ideal

Ideal horoball packings and unit-equilateral realizations of triangulated spheres: solvers, placements, homotopy continuation, and bracket proofs.

This repository contains C and Python tools for computing ideal horoball packings, certifying their existence by a sub/supersolution bracket, and continuing ideal bends to unit-equilateral Euclidean realizations.

The C tools and the ideal-packing Python tools read face lists

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

one triangulation per line. python/puffup.py reads and writes JSON.

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 three main jobs:

• compute the weights and associated flat placement
• certify existence by a bracket argument with explicit slack checks
• continue the ideal bends to equilateral realizations and check closure

The motivating combinatorial inputs in the broader project are prime 6-nets; the command-line C tools take triangulated-sphere face lists.

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

puffup

Continue the ideal bends to the unit-equilateral Euclidean problem by homotopy in the corner angle.

C version:

../clers/bin/clers decode < 20.txt | src/puffup_c

The Python version reads and writes JSON:

python3 python/puffup.py --in input.json --out output.json

The base-bend reconstruction is described in docs/puffup_pyp.md.

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. The checked-in data/v4_50_closure.txt records successful closure for all 8,239,684 prime 6-nets through v = 50; one net needed a retry.

In particular:

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

Sweep scripts

For multicore sweeps:

./scripts/run_proof.sh 81
./scripts/run_proof.sh 81 100
./scripts/find_eps.sh 81
./scripts/run_closure.sh 4 50

Notes

The separate clers repository handles combinatorial naming and decoding of triangulations. This repository handles ideal horoball packings and their continuation to equilateral realizations.

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.