30 Oct 2025

*****purpose**********************

This directory contains the computer code I used
in my paper "Vertex Minimal Paper Tori".
It is written in Java and should be compatible with
all modern Java compilers.  If you have questions
or comments please send them to Richard.Evan.Schwartz@gmail.com

****** directory contents****************

flatness_test: This has the Mathematica code that
tests that the example is flat up to 10^{-31}.

mathematica_plotter: This has a short Mathematica
program that lets you plot the pup tent and move it
around.


EmbeddingLemma: This directory contains the code we use to
prove the Robust Embedding Lemma.  It runs with
exact integer arithmetic.

Newton: This directory contains our high precision
implementation of Newton's Method.  We use this class
to generate extremely high precision versions of our
nearly flat paper torus and to compute the relevant
Jacobian matrix to high precision.   This class uses
the BigDecimal class.  Logically, this directory is
not used in our proof, except that we extract from it
a 2-digit approximation of the Jacobian. This 2-digit
approximation could practically be done on an abacus.

Visual:  This directory contains a graphical user
interface which displays our example graphically.
We used this program to generate the pictures in
the paper.  We also used it to generate the vectors
used in the proof of the Robust Embedding Lemma.
As we say repeatedly in the paper, it does not
matter logically how we generate our various
proof certificates; all that matters is their
existence.

Each of these directories has its own README file.

****more code************

This directory is part of a larger suite of programs
I have written to study this problem.  I just
extracted the parts relevant to the paper.  If you
are interested in seeing the other code, which is
still not documented well, I can send it to you.

***operating the code*******

To compile the code in any
directory use the command

javac *.java

In all but the Visual directory, run the code
using the command

java Run

In the Visual directory, you can either run the main
program with

java Main

Finally, every directory has a file called clean.
You run it using the command

./clean

This will remove all the java class files. I find
it useful to use this file to make a less cluttered
directory when I am debugging and documenting rather
than coding.


