Proof by Picture

I find proofs that rely too heavily on algebra and abstract symbol manipulation are often unsatisfactory.  I'm left with the feeling that I know that the fact to be proved is true, but I still don't understand why it's true.

Many facts in mathematics can be proved or explained in more than one way, and many things which are usually proven using abstract reasoning can also be shown to be true using pictures.  Perhaps the best-known example is the "Aha!" proof of Pythagoras's theorem.  Other facts which are easily shown via pictures include such simple stuff as (a/b)*(c/d) = (ac/bd) -- something which was taught by rote in the school I attended.

On this page I've listed a few things which I had to work to understand, and for which I eventually found satisfactory explanations which can be displayed as pictures.

A few simple things
 Focus of a Parabola A simple proof of the fact that parabolas focus light (or sound, for that matter) Focus of an Ellipse A simple proof of the fact that light from one focus of an ellipse is reflected to the other Focus of a Hyperbola A simple proof that light heading for the "back" focus of a hyperbola is reflected to the "front" focus Calculus In which we provide visual motivation for a number of facts from elementary calculus.  [This page is actually in the "Basic Stuff section; I'm linking to it from here because it also fits well with the "Proof by Picture" theme.] Why partials commute A simple fact from basic calculus which, none the less, is typically only proved symbolically Pythagoras's Theorem: The "Aha!" Proof (this one's certainly not original!) The "Aha!" Proof,Version 2 This is a slight variation on the classic "Aha!" proof, in which the triangles have been placed on the outside of the inner square, rather than the inside of the outer square.  A small difference but none the less I find it a bit clearer than the original.

 Minimization of a Path Integral The fundamental derivation of the calculus of variations. 1-Forms Fundamental element of differential geometry, the 1-form is best explained with pictures. Covariant Derivative of a Vector (along with some notes on vector components)  This is frequently presented using purely symbolic manipulations.  But it is actually rather straightforward to visualize. Covariant Derivative of a 1-Form

And one item which isn't actually a proof-by-picture at all, just because I felt like it...

The Glue Function

A good "picture proof" may take only a few seconds to understand.  However, constructing the picture to start with can be time consuming.  My plans for this page are ambitious, but it's likely to be quite a while before most of these things are actually done.  Someday, maybe, the following sections will be included:
• Lie derivative, equivalence to Lie bracket  (I had a reasonably good picture for this one as long ago as 6/22/05 but unfortunately the supporting text -- which must include a definition of the Lie derivative, of course, as well as an explanation of what the picture is supposed to show! -- turned out to be long and tedious to write.)
• Equivalence of parallel-transport and longest-path defs for geodesic
• Exterior derivative -- maybe??
• Stokes' theorem (the big one, not the little one) -- maybe??
And we have...
• Variational path principle ... done!
Page created on 8/11/04, and last updated 2/16/2008