AbstractThe spherulites are built of distinct fibrillar units and their fine‐structure presents two problems: the arrangement of these units and their internal structure. In the present paper the first has been treated in greater detail. Based on direct observations, and on earlier works on nonpolymeric spherulites, a regular periodic branching mechanism of the fibrillar units is proposed, where, in the case of maximum regularity, the position of each branch with respect to the preceding one is defined by one constant branching period and one constant angle in the two‐dimensional case, and by one constant branching period, two constant angles, and a constant direction of rotation in the three‐dimensional case. The various stages of such a spherulitic growth were directly observed in suitable samples. It is shown that this mechanism can lead to helices, to be referred to as “large‐scale helices,” which produce the previously observed zigzag extinction effects. The effects of the thickness of the crystallizing layer is followed, and possible connections between these effects and more general problems of crystal growth are indicated. Investigations of the fine‐structure of the fibrils indicate a closely coiled rope‐like arrangement in samples suitable for these observations. These are on a size of 400 A. and upward, sometimes reaching the limit of the optical microscopic resolution dependent on the preparation. They will be referred to as “small‐scale helices.” These units appear to be built of and connected by much smaller fibrils which themselves possess a periodicity of 100–200 A. or less. The possible reasons for coiling and its relation to the perpendicular orientation found in Part II are discussed, and it is suggested that branching could be a possible consequence of the coiling structures. The conclusion is reached that, as far as the morphology is concerned, crystallization in polymers represents a special case of the formation of crystalline aggregates in viscous media.