difference

Will be negative now since the hugging changes continuously this implies that the difference red minus blue will also change continuously from positive to negative and this means that at some point a difference has to be zero but

The difference being zero means that red is equal to blue that the red sides are just as long as the blue sides and of course that means that at this instant we have a hugging square and that works for any picture whatsoever not just

Homer super neat isn't it just start with any hugging rectangle start rotating and you're guaranteed to come across the hugging square what is always to do with table-turning well a very similar above than below

Argument shows why as you turn your wobbling square table through 90 degrees you can expect to come across a stable position with all four feet touching the ground okay mathematical seatbelts on then here we go say that square table is

On the ground and it's wobbling with the two blue feet touching the ground and the red feet moving up and down as the table is rocking back and forth now wobble the table so that the two red feet are exactly the same vertical

Distance from the ground we'll call that an equal hovering position now let's rotate a table 90 degrees always with the two feet in the air in an equal hovering position here we go okay how have we ended up well the table is

Exactly in its starting position except red and blue have swapped now it's the red feet on the ground with the blue feet hovering but obviously the swapping is only possible at an instant where both the blue feet and a red feet are on

The ground we can also graph this as we did for the having rectangles let's do it here red and blue stand for the vertical distances of the red and blue feet from the ground then just as before this difference function starts out

Positive and ends up negative and so should be exactly zero at some instant along the way as early we concluded both red and blue are the same at this point however since at least one of red and blue is zero at all times at

This special pink zeroing time both the red and blue distances must be zero this means that all four feet are touching the ground and our table has been stabilized fantastic okay so a square table can always be balanced by rotating

It always really hmm well let's put it like this I've been using this trick for decades to stabilize real-life tables and it's never failed me on the other hand it's quite easy to conjure up scenarios in which different parts of

Our argument break down and/or stabilizing by rotating won't work second challenge for you today try to come up with some setups that foil our argument just to get things going here's something very simple when we lower the

Table over there its top will hit the top of the mountain and the feet will never reach the ground now that we have the square table sorted give or take some finicky details let me tell you a little bit about the history

Of mathematical table balancing the broader scope of the nifty and wobbling by turning argument and the part my friends and I played in turning this intuitive argument in a nail down mathematical theorem okay

Hands up who knows this guy Marty Sanders up well out there in YouTube land not many people are nodding their heads Marty this is terrible our hero has been forgotten this is Martin Gardner by far the greatest

Popularizer of mathematics of all time starting in the 1950s and for a quarter of a century Gardner wrote the hugely influential mathematical games column in Scientific American he's also the author of more than 100 books on popular

Mathematics magic tricks and so on Gardner inspired more people of my generation to become method anybody else for example you can only watch this video because I really got into maths as a result of reading

Gardner's books and if you enjoy my follow two videos it's mostly because of the lessons I've learned from Martin Gardner he was a master at explaining real mathematics as simply as possible and no simpler in

Fact much of popular mathematics in books or on YouTube or wherever have their origin in a Martin Gardner column hexaflexagons Conway's Game of Life Dragan fractals public key cryptography wobbly tables

Yes table turning in the May 1973 issue of his mathematical games column Gardner challenged his readers to discover a version of our heuristic argument for why stabilizing a square table by

Turning should work he then supplied an answer to his puzzle in his next column Gardner credits mere drug Novakovich and can austin as his source for the wobbly table trick i actually tracked down Ken Austin about

15 years ago and he told me that in fact it was his friend mere drug who discovered a trick and how to justify it around 1950 ken only communicated the trick to Martin Gardner mere drugs version of the table turning is actually

A bit different from the one I showed you and is also well worth checking out yet another version of the argument is presented by the prominent German mathematician matias craig who is the star of numberphile

2014 table turning video also definitely check out that one now for a surprising twist have a look at the last sentence of Martin Gardner's write-up a similar argument can be applied to what we rectangular tables by giving them 180

Degree rotations now of course that's very welcome news since out in the wild most tables are rectangular but not square in shape and there are lots of other objects whose feet form rectangles to which all this applies for example I

Personally use stabilizing by rotating quite often with my trusty stepladder anyway why does everybody only go on about square tables if this all works much more generally for rectangular tables well when you try to adapt the

Slick argument for square tables to general rectangular tables you'll find that this is not nearly as straightforward as gardeners off-the-cuff comment suggests now before I show you how to extend the argument to

Rectangles let me tell you about the almost definitive paper on wobbly tables which a couple of friends and I published in 2007 in the mathematical intelligencer their mother is also taking about this

Paper was all about making the wobbly table argument into a nail down mathematical theorem by hammering out on which surfaces a perfect rectangular table can be stabilized by turning it on the spot one of the main results of our

Paper is that stabilizing by turning is guaranteed to work if the ground and the table have the following properties first the ground the ground has to be Lipschitz continuous with associated angle of at most 35 point 26 degrees

Whoa that sounds scary but it's not really that bad all it means is that the ground is given by a continuous function and that between any two points on the ground the ground slopes at an angle of at most 35 point 26 degrees another way

Of putting this is to say that when you slide around the cony 3d counterpart of the green wedge all of the ground will always be below the wedge so the scary term which is continuous amounts to a maximum slope requirement for our ground

Okay so ground continues are not too steep check and what about the table well obviously in line with our introductory disclaimer the feet should form a rectangle and we also require a minimum leg lengths if the legs are at

Least half as long as the diagonals of the tabletop then nothing can go wrong in the way of little Hills bumping into the so what bill Marty Reiner and I approve in our paper is that as long as the

Surface continues and not too sloppy and as long as the table legs are sufficiently long a perfect rectangular table can be stabilized by turning the heart of this theorem is the extension of the nice heuristic argument for

Square tables but to be able to use this argument we had to do some pretty hard work to make sure that our conditions guarantee that the table always rotates as nicely as suggested by the animations that I showed you earlier the idea

Underlying the proof and many similar proofs is the so-called intermediate value theorem this is a mathematically rigorous version of it the intuitively obvious effect captured by our positive to negative diagram the fact that if a

Continuous function changes from positive to negative it will be zero somewhere along the way in fact in the proof we use the intermediate value theorem not just once but about a zillion times and for those of you keen

On the gory details of the tricky proof and some of the other nice results in our paper I'll provide a link in the description now before we look at the proof or rectangular tables here is an interesting open problem is there any

Non rectangular four legged table which can always be stabilized by turning for example what about a half hexagon table here so is it possible to stabilize this or some other mutant table by rotating let me just tell you about one neat

Observation it turns out that it is important whether or not the feet of a table lie on a circle well that's true for our half hexagon table and of course it's also true for all rectangular tables why is the circle

Business important well it turns out that if the feet of a table are not on a circle then there is definitely and not too sloppy surface on which this table will always wobble last challenge for you today prove this non conservative

Feed implies doomed to wobble theorem okay now for the hardcore mythology pans let me sketch how our simple heuristic rotation argument for stabilizing square tables can be extended to rectangular tables

Okay deep breath as with square tables will rotate rectangular tables such at all times they are at an equal hovering position so at all times throughout the rotation either the blue pair of feet is on the ground with the red feet hovering

In equal distance above the ground as pictured over there or vice versa so if we can show that at some instant during our rotation we are guaranteed to have the blue feet on the ground and there's some other instant we're sure that red

Feet are on the ground then we can argue with continuity as before that somewhere in between all four feet have to be on the ground now an obvious difference between the square and rectangular cases is that a 90 degree turn won't work for

General rectangular table we have to rotate 180 degrees to bring it back to the starting position but now the problem is that after a 180 degree turn the same pair of feet are on the ground as at the start it's a look

Their turn same feet on the ground so unlike in the square case the end position after the half turn is not distinguished in any way from the starting position so our rotation in this case doesn't automatically provide

Us with that second crucial position where the two feet that were hovering at the start end up on the ground instead I'll show you that the crucial position always exists using a proof by contradiction so let's assume that the

Opposite of what we want to show is true let's assume that somewhere in the universe there's a rectangular table and a surface which the table cannot be stabilized by turning we're in fantasy land now and so to keep things as

Uncluttered as possible I'll only show you the table and the underlying green points on the ground now if this table cannot be stabilized by turning then the blue must stay on the ground throughout the

Rotation and the red feed will be hovering in the air all the time something like that I'd blue feet on the ground and red feet hovering all the time let me now show how this assumption leads to a

Contradiction which then finishes the proof modulo all the finicky details container our intelligence a paper let's highlight the rectangle formed by the feet and let's also highlight the z axis around which we're rotating that brown

Line is the z axis and let's get rid of the non-essential bits now we make sure that the center of the rectangle will be on the z axis throughout the rotation like that so the center just wanders up and down the z

Axis as we turn okay two important observations first since we are rotating through 180 degrees you've just seen all possible balancing positions of our table with the center on the z axis there are no other balancing positions

Like this second let's call the height of the center of the table in a certain position the elevation of the table in this position now as we rotate there will be a position of the table where the elevation is at a minimum let's see

Okay so it's wondering where's the minimum here yeah okay did you spot it well there it is we're almost ready for the punchline notice that if the diagonal connecting the red points is translated straight down the diagonal

Will connect the green dots underneath but this means that we can rotate a table into a new balancing position with the blue feed ending up at those two green points but this new balancing position is absolutely impossible why

Because the elevation of the new balancing position is lower than the supposedly minimal elevation that we started with and that's the contradiction pretty tricky but also pretty cool don't you think

And that's it for today as far as table turning is concerned I've just got one more thing to do before I sign off announce the winner of our episode of competition from last time lots of nice ideas and implementations which I've

Listed in my common pin to the top of the epicycle video comment section the one I've settled for as the winner is by tetrahedra who succeeded in doing something extra-special there epicycle systems simultaneously draws out three

Iterations of a space-filling curve how many does that check out the details of how they did it in the description of their video ok tetrahedra please send me your contact details by a comment and I'll send you Marty's and my book and

Thanks again to everybody who took part in this competition and that's it for today happy table-turning you

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