unitri

Paper: Further Bounding the Kreuzer-Skarke Landscape (arXiv:2602.16909)

Exactly counts the fine (unimodular) triangulations of a lattice polygon. For a convex polygon, hand it the polygon's lattice points and get the exact count; for more general regions it also takes an explicit upper/lower boundary description. It counts without enumerating, so it reaches regions with astronomically many triangulations that enumeration tools cannot.

Based on the original program by Stepan Orevkov (http://picard.ups-tlse.fr/~orevkov), reworked and generalized by Nate MacFadden (with Claude Opus 4.8); an earlier minor cleanup was by Michael Stepniczka and Nate MacFadden.

If there are any bugs/issues in this code, assume they are due to Nate MacFadden's rework and not Stepan Orevkov's original code.

Install

pip install -e .          # builds the Cython extension; needs a C toolchain + libgmp

libgmp is optional: it powers the fast single-run exact counter (count_triangulations, na_query). It comes from apt install libgmp-dev (Debian/Ubuntu), brew install gmp (macOS), or conda install -c conda-forge gmp, and the build finds it automatically (pkg-config, falling back to Homebrew/conda via the bundled _gmp.py). Without GMP, pip install simply skips that extension and you count via count_triangulations_parallel (mod-prime + CRT, needs only a C compiler) -- see below.

Counting a polygon's unimodular triangulations

Give count_triangulations the polygon's lattice points and it returns the exact number of fine (unimodular) triangulations:

import unitri

unitri.count_triangulations([(0,0), (4,0), (3,2), (1,2)])          # 140    (a trapezoid)
unitri.count_triangulations([(0,0), (4,0), (3,3), (1,3), (0,1)])   # 10843  (a pentagon)

There is also a command-line front end. It reads a point set from a file or stdin in almost any bracket/comma/whitespace format -- a pasted numpy array, [x, y] lists, or x y per line:

echo "[[0,0],[4,0],[3,3],[1,3],[0,1]]" | python -m unitri     # -> 10843
python -m unitri points.txt

The count is exact (arbitrary precision). The point-set path is for convex regions; for non-convex regions -- valleys, concave tops -- describe them directly with boundary profiles, below.

For large counts, or when you don't have GMP, count the same thing in parallel across cores via the mod-prime + CRT path (needs only a C compiler, no libgmp):

unitri.count_triangulations_parallel([(0,0), (4,0), (3,2), (1,2)])   # 140, in parallel

It runs the mod-prime counter for successive primes across cores and CRT-combines them into the exact integer -- the GMP-free way to get exact counts, and often faster than the single GMP run for very large ones.

The counting core (C CLI and boundary profiles)

Under the point-set convenience is a single-header C counter, na_query.h (stb-style), driven by an m x n bounding box and upper/lower boundary profiles. This is the lower-level, more general interface -- it handles any region between a lower and an upper boundary, including non-convex ones -- and it needs no Python. na-query.c is a thin CLI that pulls in the implementation; width m and height n are runtime arguments, so one binary handles every size:

gcc -O2 -o na-query unitri/na-query.c                 # default: counts modulo a prime
gcc -O2 -DGMP -o na-query unitri/na-query.c -lgmp     # big-integer: the whole count

If GMP isn't on the compiler's default path (Homebrew, or a conda env), splice in the bundled locator's flags: gcc -O2 $(python3 _gmp.py) -DGMP -o na-query unitri/na-query.c -lgmp. Or just use the Makefile: make na-query-mod (mod-prime), make na-query (GMP), or make both.

Run ./na-query <m> <n> [prime_index] and pipe the region to stdin, one profile per line:

Line 1 -- upper boundary (the query): m+1 heights h_0 h_1 ... h_m, each an integer in [0, n], or . for an absent vertex (the boundary passes between lattice points there). The endpoints h_0 and h_m must be present.
Line 2 -- lower boundary / floor (optional): same format; a blank line or Ctrl-D leaves the floor flat at 0. The upper profile must lie on or above it.

echo "4 4 4 4 4" | ./na-query 4 4                  # query_value 736983568  (full 4x4 square)
printf '4 4 4 4 4\n0 1 0 1 0\n' | ./na-query 4 4    # query_value 14032211   (over a non-flat floor)
echo "0 . 3 . 0" | ./na-query 4 4                  # query_value 35          (an absent vertex)

The result prints as query_value <count> -- the whole integer under -DGMP, or a residue mod the chosen prime in the default build (combine several primes with unitri/crt_combine.py to recover the exact count). With no input at all, na-query <m> <n> prints the flat-rectangle f(m,k) table for k = 1..n.

The same profile interface is available in-process from Python as unitri.na_query(m, n, upper, lower=None) -- what count_triangulations calls under the hood; upper/lower are the m+1 boundary heights, omit lower for a flat floor at 0:

unitri.na_query(4, 4, [4, 4, 4, 4, 4])              # 736983568  (the 4x4 square)
unitri.na_query(3, 12, [12, 8, 4, 0])               # 668517487  (a base-3 triangle)
unitri.na_query(4, 4, [4,4,4,4,4], [0,1,0,1,0])     # 14032211   (over a non-flat floor)

Run the tests with pip install -e .[test] (adds pytest, plus cytools for the TOPCOM cross-checks) then pytest tests/.

Performance

na_query counts triangulations with a dynamic-programming recurrence -- it never enumerates them -- so its cost depends only on the bounding box (m, n), not on the (often astronomically large) number of triangulations. TOPCOM, by contrast, enumerates one triangulation at a time, so its cost scales with the count and cannot reach large regions at all.

Exact (GMP) build vs TOPCOM (via CYTools) on an Intel Core Ultra 7 270K Plus, Ubuntu 26.04, gcc 15.2. Both columns are warmed up once, then reported as the per-call mean ± stdev: na_query in process through the compiled extension (not a subprocess), with auto-scaled repetitions over 7 batches; TOPCOM one run per batch, its repeats stopping once their cumulative time exceeds a 60 s budget. The vertices column gives each region's convex-hull corners; pass them to count_triangulations (or count_triangulations_parallel) to reproduce.

region	vertices	triangulations	`na_query`	TOPCOM
3x2 rectangle	(0,0),(0,2),(3,0),(3,2)	852	0.020 ± 0.000 ms	0.04 ± 0.01 s
polygon	(0,2),(1,3),(2,0),(3,0),(3,3),(4,1),(4,2)	10,653	0.190 ± 0.001 ms	0.56 ± 0.00 s
polygon	(0,3),(1,1),(1,4),(2,0),(3,0),(3,4),(4,1),(4,3)	840,021	0.517 ± 0.001 ms	55.0 ± 0.3 s
4x4 square	(0,0),(0,4),(4,0),(4,4)	736,983,568	0.330 ± 0.004 ms	infeasible
4x10 square	(0,0),(0,10),(4,0),(4,10)	~5.8e23	13.10 ± 0.04 ms	infeasible
triangle, height 84	(0,0),(0,84),(3,0)	~7.6e65	1.26 ± 0.003 s	infeasible

Counts agree exactly with TOPCOM wherever TOPCOM can finish. Reproduce with pip install -e . && python benchmarks/benchmark.py.

Organization

unitri/
├── unitri/
│   ├── profiles.py     # point set -> (m,n,upper,lower); count_triangulations (the main entry)
│   ├── na_query.pyx    # Cython binding: in-process na_query(m, n, upper, lower)
│   ├── na_query.h      # the counter: stb-style single header, mod-prime (default) or GMP (-DGMP)
│   ├── na-query.c      # thin CLI wrapper around na_query.h
│   ├── crt_combine.py  # combine the default build's per-prime residues into the exact count
│   ├── crt_parallel.py # GMP-free exact counts: parallel mod-prime runs + CRT
│   ├── __main__.py     # CLI: python -m unitri (count a lattice point set)
│   └── __init__.py
├── tests/
│   ├── test_topcom_convex.py    # convex polygons vs TOPCOM: curated + randomized (the main use-case)
│   ├── test_counts.py           # exact counts vs literature / TOPCOM; f(m,k) table mode
│   ├── test_mod_prime.py        # default mod-prime backend + crt_combine (CRT reconstruction)
│   ├── test_symmetry.py         # hard x<->m-x reflection cases
│   ├── test_unimodular.py       # unimodular invariance
│   ├── test_readme_examples.py  # the examples in this README, asserted
│   ├── check_topcom.py          # standalone TOPCOM cross-check (fixed convex regions)
│   ├── conftest.py              # shared fixtures (builds the GMP/mod-prime binaries)
│   ├── _cli.py                  # shared "run the na-query CLI" helper
│   ├── _topcom.py               # shared TOPCOM enumeration helper
│   └── transforms.py            # GL(2,Z) invariance helpers
├── benchmarks/
│   ├── benchmark.py             # na_query vs TOPCOM timing (the Performance table)
│   └── profile.sh               # wall-time + peak-memory profiler for any command
├── Makefile                     # build the na-query CLI (make both / na-query / na-query-mod)
├── pyproject.toml
├── setup.py
├── environment.yml              # conda env with the GMP backend (recommended)
├── environment-nogmp.yml        # conda env without GMP (mod-prime + CRT path)
├── _gmp.py                      # locate GMP (pkg-config -> Homebrew/conda); shared by setup.py + tests
├── MANIFEST.in
├── CITATION.cff
├── NOTICE                       # provenance: Orevkov's original code, shared with permission
└── LICENSE                      # GPLv3

Provenance

The core counting algorithm and original code are by Stepan Orevkov, included and distributed with his permission; see NOTICE and the attribution notes in unitri/na_query.h and CITATION.cff. The modifications, generalizations, and packaging are licensed under GPLv3 (see LICENSE).

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This release

0.2.0 This release

Sep 9, 2026 11 files

unitri 0.2.0

unitri

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Counting a polygon's unimodular triangulations

The counting core (C CLI and boundary profiles)

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