  An equiangular tight frame (ETF) is a type of optimal packing of lines in
Euclidean space. A regular simplex is a special type of ETF in which the number
of vectors is one more than the dimension of the space they span. In this
paper, we consider ETFs that contain a regular simplex, that is, have the
property that a subset of its vectors forms a regular simplex. As we explain,
such ETFs are characterized as those that achieve equality in a certain
well-known bound from the theory of compressed sensing. We then consider the
so-called binder of such an ETF, namely the set of all regular simplices that
it contains. We provide a new algorithm for computing this binder in terms of
products of entries of the ETF's Gram matrix. In certain circumstances, we show
this binder can be used to produce a particularly elegant Naimark complement of
the corresponding ETF. Other times, an ETF is a disjoint union of regular
simplices, and we show this leads to a certain type of optimal packing of
subspaces known as an equichordal tight fusion frame. We conclude by
considering the extent to which these ideas can be applied to numerous known
constructions of ETFs, including harmonic ETFs.
