Vertex-Minimal Paper Tori
A paper torus is an embedded polyhedral torus isometric to a flat torus. Schwartz proves no paper torus with 7 vertices exists and constructs one with 8, settling the minimum-vertex question in the flat-torus embedding tradition of Császár-torus combinatorics and the Lazarus-Tallerie universal triangulation.
- Result
- Proved
- Status
- Resolved
- AI contribution
- AI-assisted
- Method
- Computation
- Field
- Polyhedral geometry
- Posed by
- —
- Year posed
- —
- Years open
- —
- Solved
- 2025-07-20
- Model
- ChatGPT-4o, ChatGPT-5
- Vendor
- OpenAI
- Collaborators
- Richard Evan Schwartz
- Verification
- Unreviewed
- Publication
- Preprint
- Significance
- 10 / 100
- Disclosed cost
- —
- Wikipedia
- No dedicated article
What the AI did
Used for high-level idea discussion, error-finding, data formatting and program documentation across a heavily computer-assisted proof - plus one moment the author singles out: "In a stroke of brilliance, it found on its own the expression for the denominator... I might not have arrived at that."
Verification
Computer-assisted with exact integer arithmetic; all code public. No independent review.