We describe a theory of packing hyperboloid “diabolic” domains in bend-free textures of liquid crystals. The domains sew together continuously, providing a menagerie of bend-free textures akin to the packing of focal conic domains in smectic liquid crystals. We show how distinct domains may be related to each other by Lorentz transformations and that this process may lower the elastic energy of the system. We discuss a number of phases that may be formed as a result, including splay–twist analogues of blue phases. We also discuss how these diabolic domains may be subject to “superluminal boosts,” yielding defects analogous to shock waves. We explore the geometry of these textures, demonstrating their relation to Milnor fibrations of the Hopf link. Finally, we show how the theory of these domains is unified in four-dimensional space.

我们描述了在液晶的无弯曲纹理中填充双曲面“双曲面”结构域的理论。这些畴连续缝制在一起，提供了类似弯角液晶畴中堆积的局部圆锥畴的无弯曲纹理的痕迹。我们展示了不同的域如何通过Lorentz变换彼此关联，并且该过程可能降低系统的弹性能。我们讨论了可能由此形成的多个相位，包括蓝色相位的张开-扭曲类似物。我们还将讨论如何对这些合成代谢域进行“超腔增强”，从而产生类似于冲击波的缺陷。我们探索了这些纹理的几何形状，证明了它们与Hopf链的Milnor纤维的关系。最后，我们展示了这些域的理论如何在四维空间中统一。