The system's internal machinery is worth setting out because it is what distinguishes the Enneagram from a simple list of nine types. Wings are the two adjacent numbers on the circle, one of which is normally held to colour the type. The arrows connect the nine points in two paths, the triangle 9-3-6 and the hexad 1-4-2-8-5-7, derived from the properties of the number seven in decimal division, and the system reads movement along them as integration or disintegration depending on direction. The instinctual subtypes, self-preservation, social and sexual or one-to-one, multiply the nine into twenty-seven and are largely Naranjo's development. Levels of development, a nine-step scale within each type, are Riso and Hudson's addition from the late 1980s. Each of these layers has an identifiable modern author and date.
A further layer built on top of type and instinctual subtype is the tritype, developed in 1994 and 1995 by Katherine Chernick Fauvre, drawing on scattered remarks by Ichazo and Naranjo but organized into a formal, published theory only by her; it holds that a person carries one dominant type from each of the three triads, body, heart and head, for twenty-seven possible three-number combinations said to describe a fuller personality than a single type alone. Like the wings, arrows and instinctual subtypes, it is a distinct, datable addition to the base system rather than part of Ichazo's or Naranjo's original teaching.