  In addition to the vertices of the regular hexagon and icosahedron, there are
precisely two strictly projective tight 5-designs: one constructed from the
short vectors of the Leech lattice and the other corresponding to a generalized
hexagon structure in the octonion projective plane. This paper describes a new
connection between these two strictly projective tight 5-designs -- a common
construction using octonions. Certain octonion involutionary matrices act on a
three-dimensional octonion vector space to produce the first 5-design and these
same matrices act on the octonion projective plane to produce the second
5-design. This result uses the octonion construction of the Leech lattice due
to Robert Wilson and provides a new link between the generalized hexagon
Gh(2,8) and the Leech lattice.
