Adv. Geom. 3 (2003), 263–286 Advances in Geometry ( de Gruyter 2003 Multiple spread retraction Norman L. Johnson and Keith E. Mellinger* (Communicated by G. Korchma ´ros) Abstract. Translation planes of order q t and kernel containing K G GFðqÞ admitting fixed- point-free collineation groups GK  , each of whose point orbits is the set of nonzero vectors of a 2-dimensional K-subspace, are shown to permit spread-retraction and produce either Baer subgeometry or mixed partitions of a corresponding projective space. When the same transla- tion plane or spread produces a number of partitions of isomorphic projective spaces, we call this multiple spread-retraction. This analysis is used to describe triply-retractive spreads, in general, and to consider the triply-retractive spreads of order 16, in particular. Key words. retraction, multiple, spread. 2000 Mathematics Subject Classification. Primary 51E23, Secondary 51A40 1 Introduction In Mellinger [14], there is a complete enumeration of the mixed partitions in PGð3; 4Þ. Each partition produces a translation plane of order 16. In fact, it is shown, in par- ticular, that these partitions produce all of the translation planes of order 16. How- ever, several inequivalent partitions produce the same translation plane. The question is, why is this so? In Johnson [12], it is shown that given a translation plane of order q 4 with spread in PGð7; qÞ that may be considered a GFðq 2 Þ-vector space, which admits the GFðq 2 Þ  group as a collineation group with component orbits of lengths 1 or q þ 1, there is a retraction which produces a mixed partition in PGð3; q 2 Þ. In the case of order 16, this would mean that a translation plane of order 16 admitting a GFð4Þ  group as a col- lineation group would correspond to a mixed partition in PGð3; 4Þ. In this article, we consider the implication of the results of Mellinger [14] from the standpoint of the translation plane and generalize the ideas for spreads of order 2 n admitting fixed-point-free groups of order 3. In particular, we are able to show that the set of partitions arising from a given translation plane is in bijective corre- spondence to the set of fixed-point-free groups of order 3 acting on the plane. More * The authors gratefully acknowledge the help of the referee in the writing of this article.