  The Hopf fibration has inspired any number of geometric structures in
physical systems, in particular in chiral liquid crystalline materials. Because
the Hopf fibration lives on the three sphere, $\mathbb{S}^3$, some method of
projection or distortion must be employed to realize textures in flat space.
Here, we explore the geodesic-preserving gnomonic projection of the Hopf
fibration, and show that this could be the basis for a new liquid crystalline
texture with only splay and twist. We outline the structure and show that it is
defined by the tangent vectors along the straight line rulings on a series of
hyperboloids. The phase is defined by a lack of bend deformations in the
texture, and is reminiscent of the splay-bend and twist-bend nematic phases. We
show that domains of this phase may be stabilized through anchoring and
saddle-splay.
