An App Proof
That is “app” as in an on-line application
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| [ Leo Stein ] |
Leo Stein is an assistant professor in the department of Physics and Astronomy at the University of Mississippi. His research interests include general relativity from an astrophysical standpoint.
Today I want to share an unusual proof of his.
Mathematics and complexity theory are all about proving theorems. Most of the time, so far, we prove the old way: we write out a humanly readable proof. At least we hope the proof is readable. Some of the time, we use a computer to check or even create the proof. Sometimes we do extensive numerical computations, but these are not proofs.
Solving Quadratic Equations
I have known, as I am sure you do, forever that a quadratic equation can be solved in closed form. That is
has the two solutions
I have discussed this before here and its relationship to the World’s Fair in Flushing Meadows.
A natural question is: Are square roots needed in any formula for quadratic equations? The answer is “Yes”.
Theorem 1 There does not exist any continuous function from the space of quadratic polynomials to complex numbers which associates to any quadratic polynomial a root of that polynomial.
Corollary 2 There is no quadratic formula built out of a finite combination of field operations and the functions
, and the coefficients of the polynomial.
The corollary uses the basic fact that 
An App Based Proof
Here is a novel, I think, proof that uses an app. Stein has written the app and it is here. He explains how to use it. I strongly suggest that you try this yourself.
To get a feel for all this, drag the
coefficient to
and the
coefficient to
. You should have two real roots in root space (one at
, the other at
). Let’s call
the negative root, and
the positive root. Now move the coefficient
Leo Goldmakher says here:
Pause and think about this for a second. This is really, really weird.
Here is one immediate consequence of this observation:
Theorem 3 There does not exist any continuous function from the space of quadratic polynomials to complex numbers which associates to any quadratic polynomial a root of that polynomial.
And so the corollary follows.
A Standard Proof
Goldmakher writes out a more conventional proof in his paper titled Arnold’s Elementary Proof Of The Insolvability Of The Quintic. He also shows the following theorem:
Theorem 4 Fix a positive integer
. Any quintic formula built out of the field operations, continuous functions, and radicals must have nesting of level more than
This says that there can be no fixed formula for fifth degree, quintic, polynomials. Of course, this follows from Galois theory, but his proof uses just calculus. The Arnold is Vladimir Arnold.
Open Problems
Do you know other cases of an app with animation conveying the essence of a mathematical proof? This means more than “proofs in pictures” or “proofs without words”—the animation and interactivity are crucial.
Does a counting frame, as used in elementary school to teach addition, qualify?
Dear Abigail:
I would think that does count…oops bad pun? Did you use one in school?
Best
I don’t remember whether we used one — it’s been more than 50 years since I first went to elementary school.
Here’s a few Animated Proofs of theorems in propositional calculus, using a variant of C.S. Peirce’s logical graphs. There’s no need for user interaction in these very simple, purely routine proofs, but it would be a good project for someone to supplement the underlying algorithm with user guessing and guidance in more complicated problems.
☞ Proof Animations
Dear Jon Awbrey:
I like these animations too. There is the theory that when reading math you should create your own pictures. This helps you learn and understand them. Terry Tao has said this many times. His brilliant books rarely have pictures.
I do like pictures. Thanks for your examples.
Best
Dear Dr Lipton,
I am pleased you like the pictures. More information about this style of logical graphs, extending the ones C.S. Peirce used for propositional logic and Spencer Brown revived back in the 60s, can be found on this page:
☞ Survey of Animated Logical Graphs
Here is a piece with a couple of interactive gadgets conveying the essence of an interesting calculus problem: http://hardmath123.github.io/envelope.html
This is beyond superb!
Galois Theory brought alive.
Bravo!!!
Dear Math Boy:
Very nice comment. Glad you like this. I really enjoyed the app too.
Best
“Next, move {a_{0}} in a big loop (big enough that it orbits around {r_{2}}). Something funny happens: the roots {r_{1}} and {r_{2}} switch places.”
Presumably that should be “… orbits around {a_{1}} …”.
But actually, that’s neither sufficient nor necessary. The critical item is to orbit around the origin; then the roots swap places. You can even drag a_0 in a closed path that encloses a_1 and the roots don’t swap places as long as you avoid enclosing the origin.
Hi, Delta. I don’t think it’s the origin that’s the important point. Rather, we need to study the discriminant D of the polynomial P. The discriminant is a polynomial in n variables: the coefficients a_0 … a_{n-1} (since we’ve set a_n=1). Let’s pick one particular a_i and fix all the others, so we can treat the discriminant D(a_i) as a polynomial of at most degree n in the single variable a_i. The roots of the discriminant tell us points in a_i’s complex plane that will lead to P having degenerate roots. Every loop around a root of D generates a transposition of two roots. If you loop around more roots of D, you can generate more complicated permutations.
(longtime reader, first time posting!) This is a great app! Some related links you might enjoy:
1) Frank Sottile and his collaborators have done very interesting work pushing the limits of the numerical approach to computing Galois groups in their work on numeric homotopy continuation. A fun example is the following paper, where they show the Galois group of a natural Schubert problem is the symmetric group on 6006 elements! https://www.math.tamu.edu/~frank.sottile/research/abstracts/Galois.html
This one looks fun too: https://www.math.tamu.edu/~frank.sottile/research/pdf/NGalois.pdf
2) Not knowing about this great app, over the past few years I’ve developed a similar Mathematica script to use for talks to high school students; you might like it for computing more complicated galois groups, as it has the advantage of numbering roots and drawing pretty colored paths between them [at the price of inconvenience and installing Mathematica!] – web.stanford.edu/~seanpkh/rootloops
Dear seanpkh:
Thanks for viewing and thanks for comment. I like the idea of coding the roots. The one thing that should be fixed in the root app is that. If the roots were even just colored coded that would help us see the point.
The Galois group calculations are quite interesting. I will check them out. Thanks for pointing them out.
Best
and keep commenting