FOM: Visual proofs -- two examples
Joe Shipman
shipman at savera.com
Fri Feb 26 10:47:16 EST 1999
Discussion of "geometric proofs" here has mostly been restricted to
"diagrammatic" proofs, where mathematical entities and their
relationships are summarized in diagrams and further operations on those
diagrams lead to propositional conclusions in a valid, rigorous, but
*not* sentential way. There seems to be agreement that this type of
mathematical argument can be formalized in the predicate calculus, but
that the formalization may result in a loss of understandability or
conciseness.
But Reuben Hersh pointed out that some (real, published) proofs use
diagrams and pictures in a more informal but essential way. He pointed
to the "proofs without words" that regularly appear in Mathematics
magazine, the book "Visual Complex Analysis", and Arnold's books on
ODE's. I propose to call such proofs "visual proofs" (the term
"geometric proof" suggests that the subject matter must be geometry and
the term "diagrammatic proof" suggests that the diagram does all the
work).
I will examine two well-known examples. Because FOM only allows one to
post text, these will be visual proofs without pictures! But the point
is that the informal proofs I give will only be understandable (or at
least, will only be *easily* understandable) to someone who is
visualizing the pictures I am omitting in his head. Any reasonable
attempts to translate my informal proofs (which are simply text in
English) into formal ones will use pictures in some way that I would
like FOMers to try to elucidate. Certainly, the way I remember these
proofs is as pictures, and if I were presenting them in a seminar
drawing the pictures would be most of the work.
First example: Any two polygons with the same area are equivalent under
piecewise linear decomposition.
Proof: First show this is an equivalence relation (intuitively obvious
if you see a picture showing the superposition of two separate
decompositions of a polygon). Next reduce the polygons to collections
of triangles by drawing diagonals (an appropriate diagram makes it
obvious that this is always possible). Next reduce each triangle to a
rectangle (diagram to show this is always possible draws altitude to the
longest side, perpendicular bisector of this altitude, and rotation of
the two pieces that don't intersect the long side by 180 degrees around
the corners that aren't on the altitude). Next reduce each rectangle to
a not-too-skinny rectangle (diagram to show this is always possible just
chops a long rectangle in half by bisecting the long sides and placing
the two pieces side by side and repeats). Next reduce the
not-too-skinny rectangles to squares (diagram to show this is always
possible shows the rectangle standing on its short side, a segment
between the upper left corner and a point in the lower half of the right
edge dividing the rectangle into a triangle and a trapezoid, sliding of
the triangle along the segment until the lower corner touches the
extension of the rectangle's base, extension leftward of the top edge of
the shifted triangle to cut off another triangle from the top of the
trapezoid, and movement of that triangle around to fill the hole at the
bottom between the lower part of the trapezoid and the shifted
triangle). Now you've got two collections of squares equivalent to the
two original polygons. The last step is to show that two squares are
equivalent to a larger square (there are many "picture proofs" of the
Pythagorean Theorem that do this--one nice one places four right
triangles with sides a and b inside a square with side a+b in two
different ways, so that the remaining area is either a square on the
hypotenuse or two squares with sides a and b). This finishes the proof
because it shows each polygon is equivalent to a single square, and
since the transformations were area-preserving the polygons will be
equivalent iff they had the same area to begin with.
Second example: The non-cancellation theorem for knots.
The connected sum operation # is commutative and associative (proof of
associativity is trivial; to prove commutativity imagine shrinking the
first knot real small and moving it along the loop around and past the
second knot). Given K#L, make the wild knot K#L#K#L#K#.... converging
at a point p. If K#L is the unknot, there are local isotopies around
K#L and L#K that trivialize these knots. You can piece these together
in two ways: K#(L#K)#(L#K)#... and (K#L)#(K#L)#(K#L)#... to reduce the
wild knot to both K and the unknot and compose them to get an isotopy
between K and the unknot. Furthermore, p will be a fixed point, so the
trace under this isotopy of a closed ball neighborhood of K that doesn't
include p will stay outside a neighborhood of p and the (wild) isotopy
can be replaced by a tame one.
Can anyone tell me who first discovered each of these proofs? I'm not
surprised that I don't remember the names, because both proofs are so
remarkably memorable I only needed to see each of them in a book once,
though I've since demonstrated them to people many times. Would they be
so memorable in a form with no implicit or explicit pictures?
-- Joe Shipman
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