Wednesday, December 24, 2008

Church Windows in Curved Space

The notion that church windows are thicker at the base because glass is a viscous liquid is an urban legend. In fact, glass probably isn't a liquid and church windows are thicker at the base because they were cooled while upright. Probably because no one really understands the physics of glass, though no one thinks it's useful to call it a liquid -- glass doesn't flow on observable timescales.

This is by way of preamble to a neat idea in David Nelson's book Defects and Geometry in Condensed Matter Physics, viz. that glassiness is due to geometrical frustration. When liquids crystallize quickly (upon rapid cooling, say), they form the locally optimal structure, which, for spherical atoms, is an equilateral triangle in two dimensions and a regular tetrahedron in three dimensions. Equilateral triangles are a tiling of the plane, so the local crystals line up and there are no glasses in two dimensions; regular tetrahedra are not a "tiling" of space, so the tetrahedra can't line up to form a global crystal arrangement, and get jammed in place to form a glass. One fascinating consequence of this is that glasses should (1) not exist in some uniformly curved 3D spaces, like the 3-sphere, that are tiled by tetrahedra; (2) exist in curved two-dimensional spaces, like the surface of saddle, where equilateral triangles are not a tiling. Apparently this is consistent with computer simulations.

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