The head triad, also called the thinking or fear triad, comprises types Five, Six and Seven, and the emotion the system assigns to it is fear. The three are held to handle it differently: Five withdraws and accumulates knowledge as a defence, Seven runs ahead into options and possibilities, and Six at the centre engages the fear directly, either by seeking security or by attacking what frightens it. The triad's concern is with security and guidance, and its time orientation is the future. The three triads together are one of the system's more useful organising ideas, and they are also a good place to see its construction: the assignment of anger, shame and fear to three groups of three is a modern psychological scheme mapped onto a diagram Gurdjieff used for something else entirely, and it dates from the 1970s.
The same holds for the head triad under Riso and Hudson's second system, the Hornevian groups, set out in The Wisdom of the Enneagram (1999) and drawn from Karen Horney's three neurotic trends of moving toward, against and away from people: Five falls among the withdrawn types, Six among the compliant, and Seven among the assertive. That every emotional triad contains exactly one member of each Hornevian group is a structural feature of how Riso and Hudson built the second scheme, not a coincidence, and it is a useful check for a reader trying to keep the system's several overlapping groupings straight: the two triads answer different questions and are not the same grouping under different names.
Sources The Enneagram: G. I. Gurdjieff's Esoteric SymbolCarole M. Cusack with Personality Types: Using the Enneagram for Self-DiscoveryDon Richard Riso and Russ Hudson