impossible-figures 0.1.0

Detect, measure, localize, generate and render impossible figures via the cohomology of depth offsets.

impossible-figures

A Python library for the geometry of impossible figures — the Penrose triangle, Reutersvärd bars, Escher-style staircases, and friends.

It can:

• detect whether a figure is geometrically realizable,
• measure how impossible it is with a principled index in [0, 1],
• localize which loops cause the paradox,
• generate new impossible figures with a target holonomy, and
• render them to SVG.

Honest positioning

The mathematics here is classical, not new: it is the cohomology of impossible figures (Penrose, Pictures of Impossible Worlds and the Cohomology of Pictures, 1992 — building on Penrose & Penrose, 1958). This library implements that theory cleanly; it does not invent it.

What is actually missing in the Python ecosystem — and what this package provides — is a single, maintained library that bundles detection, an impossibility measure, paradox localization, generation, and rendering together. The existing Python material is mostly one-off educational scripts. This is that library, not a "first ever" anything.

Install

pip install impossible-figures

Requires Python ≥ 3.10. Runtime dependencies: numpy, networkx.

Quickstart

from impossible_figures import penrose_triangle, consistent_triangle

fig = penrose_triangle()
print(fig.impossibility_index())   # 1.0  -> maximally impossible
print(fig.is_possible())           # False

for cycle, holonomy in fig.offending_cycles():
    print(cycle, "closes with offset", holonomy)   # ['A', 'B', 'C'] ... 3.0

ok = consistent_triangle()
print(ok.impossibility_index())    # 0.0
depths, residual = ok.realize_depths()
print(depths)                      # a consistent 3D depth per part

Render the tribar to SVG (the occlusion is driven by the depth offsets, so the loop comes out impossible on its own):

from impossible_figures import penrose_triangle, to_svg

svg = to_svg(penrose_triangle())
open("penrose_triangle.svg", "w").write(svg)

See examples/ for runnable scripts and sample images.

Generate a figure with a prescribed paradox (the measure, inverted):

from impossible_figures import generate

fig = generate(holonomy=2.0, index=0.5)   # target loop sum and impossibility
print(fig.impossibility_index())          # 0.5

The math

A figure is a set of parts (faces or bars) joined at junctions. Local occlusion in the drawing tells us, at each junction, the relative depth offset between two parts: w(u → v) = depth(v) − depth(u).

Model this as a directed graph (nodes = parts, edges = junctions). A globally consistent 3D interpretation exists iff there is a depth d(part) for every part with d(v) − d(u) = w(u → v) on every junction — equivalently, iff the offsets sum to zero around every cycle.

The offsets form a 1-cochain w. It is realizable iff it is a coboundary (w = grad d); the obstruction lives in H¹, the cycle space (cokernel of the oriented incidence operator). The figure is impossible exactly when w has a non-zero component there, and the size of that component — ‖residual‖ / ‖w‖ — is the impossibility index in [0, 1]. The non-zero cycles are the loops that witness the paradox.

Roadmap

This is an incremental build. Available now:

• the math core (detection, impossibility index, offending-cycle localization),
• the catalogue (penrose_triangle, consistent_triangle, reutersvard, penrose_staircase, waterfall, impossible_cube),
• SVG rendering (to_svg) with occlusion driven by the depth offsets,
• target-holonomy / target-index generation (generate).

Coming next:

• renderable geometry for more of the catalogue,
• generation of multi-loop figures (several prescribed offending cycles).

Development

python -m venv .venv
.venv/Scripts/python -m pip install -e ".[dev]"   # Windows
# source .venv/bin/activate && pip install -e ".[dev]"   # POSIX

ruff check src tests        # lint
ruff format --check src tests
mypy src                    # types (strict)
pytest                      # tests + coverage

License

MIT — see LICENSE.