  We describe the homotopy types of Vietoris-Rips complexes of hypercube graphs
at small scale parameters. In more detail, let $Q_n$ be the vertex set of the
hypercube graph with $2^n$ vertices, equipped with the shortest path metric.
Equivalently, $Q_n$ is the set of all binary strings of length $n$, equipped
with the Hamming distance. The Vietoris-Rips complex of $Q_n$ at scale
parameter zero is $2^n$ points, and the Vietoris-Rips complex of $Q_n$ at scale
parameter one is the hypercube graph, which is homotopy equivalent to a wedge
sum of circles. We show that the Vietoris-Rips complex of $Q_n$ at scale
parameter two is homotopy equivalent to a wedge sum of 3-spheres, and
furthermore we provide a formula for the number of 3-spheres. Many questions
about the Vietoris-Rips complexes of $Q_n$ at larger scale parameters remain
open.
