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| Format: | Recurso digital |
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Zenodo
2025
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| Online Access: | https://doi.org/10.5281/zenodo.17818471 |
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Table of Contents:
- <p>Abstract<br>We propose that the Viviani curve (sphere–cylinder intersection, 1724), the brachis-<br>tochrone cycloid (Bernoulli 1696), and the modern infinity symbol ∞ (Wallis 1655) are the<br>same geometric object up to affine transformation.<br>When this curve is used as the local transverse-field direction in a strongly-coupled<br>(J/h → ∞) cyclic XX spin-1/2 chain and sampled at exactly four sites per full 4π parametric<br>traversal (N ≡ 0 (mod 4)), numerical evidence up to N = 50 suggests the ground state<br>acquires exact 2N/4 -fold degeneracy with polarization along the local normals vanishing<br>to machine precision (typically |⟨m · n⟩| ≲ 10−24 , variance at double-precision noise floor<br>∼ 10−24 –10−29 ).<br>We present an analytic argument that period-4 sampling renders the lattice discretization<br>mathematically lossless and propose that the Viviani/brachistochrone/∞ manifold is the<br>unique lowest-action closed curve permitted by Euclidean geometry — the “laziest” way for<br>the universe to close a loop without energy cost.<br>The symbol has been staring at us from every mathematics book for 370 years. We<br>finally followed the line.</p>