Dr. Gareth Alexander: Geometry and topology in soft matter: elucidating the hidden symmetries in focal conics

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    Geometry and topology in soft matter: elucidating the hidden symmetries in focal conics

    Dr. Gareth. P. Alexander, Department of Physics & Centre for Complexity Science, University of Warwick, Coventry, UK.

The combination of long-range order and fluid softness in liquid crystals goes hand-in-hand with intricate and exquisite textures of a distinctive geometric, or topological, nature. One of the most remarkable of these are focal conics -- the hallmark of layer order in smectic materials. These are beautiful and geometrically precise conic sections that are observed to occur in pairs of perfect confocal partners and with adjacent domains often exhibiting a more widespread level of geometric organisation, forming intricate networks and trellises, grain boundaries, and Apollonian packings.

I will survey the history of focal conics from the pioneering work of Friedel and Grandjean, who elucidated their pristine geometry, before describing our own contributions that reveal an enhanced conformal symmetry of focal conic domains coming through an analogy with light-like surfaces in special relativity. I will use this perspective to shed light upon several classic focal conic textures, including the concentric cyclides of Dupin, and in doing so illustrate the uses of geometric and topological constructions in understanding the properties of soft materials.