We construct all harmonic maps from the two-dimensional sphere to a quaternionic projective space \({\mathbb{H}}P^{n-1}\) from holomorphic data in an explicit fashion by a purely algebraic algorithm. This gives explicit formulae for the harmonic maps as rational functions. The method is a development of the ideas of K. K. Uhlenbeck [J. Differ. Geom. 30, 1-50 (1989)] who showed that any harmonic map of the two-sphere to the unitary group U(n) can be obtained from a constant map by a finite number of operations called adding a uniton (generalized under the name flag transform [F. E. Burstall and J. Rawnsley, Twistor theory for Riemannian symmetric spaces (with applications to harmonic maps of Riemann surfaces), preprint, Universities of Bath and Warwick]. The method gives a parametrization of all non-isotropic harmonic maps, the isotropic ones having been discussed by S. Erdem and the second author [J. Lond. Math. Soc., II. Ser. 28, 161-174 (1983; Zbl 0492.58013)] and J. Glazebrook [Contemp. Math. 49, 51-61 (1986; Zbl 0588.58016)]. In the case \(n-1=1\), our result reduces to that of R. Bryant [J. Differ. Geom. 17, 455-473 (1982; Zbl 0498.53046)] who shows that, for every harmonic map \(\phi\) : \(S^ 2\to {\mathbb{H}}P^ 1=S^ 4\), \(\phi\) or \(\phi^{\perp}\) arises from a horizontal holomorphic map of \(S^ 2\) into \({\mathbb{C}}P^ 3\) via the twistor fibration \({\mathbb{C}}P^ 3\to {\mathbb{H}}P^ 1\). In the case \(n-1=2\), our construction is equivalent to that of A. R. Aithal [Osaka J. Math. 23, 255-270 (1986; Zbl 0619.58017)].
A similar treatment, but with interesting differences, for the case of harmonic two-spheres in the complex quadric or real projective space \(G_ 2({\mathbb{R}}^ n)\) was given by the authors [J. Reine Angew. Math. 398, 36-66 (1989; Zbl 0665.58009)].