Symmetry-Structured Neural Completion of Islamic Geometric Patterns from Sparse Control Geometry

2026-06-30

AI Summary

This work addresses the challenge of accurately and efficiently generating strictly rotationally symmetric Islamic geometric patterns under sparse control conditions. By formalizing the construction rules of Islamic patterns as geometric priors within a neural completion framework, the authors introduce an orbit-binding mechanism that jointly predicts edges and refines curves within cyclic group orbit structures. This ensures generated outputs consistently satisfy exact N-fold rotational symmetry, anchor-point preservation, and boundary controllability. Integrating group theory, symmetry-aware projection reasoning, and training on a procedurally generated dataset, the method achieves lossless fidelity with complete inputs and, under partial or missing inputs, maintains zero symmetry violations—outperforming unstructured baselines—while simultaneously preserving both generation validity and geometric fidelity.

Abstract

Islamic geometric patterns are governed by exact rotational symmetry and strict construction rules. This paper treats these rules as formal geometric knowledge and embeds them in a neural completion framework, rather than leaving them to be learned statistically from data. Given sparse control geometry and a target symmetry order, the system completes the pattern as a vector graph by predicting edges and refinements of bounded curves over a candidate lattice whose edges are organised into rotational orbits under the cyclic group. Symmetry is enforced either by constraining predictions within these orbits or by projecting them onto them during inference. The orbit-tied variant provides a constructive guarantee: for any input and any orbit-level selection rule, it produces exact N-fold symmetry, preserves anchor points, and keeps all refinements within prescribed bounds. These properties are verified numerically. The study focuses on rotational symmetry, and all quantitative results are obtained from procedurally generated graphs inspired by Islamic geometric design rather than from a historical corpus. On clean inputs, enforcing exact validity produces no measurable loss in fidelity. When control geometry is missing, an unstructured decoder loses fidelity and breaks symmetry; retraining on corrupted inputs recovers much of the fidelity but not exact validity. Symmetry-structured inference, by contrast, keeps violations at zero throughout. The results show that augmentation and symmetry structure address distinct failure modes: augmentation improves fidelity under corruption, while symmetry structure guarantees validity. The framework therefore provides a knowledge-constrained, guarantee-backed approach to neural completion for scalable vector ornaments whose validity depends on exact geometric structure.

Problem

• Islamic geometric patterns
• rotational symmetry
• neural completion
• geometric constraints
• vector graphics

Innovation

• symmetry-structured inference
• rotational orbits
• neural completion
• geometric validity guarantee
• vector graph generation