Constructions of biangular tight frames and their relationships with equiangular tight frames

We study several interesting examples of Biangular Tight Frames (BTFs) - basis-like sets of unit vectors admitting exactly two distinct frame angles (ie, pairwise absolute inner products) - and examine their relationships with Equiangular Tight Frames (ETFs) - basis-like systems which admit exactly one frame angle (of minimal coherence). We develop a general framework of so-called Steiner BTFs - which includes the well-known Steiner ETFs as special cases; surprisingly, the development of this framework leads to a connection with famously open problems regarding the existence of Mersenne and Fermat primes. In addition, we demonstrate an example of a smooth parametrization of 6-vector BTFs in R-3, where the curve "passes through" an ETF; moreover, the corresponding frame angles "deform" smoothly with the parametrization, thereby answering two questions about the rigidity of BTFs. Finally, we generalize from BTFs to (chordally) biangular tight fusion frames (BTFFs) - basis-like sets of orthogonal projections admitting exactly two distinct trace inner products - and we explain how one may think of them as generalizations of BTFs. In particular, we construct an interesting example of a BTFF corresponding to 16 2-dimensional subspaces of R4 that "Plucker embeds" into a Steiner ETF consisting of 16 vectors in R-6, which we refer to as a Plucker ETF.