didericis
44d0ed12b0
Previous Figure 4 had two bugs:
(1) Dual vertices were placed in arbitrary positions, not at
annular triangle centroids.
(2) The "bridge" chord didn't actually correspond to a bridge,
since B_in was drawn as a single hexagonal cycle (which has
no bridges). For a real bridge, O needs to be a barbell.
Redrawn as a clean spoke-only example:
- B_out: hexagon (6 outer vertices u_0..u_5, red).
- B_in: triangle (3 inner vertices w_0, w_1, w_2, light red).
- V(O) = V(B_in), no chord of O, no bridge.
- Triangulation: 9 spokes between outer and inner.
- 9 annular triangles: 6 "outer-cap" + 3 "inner-cap".
- Dual vertices placed using TikZ barycentric coordinates at
each triangle's exact centroid.
- Dual graph Γ ≅ C_9 (just a cycle, no chords for spoke-only).
The chord/bridge case isn't drawn directly in the figure but is
referenced via Remark 1.14, which already discusses the bridge
case (Θ(1,b,c) = Hamilton cycle + length-1 chord) textually.
This keeps the figure correct and unambiguous; readers wanting
the chord case can refer to the remark or the dual paper.
Page count: 9 → 10.
Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
math-research
Personal mathematics research repository by Eric Bauerfeld. Papers are written in AMS-LaTeX using the amsart document class.
Papers
kempe_style_search_for_smaller_contradiction
Humans Suffice: A Novel Proof of the Four Color Theorem
An in-progress proof of the Four Color Theorem via a minimal counterexample argument. The paper builds on Kempe's 1879 strategy — establishing valid cases for vertices of degree ≤ 4, then extending the argument to the degree-5 case using properties of non-adjacent degree-5 vertices, merged subgraphs, and locked colorings.
plane_depth_labelling
Plane Depth Labelling
Early-stage paper. Title and author information set; content in progress.
Creating a New Paper
Use run.sh to scaffold a new paper from the AMS-LaTeX template:
./run.sh init_paper "Your Paper Title"
This creates a new directory (name derived from the title) containing a paper.tex pre-filled with the title and author.
Setup
The Python library code in lib/ requires SageMath. Run setup once per machine:
./run.sh setup <sage_python_path> <sage_site_packages> [system_name]
sage_python_path— path to the SageMath Python interpreter (e.g./opt/sage/local/bin/python3)sage_site_packages— path to SageMath's site-packages directorysystem_name— optional label for this machine (defaults tohostname -s); used to store per-machine env files as.env.<system_name>
On subsequent runs the paths default to whatever was saved in .env, so ./run.sh setup alone re-runs setup with the existing configuration.
Setup also compiles the plantri submodule via make.
Running Sage
To run a Sage script with plantri available on PATH:
./run.sh sage <script.py> [args...]
Or to open an interactive Sage session:
./run.sh sage
Linting
./run.sh lint
Runs pyright and pylint on lib/ using the SageMath Python interpreter.
Shell Completion
To enable tab-completion for run.sh in zsh, add this to your .zshrc:
eval "$(path/to/run.sh completion)"
Or source it once in the current shell session:
eval "$(./run.sh completion)"
Building
Papers are compiled with LaTeX. From within a paper directory:
latexmk -pdf paper.tex