What an Incarnation Cross is
The Incarnation Cross is a person's life theme — their purpose in the broad sense. It is made of four gates: the Sun and Earth in the Personality (conscious) and the Sun and Earth in the Design (unconscious). Together the Sun and Earth carry about 70% of all the influences in a chart, which is why the cross describes a person so strongly. The cross isn't something to "achieve": it unfolds by itself when a person lives by their strategy and authority.
Angles
The angle of the cross depends on the profile. Right Angle (profiles 1/3, 1/4, 2/4, 2/5, 3/5, 3/6, 4/6) — a personal path. Left Angle (5/1, 5/2, 6/2, 6/3) — a path lived through relationships with others. Juxtaposition (4/1 only) — the fixed path of a single theme.
Quarters
The wheel of 64 gates is divided into four quarters of 16 gates: Initiation, Civilization, Duality and Mutation. The quarter of the Personality Sun sets the general direction through which a person looks for meaning: the mind, form, relationships or transformation.
192 crosses, 112 names
Each of the 64 Personality Sun gates gives three crosses, one per angle — 192 in all. Right Angle crosses group into 16 "families" of four gates (the Sphinx, for example, is gates 1, 2, 7 and 13), Left Angle crosses into 32 pairs, and the 64 Juxtaposition crosses are each unique.