The pulvinar is an 'associative' thalamic nucleus, meaning that most of its input and output relationships are formed with the cerebral cortex. The function of this circuitry is little understood and its anatomy, though much investigated, is notably recondite. This is because pulvinar connection patterns disrespect the architectural subunits (anterior, medial, lateral and inferior pulvinar nuclei) that have been the traditional reference system. This article presents a simplified, global model of the organization of cortico-pulvinar connections so as to pursue their structure-function relationships. Connections between the cortex and pulvinar are topographically organized, and as a result the pulvinar contains a 'map' of the cortical sheet. However, the topography is very blurred. Hence the pulvinar connection zones of nearby cortical areas overlap, allowing indirect transcortical communication via the pulvinar. A general observation is that indirect cortico-pulvino-cortical circuits tend to mimic direct cortico-cortical pathways: this is termed 'the replication principle'. It is equally apt for certain pairs (or groups) of nearby cortical areas that happen not to connect with each other. The 'replication' of this non-connection is achieved by discontinuities and dislocations of the cortical topography within the pulvinar, such that the associated pair of connection zones do not overlap. Certain of these deformations can be used to divide the global cortical topography into specific sub-domains, which form the natural units of a connectional subdivision of the pulvinar. A substantial part of the pulvinar also expresses visual topography, reflecting visual maps in occipital cortex. There are just two well-ordered visual maps in the pulvinar, that both receive projections from area V1, and several other occipital areas; the resulting duplication of cortical topography means that each visual map also acts as a separate connection domain. In summary, the model identifies four topographically ordered connection domains, and reconciles the coexistence of visual and cortical maps in two of them. The replication principle operates at and below the level of domain structure. It is argued that cortico-pulvinar circuitry replicates the pattern of cortical circuitry but not its function, playing a more regulatory role instead. Thalamic neurons differ from cortical neurons in their inherent rhythmicity, and the pattern of cortico-thalamic connections must govern the formation of specific resonant circuits. The broad implication is that the pulvinar acts to coordinate cortical information processing by facilitating and sustaining the formation of synchronized trans-areal assemblies; a more pointed suggestion is that, owing to the considerable blurring of cortical topography in the pulvinar, rival cortical assemblies may be in competition to recruit thalamic elements in order to outlast each other in activity.