Home > General > The Hidden Symmetry Behind Elevator Etiquette

The Hidden Symmetry Behind Elevator Etiquette

January 24, 2012 Leave a comment Go to comments

As humans, we tend to respect each others’ personal space to the extent that we take actions to optimize it. We see this every day in public bathroom stalls and urinals. For instance, if there are five stalls/urinals, and they’re numbered in order [1,2,3,4,5], the order in which they fill up is usually 1->5->3->x->x or 5->1->3->x->x. We even find humor when a person is out of touch with this natural law of optimized personal space. This image, for example, has been floating around the internet for some time now and is called ‘Bad Urinal Etiquette’:

But what about elevators? Does the Law of Optimal Personal Space (LOOPS) apply here as well? I live on the 17th floor of a high-rise building and work downtown on the 23rd floor of an office building and have observed, over a very large sample space of data, that this law does indeed hold with great symmetry. Let me break down the elevator geometry here:

n = 1
on a 2-dimensional plane, this is just a dot or single node. (I’ve denoted the rider as a smiley and the elevator as a rectangle, obviously.)

This person doesn’t necessarily have to stand in the middle. Often, this person will either lean against the back wall or one of the back corners for support. They choose the back in anticipation of more people entering the elevator.

n = 2
this is a line.

This is the default positioning that I most often encounter with a stranger in an elevator. Sometimes we’ll both be leaning against our respective back corners on a particularly wearisome day. Clever readers will point out that the optimal space between the two would actually be the diagonal of this rectangle. While this is true, and does occasionally happen, we tend to default towards the back in case we pick up more passengers along the ride.

n = 3
this is a triangle.

The above figure may not be drawn to an exact scale, but this equilateral triangle is the most common configuration I’ve seen among three riders. However, a reasonably large number of the time, a right triangle will form when the top person decides they shouldn’t obstruct the doorway and chooses to position themselves on either the upper right or upper left corner. If the buttons are only on one side, they’ll usually end up in that corner after pressing for their floor. In a true vacuum experiment where there is no doorway obstruction to consider, I firmly believe that this equilateral triangle would result every time as it optimizes personal space equally.

n = 4
this is a rectangle.

With four riders, this configuration happens over 90% of the time assuming there are no drunkards on board. Basically, everyone intrinsically understands that with four people within a four-sided elevator, this setup provides optimal personal space for all. There is no doorway obstruction to consider, and everybody gets two corners to support their weight instead of just one.

n = 5
a single person nested within a rectangle

Now when you add a fifth person to the previous configuration, where else can this person stand without creeping anybody out? The middle – exactly equidistant from all riders to signify absolutely no bias towards any one stranger. Let’s hope for no turbulence because the fifth wheel here gets no corners to lean against and must constantly stay alert at every stop in the case where a rider behind him needs to get out. Being fifth in an elevator sure can be stressful.

n = 6
a rectangle with six nodes

Once a sixth person enters, we fall back into the elegant symmetry of the rectangle. Person#5 is allowed to share the back wall to make room for person#6 who occupies the spot right in front of the doorway which optimizes personal space and also requires the least effort in travel.

And there you have it. People are subconsciously positioning themselves like the six faces of dice. At n > 6, I find that the elevator is generally too crammed and chaos begins to overwhelm order. Oftentimes, an unforeseen variable like one person being overweight or having pungent body odor results in asymmetry before n > 6. Alas, we are only human.

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  1. January 24, 2012 at 2:09 am

    Hi, this is a comment.
    To delete a comment, just log in, and view the posts’ comments, there you will have the option to edit or delete them.

  2. January 24, 2012 at 5:16 am

    this is a test comment… asfl;fd

  3. February 2, 2012 at 4:42 am

    Dude, I love this

  4. Daniel
    February 2, 2012 at 11:22 am

    ha ha cool Taka ! I think you are on a point of discovering something like Davinci code for common space… you should buy the rights for this and write a handbook, and then buy a privet jet with the profit ;)

  5. February 2, 2012 at 12:56 pm

    Tak, really interesting. And the diagrams really help those of us are less mathematically inclined. I can’t wait for more entries, keep them coming :)

  1. March 12, 2012 at 10:21 pm

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