  We introduce a new class of division algebras, the hyperpolyadic algebras,
which correspond to the binary division algebras $\mathbb{R}$, $\mathbb{C}$,
$\mathbb{H}$, $\mathbb{O}$ without considering new elements. First, we use the
matrix polyadization procedure proposed earlier which increases the dimension
of the algebra. The algebras obtained in this way obey binary addition and a
nonderived n-ary multiplication and their subalgebras are division n-ary
algebras. For each invertible element we define a new norm which is
polyadically multiplicative, and the corresponding map is a $n$-ary
homomorphism. We define a polyadic analog of the Cayley-Dickson construction
which corresponds to the consequent embedding of monomial matrices from the
polyadization procedure. We then obtain another series of n-ary algebras
corresponding to the binary division algebras which have a higher dimension,
that is proportional to the intermediate arities. Second, a new polyadic
product of vectors in any vector space is defined. Endowed with this product
the vector space becomes a polyadic algebra which is a division algebra under
some invertibility conditions, and its structure constants are computed. Third,
we propose a new iterative process ("imaginary tower"), which leads to
nonunital nonderived ternary division algebras of half the dimension, which we
call "half-quaternions" and "half-octonions". The latter are not subalgebras of
the binary division algebras, but subsets only, since they have different
arity. Nevertheless, they are actually ternary division algebras, because they
allow division, and their nonzero elements are invertible. From the
multiplicativity of the introduced "half-quaternion" norm we obtain the ternary
analog of the sum of two squares identity. We show that the ternary division
algebra of imaginary "half-octonions" is unitless and totally associative.
