TongGeometry - A geometric model jointly developed by Beijing Academy of Telecommunication Research and Development (BAST) and Peking University AI Research Institute.

What is TongGeometry?

TongGeometry, jointly developed by the Beijing General AI Research Institute and the AI Institute of Peking University, is a tree-search-based geometric model specifically designed for posing and solving Olympiad-level geometric problems. Based on efficient search algorithms and massively parallel computing, TongGeometry has built the most extensive library of geometric theorems to date, discovering 6.7 billion geometric theorems requiring auxiliary construction, of which 4.1 billion possess geometric symmetry. TongGeometry can generate high-quality competition problems and autonomously complete proofs. For the first time, TongGeometry surpassed the performance of gold medalists in solving geometric problems at the International Mathematical Olympiad (IMO) and can run efficiently on consumer-grade computers.

TongGeometry's main functions

• Geometric problem generationAutomatically generates high-quality Olympic geometry problems, covering a wide range of types from basic to complex, and supports the generation of symmetry and auxiliary constructions.
• Theorem discovery and proofBased on tree search and neural symbolic reasoning, it autonomously discovers and proves geometric theorems, including complex auxiliary constructions.
• Problem assessment and screeningProblems suitable for the competition are selected based on the scoring criteria and evaluated according to their difficulty and innovation.
• Educational resource generationIt provides a wealth of geometric problems and proof examples for education and research, promoting the popularization of geometry teaching.

TongGeometry's technical principles

• Tree search and guided question generationThis approach uses a tree search algorithm to gradually construct complex geometric problems starting from basic geometric elements. It generates geometric problems requiring auxiliary construction by combining backward tracing (starting from the target) and forward reasoning (gradually building auxiliary structures).
• Neurosymbolic reasoningCombining neural networks and symbolic reasoning, the policy model generates auxiliary constructs, and the value model estimates the problem-solving steps.
• Massive parallel computingUtilizing massively parallel computing resources (such as thousands of CPU cores), it explores the geometric problem space within a finite time, generating billions of problems. Based on efficient search strategies and caching mechanisms, it accelerates the problem generation and selection process.
• Auxiliary constructions and theorem proofsAutomatically identifies and generates auxiliary structures (such as auxiliary lines and circles) to fill in key steps in geometric proofs. Uses a deductive database approach for theorem proofs, combined with a full-angle method to generate human-readable proofs.

TongGeometry's project address

• arXiv technical paper:https://arxiv.org/pdf/2412.10673

Applications of TongGeometry

• Mathematics CompetitionGenerate high-quality geometry competition problems to assist in competition problem setting and contestant training.
• Mathematics EducationProvides teaching resources and supports personalized learning and online education platforms.
• Artificial intelligence research: Serving as an experimental platform for geometric reasoning research, it promotes algorithm development and model optimization.
• Mathematical research: To assist in the discovery and verification of geometric theorems and to build a problem library.
• Educational assessmentSupports competition question design and student ability assessment.