Parallel Parking at Rigorous Trivialities
October 1st, 2007 by WaltAt Rigorous Trivialities, a relatively new math weblog, there is a post that is an early contender for the title of Most Important Blog Post of All Time. It’s a sketch of a rigorous proof that parallel parking is always possible (assuming the space is longer than your car). So if you can’t fit your car in that space, it’s your fault. Mathematics demonstrates its practical usefulness once again.
The post’s argument relies on an appeal to Lie algebras. If you take infinitesimal versions of the two basic motions of a car — turning the steering wheel and going forward — these generate a Lie algebra. Contained in the Lie algebra is the vector field that corresponds to going sideways. This implies that you can parallel park any car if you can switch between going forward and turning rapidly enough.
October 2nd, 2007 at 6:36 am
i am afraid it is ont a very good post. it should have at least mentioned the buzz-words for this kind of arguments, i.e. holonomy (systems) and h-principle.
October 2nd, 2007 at 11:04 am
I thought this was simpler - isn’t the commutator product of two rotations a translation, and isn’t that all there is to it?
October 2nd, 2007 at 1:22 pm
Again this demonstrates the significant difference between theory and practice. We all know this is pretty much impossible all the time
So, maybe the practical theorem would be of the form — assuming reasonable lower bounds on the distances moved, turn radius etc, whats the factor $\delta$ such that it is possible to park car in $L \delta $ gap.
October 2nd, 2007 at 5:14 pm
> Again this demonstrates the significant difference between theory and practice.
I don’t know about that. I think it works quite well. I know my wife starts getting a bit annoyed with me when I’m on around the 20th iteration so maybe impatience should be factored in somewhere…
October 4th, 2007 at 1:03 am
Years ago I encountered this argument in Nelson’s book (as quoted in the blog post). I was teaching a class on mathematical physics at Swarthmore College and had introduced Lie algebras so I thought it was the perfect, cute example of the utility of the formalism.
Don’t quote me on this, but I remember trying to check out if the operators as defined where indeed a Lie algebra by making sure that their commumators were closed. But I failed to convince myself that this was indeed the case.
Has someone gone to the trouble of confirming this? I would be interested in seeing where my calculations went wrong, since the example is indeed very compelling.
October 4th, 2007 at 4:16 pm
I have confirmed that the commutator of Drive and Slide is not a linear combination of Steer, Drive, Wriggle and Slide. This means that either the operators are wrong or they do not form a Lie algebra as claimed. I’ve posted this problem to the original blog in “Rigorous Trivialities”. Can someone clarify where the problem lies?
October 8th, 2007 at 9:04 am
In the state of Pennsylvania, you a re required to parallel park in order to be issued a driver’s license, and—this is the important part—you are not allowed to reverse direction more than three times while doing so.
So I must agree with the comments that there is an important disconnect here between theory and practice.
Moreover, the theory completely fails to explain why it is easier to parallel-park a car by backing into the space than by driving forward into the space.