| _version_ | 1866901656160960512 |
|---|---|
| author | Keeble, Clifford |
| author_facet | Keeble, Clifford |
| contents | <p>For seven decades, materials scientists have observed that the glass<br>transition occurs at approximately 2.5% critical free volume across<br>virtually all glass-forming materials—polymers, metallic glasses,<br>molecular liquids, and oxide glasses. Despite this remarkable<br>universality, no fundamental explanation has been proposed for why<br>this specific value governs the transition. We present a derivation<br>from icosahedral geometry: the critical free volume fraction equals<br>1/41 = 2.439%, where 41 = (V − 1) + E represents the structural<br>degrees of freedom accessible from a local perspective within an<br>icosahedral cluster (V = 12 vertices, E = 30 edges). The (V − 1)<br>term follows from Klein's icosahedral geometry: from any vertex,<br>only 11 others are visible—you cannot use the vertex you occupy as<br>a rearrangement coordinate. The theoretical value 1/41 = 2.439%<br>falls within the experimental range of 2.4–2.6%. The icosahedron<br>cannot tile 3D space, and this frustration sets a universal threshold.</p> |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_18328719 |
| institution | Zenodo |
| language | |
| publishDate | 2026 |
| publisher | Zenodo |
| record_format | zenodo |
| spellingShingle | Klein's Icosahedron and the 1/41 Glass Transition Threshold Keeble, Clifford glass transition, critical free volume, icosahedral geometry, geometric frustration, Frank clusters, condensed matter physics, Klein icosahedron <p>For seven decades, materials scientists have observed that the glass<br>transition occurs at approximately 2.5% critical free volume across<br>virtually all glass-forming materials—polymers, metallic glasses,<br>molecular liquids, and oxide glasses. Despite this remarkable<br>universality, no fundamental explanation has been proposed for why<br>this specific value governs the transition. We present a derivation<br>from icosahedral geometry: the critical free volume fraction equals<br>1/41 = 2.439%, where 41 = (V − 1) + E represents the structural<br>degrees of freedom accessible from a local perspective within an<br>icosahedral cluster (V = 12 vertices, E = 30 edges). The (V − 1)<br>term follows from Klein's icosahedral geometry: from any vertex,<br>only 11 others are visible—you cannot use the vertex you occupy as<br>a rearrangement coordinate. The theoretical value 1/41 = 2.439%<br>falls within the experimental range of 2.4–2.6%. The icosahedron<br>cannot tile 3D space, and this frustration sets a universal threshold.</p> |
| title | Klein's Icosahedron and the 1/41 Glass Transition Threshold |
| topic | glass transition, critical free volume, icosahedral geometry, geometric frustration, Frank clusters, condensed matter physics, Klein icosahedron |
| url | https://doi.org/10.5281/zenodo.18328719 |