Symmetric Graph Drawing

Quotient-driven layouts for symmetric graphs, using automorphism group orbits to determine vertex placement on concentric rings with rotational symmetry.

Pipeline: GAP computes Aut(Γ) and orbit quotients → Python scores quotients, minimises crossings, places vertices → TikZ renders PDF (or D3.js renders interactive HTML).

Quickstart

# Single graph → individual_graphs/auto-dodecahedron.pdf
python3 src/draw_graph.py dodecahedron

# Full gallery → gallery.pdf
python3 src/draw_all.py

# Orbit quotient diagrams → quotients/
python3 src/draw_quotients.py

# Visual catalogue (all individual drawings, one per page) → catalogue.pdf
python3 src/make_catalogue.py

Requires GAP with the GRAPE package, Python 3, and pdflatex.

Repository layout

Directory	Contents
src/	The pipeline: draw_graph.py (layout engine), draw_all.py (gallery), draw_quotients.py, symmetric-layout.g (GAP: Aut, orbits, quotients), enumerate-quotients.g, code-construction scripts
src/census/	Standalone GAP research scripts: PSL(2,q) cubic graph censuses, M₁₁ searches, Macbeath graph, Klein quartic codes
docs/	Implementation & theory, research program, algebraic background, research notes, Eades & Hong reference (thesis.pdf)
tex/	Hand-written papers/diagrams: cubic-bi-circulant.tex, Fano plane figures, quotient unfoldings (compile with pdflatex from inside tex/)
individual_graphs/	Generated + hand-tuned TikZ drawings of single graphs
quotients/	Generated orbit-quotient diagrams (output of draw_quotients.py)
named_graphs_tikz/	Curated TikZ sources for named graphs
assets/	Reference images (downloaded comparisons, JPEG exports)
bolza-surface/	Genus-2 Bolza surface: cohomology, Poincaré disk tilings, interactive foldings (docs, maths)
klein-quartic/	Genus-3 Klein quartic: PSL(2,7) cohomology, 3D visualization (notes)
clayworth/	CSS quantum codes from graph tilings, Lean 4 formalization (docs)
clifford-torus/	Interactive Clifford torus homotopies (docs)
surfaces/	G₂(2) triality hexagon, Clebsch surface (readme)
hecke/	Clebsch surface and three Hecke algebras (notes)
lean/	Lean 4 / mathlib notes and named graph images

How it works

1. GAP computes Aut(G), enumerates subgroups, finds orbit quotients with low degree
2. Python selects the best quotient (by symmetry score), detects cyclic vs cylindrical layout, and places vertices on concentric rings
3. Crossing minimisation: path reordering of orbits (Schlegel-like), global angular offset search with rotations and reflections
4. TikZ renders the final PDF

For the full theory (Eades & Hong Theorem 3.2, layout modes, adding new graphs), see docs/SYMMETRIC_EMBEDDING.md.

Graphs

All graphs are cubic (3-regular). The collection is the complete Foster census of cubic arc-transitive (symmetric) graphs up to 30 vertices — all 13 of them. Cubic graphs are the simplest non-trivial regular graphs, and their automorphism groups are large relative to their size, making them a natural testbed for symmetry-aware drawing.

The list is exhaustive for n ≤ 30: every cubic arc-transitive graph on at most 30 vertices appears (there are exactly 13). No cubic arc-transitive graph exists on 12 or 22 vertices.

Given a graph Γ, we compute G = Aut(Γ) and select a subgroup H ≤ G whose orbits give a clean quotient. The layout places each H-orbit on a concentric ring.

Foster	Graph	n	G = Aut(Γ)	|G|	H	|H|	Orbits
F004	K₄	4	S₄	24	C₂	2	[2, 2]
F006	K₃,₃	6	(S₃ × S₃) : C₂	72	C₂	2	[2, 2, 2]
F008	Cube = GP(4,1)	8	C₂ × S₄	48	C₄	4	[4, 4]
F010	Petersen = GP(5,2)	10	S₅	120	D₁₀	10	[5, 5]
F014	Heawood	14	PSL(3,2) : C₂	336	C₂ × C₂	4	[2, 2, 2, 4, 4]
F016	Mobius-Kantor = GP(8,3)	16	GL(2,3) : C₂	96	C₈	8	[8, 8]
F018	Pappus	18	((C₃ × C₃) : C₃) : D₈	216	S₃	6	[6, 6, 6]
F020A	Dodecahedron	20	C₂ × A₅	120	C₅	5	[5, 5, 5, 5]
F020B	Desargues = GP(10,3)	20	C₂ × S₅	240	C₁₀	10	[10, 10]
F024	Nauru = GP(12,5)	24	S₄ × S₃	144	C₁₂	12	[12, 12]
F026A	F26A	26	C₁₃ : C₆	78	C₁₃ : C₃	39	[13, 13]
F028	Coxeter	28	PSL(3,2) : C₂	336	D₁₄	14	[7, 7, 7, 7]
F030	Tutte-Coxeter	30	(A₆ . C₂) : C₂	1440	C₁₀	10	[10, 10, 10]

All group structures computed by GAP (GRAPE package). K₄ and K₃,₃ are included in the code but omitted from the gallery as trivially small. gallery.pdf shows the 11 graphs from F008 to F030.

The next entry after F030 is at n=32.

Going deeper

• docs/SYMMETRIC_EMBEDDING.md — implementation: pipeline architecture, graph library, layout modes, how to add a graph
• docs/RESEARCH_PROGRAM.md — the research line: Biggs-Smith as "a picture of SU(2) over F₁₇", PSL(2,q) censuses, towers of covers, open questions
• docs/CLASSIFICATION.md — algebraic background: the GL(2,q) family, PSL(2,q) ≅ SO(3,q), split vs non-split tori
• docs/RESEARCH_NOTES.md — triangle quotient spectra, strong symmetric genus

Interactive visualizations

Square → Torus folding

Watch a unit square fold into a torus in two phases: rolling into a cylinder (identifying a ↔ a⁻¹), then bending into a torus (identifying b ↔ b⁻¹). Grid lines and colored loops show how the square's coordinate system wraps onto the surface. Drag the slider or hit Play.

Bolza surface (genus 2)

Cohomology and Poincaré disk tiling of the Bolza surface — the most symmetric genus-2 Riemann surface, with Aut = GL(2,3) of order 48. The (2,3,8) triangle group tiles the hyperbolic plane with 48 fundamental domains per copy of the surface.

• Octagon → Torus pipeline — Full animated pipeline: hyperbolic octagon → Euclidean octagon → cut into two pentagons → deform to cut squares → roll into cylinders → bend into two wounded tori → glue. All 48 (2,3,8) triangles tracked through every transformation. Drag the slider or hit Play.
• Hyperbolic octagon explorer — Interactive viewer of the 48 fundamental domain triangles. Press 1-5 to switch between: hyperbolic octagon (geodesic arcs), Euclidean octagon, half-octagon, cut square, wounded torus.
• Wounded torus fold — Half-octagon deforms into a cut square, then folds into a wounded torus via progressive bending (no self-intersection).
• bolza-surface/bolza_cohomology.py — oriented Lefschetz traces, H¹ = V₂ ⊕ V̄₂
• bolza-surface/bolza_3d.html — Poincaré disk tiling + 3D genus-2 surface
• bolza-surface/bolza_disk.pdf — publication-quality TikZ rendering of the tiling

Klein quartic (genus 3)

Cohomology of the Klein quartic via dual PSL(2,7) cell decompositions. The (2,3,7) triangle group gives the Hurwitz-maximal automorphism group of order 168.

• klein-quartic/klein_quartic_cohomology.py — H¹ = V₃ ⊕ V̄₃, oriented Lefschetz traces
• klein-quartic/klein_quartic_3d.html — Poincaré disk tiling + 3D genus-3 surface
• klein-quartic/verify_cohomology.g — GAP verification