Power of the Poincaré group: elucidating the hidden symmetries in focal conic domains.

Focal conic domains are typically the "smoking gun" by which smectic liquid crystalline phases are identified. The geometry of the equally spaced smectic layers is highly generic but, at the same time, difficult to work with. In this Letter we develop an approach to the study of focal sets in smectics which exploits a hidden Poincaré symmetry revealed only by viewing the smectic layers as projections from one-higher dimension. We use this perspective to shed light upon several classic focal conic textures, including the concentric cyclides of Dupin, polygonal textures, and tilt-grain boundaries.

[1]  R. Kamien,et al.  Symmetry breaking in smectics and surface models of their singularities , 2009, Proceedings of the National Academy of Sciences.

[2]  O. Lavrentovich,et al.  Liquids with conics , 2009 .

[3]  C. Brenner.  Memoir , 2009 .

[4]  Douglas N. Arnold,et al.  Möbius transformations revealed , 2008 .

[5]  Xiangjun Xing.  Topology of smectic order on compact substrates. , 2007, Physical review letters.

[6]  C. Santangelo,et al.  Geometric theory of columnar phases on curved substrates. , 2007, Physical review letters.

[7]  Blanc,et al.  Tiling the plane with noncongruent toric focal conic domains , 2000, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[8]  O. Lavrentovich,et al.  Grain boundaries and the law of corresponding cones in smectics , 2000 .

[9]  J. Sethna,et al.  Spheric domains in smectic liquid crystals , 1982 .

[10]  F. G. Friedlander,et al.  The Wave Equation on a Curved Space-Time (Book Review) , 1976 .

[11]  George F. R. Ellis,et al.  The Large Scale Structure of Space-Time , 2023 .

[12]  P. Gennes,et al.  Statistical properties of focal conic textures in smectic liquid crystals , 1973 .

[13]  Y. Bouligand,et al.  Recherches sur les textures des états mésomorphes - 1. Les arrangements focaux dans les smectiques : rappels et considérations théoriques , 1972 .

[14]  P. D. Gennes,et al.  An analogy between superconductors and smectics A , 1972 .

[15]  F. G. Friedlander.  Simple progressive solutions of the wave equation , 1947, Mathematical Proceedings of the Cambridge Philosophical Society.

[16]  Jyoichi Kaneko.  WAVE EQUATION AND DUPIN HYPERSURFACE , 1986 .

[17]  F. Grandjean,et al.  Observations géométriques sur les liquides à coniques focales , 1910 .

[18]  A. Einstein.  Zur Elektrodynamik bewegter Körper , 1905 .

[19]  J. Larmor.  A dynamical theory of the electric and luminiferous medium. Part III. Relations with material media , 1897, Proceedings of the Royal Society of London.

[20]  A. Cayley.  IV. Note on a quartic surface , 1865 .