ideal

Ideal horoball packings of triangulated spheres: solvers, placements, and bracket proofs.

This repository contains small C and Python tools for computing ideal horoball packings of triangulated spheres and for certifying existence by a sub/supersolution bracket. It is a project tool rather than a polished package.

Input is a face list

a,b,c;d,e,f;...

one triangulation per line.

Scope

The underlying geometric model is the usual product metric: adjacent vertices i and j have edge length u[i] u[j], with one vertex sent to infinity and its neighbors pinned on the boundary. The unknowns are the remaining positive vertex weights.

This repository has two main jobs:

• compute the weights and associated flat placement
• certify existence by a bracket argument with explicit slack checks

The motivating combinatorial inputs in the broader project are prime 6-nets, but the basic tools take triangulated-sphere face lists as input.

Layout

ideal/
├── Makefile
├── data/
│   └── eps_needed.txt
├── python/
│   ├── horou.py
│   ├── horoz.py
│   └── proof.py
├── scripts/
│   ├── run_proof_doob.sh
│   └── find_eps.sh
└── src/
    ├── horou_c.c
    ├── horoz_c.c
    └── proof_c.c

Build

make
make clean

Requires a C compiler and libm (standard on macOS and Linux).

Main tools

horou

Solve for the horoball weights u[v] of a triangulation.

C version:

../clers/bin/clers decode < 20.txt | src/horou_c > horou/20.bin

Python version:

echo "CCAE" | ../clers/bin/clers decode | python3 python/horou.py

horoz

Solve for the weights and place vertices in the upper half-plane.

C version:

../clers/bin/clers decode < 20.txt | src/horoz_c > horoz/20.bin

Python usage:

from horou import horou
from horoz import horoz
u = horou(poly)
pos = horoz(poly, u)

proof

Certify existence by bracketing the solution between a sub-solution and a super-solution.

The proof code checks five positive-slack conditions:

• mono
• excess
• triangle
• convex
• boundary

Python examples:

echo "CCAE" | ../clers/bin/clers decode | python3 python/proof.py --verbose

../clers/bin/clers decode < 20.txt | python3 python/proof.py

C example:

../clers/bin/clers decode < 20.txt | src/proof_c > proof_20.bin

Results currently checked in

The checked-in table data/eps_needed.txt records the smallest tested eps proving all prime 6-nets for each v through v = 80.

In particular:

• the checked-in results currently go through v = 80
• 1/8000 suffices from v = 60 through v = 80

Parallel scripts

For server-side runs:

./scripts/run_proof_doob.sh 81
./scripts/run_proof_doob.sh 81 100
./scripts/find_eps.sh 81

These are project scripts for batch runs, not general installation machinery.

Notes

This repository sits in the circle-packing / discrete-conformal / ideal-polyhedral orbit of ideas, but it is not meant to be a survey or exposition. The point is to keep the computational core small, explicit, and inspectable.

Where the separate clers repository handles combinatorial naming and decoding of triangulations, this repository handles the ideal horoball-packing side.

Provenance

The code and documentation in this repository were drafted primarily with Claude Code under the author's direction, with additional advice, review, and supervision from ChatGPT.

License

See LICENSE.