What the incarnation cross is
In Human Design the incarnation cross is read as the theme of a whole life, the purpose the rest of the chart serves. It is built from four activations: the Sun and the Earth at birth, the conscious or Personality side, and the Sun and the Earth at the Design moment, about three months earlier, when the Sun stood 88° behind its place at birth. The Earth is always exactly opposite the Sun, so the four gates are two opposite pairs. The system teaches that the Sun and the Earth carry most of a chart's imprint, which is why the cross they make is read as its largest theme. A cross is written with the conscious gates first: 13/7 | 1/2 means the Sun in Gate 13 and the Earth in Gate 7 at birth, and the Sun in Gate 1 and the Earth in Gate 2 at the Design moment.
How the profile sets the angle
The first number of a profile is the line of the conscious Sun and the second the line of the Design Sun, and the profile decides the cross's angle. Profiles 1/3, 1/4, 2/4, 2/5, 3/5, 3/6 and 4/6 have a Right Angle cross, the 4/1 alone a Juxtaposition, and 5/1, 5/2, 6/2 and 6/3 a Left Angle. Measured on 10,000 birth moments spread evenly over 1950 to 2009, 64% of charts have a Right Angle cross, 34% a Left Angle and only 2.5% a Juxtaposition. The Juxtaposition is rare for the same reason the 4/1 profile is: it needs the conscious Sun in the last sliver of its fourth line, just after the Design Sun has crossed into the next gate.
What each angle means
Right Angle crosses are read as a personal destiny: a life that unfolds through its own process, with other people in supporting roles. Left Angle crosses are read as a transpersonal destiny, lived through relationships and encounters, where other people are part of the purpose itself. The Juxtaposition, between the two, is read as a fixed and specific path, less a story of growth than a steady role that others can rely on. The tradition does not rank them; it treats the three as different geometries of the same wheel.
How the 192 crosses are named
The conscious Sun can sit in any of 64 gates, under any of three angles, which makes 192 crosses, but they share only 112 names. Each Juxtaposition cross has a name of its own, one for each gate, often echoing that gate's keynote: a conscious Sun in Gate 13, the Listener, gives the Juxtaposition Cross of Listening. A Left Angle name is shared by a gate and the gate opposite it, so Gates 13 and 7 both give the Left Angle Cross of Masks. A Right Angle name is shared by four gates a quarter of the wheel apart: Gates 1, 2, 7 and 13 all give the Right Angle Cross of the Sphinx, and its four versions use the same four gates in a different order. The tables below list the 16 Right Angle crosses and every cross by gate.
Birth time and the cross
The whole cross follows from the Sun at birth, because the Design Sun is always 88° behind it. The Sun moves about a degree a day, spending 22 to 24 hours in each line and five and a half to six days in each gate, so on most birth dates any time of day gives the same cross. Measured over every day from 1950 to 2009, it changes at some moment on 37% of days, either where the Sun enters a new gate or where the angle changes with the profile. On those days the birth time decides the cross. Without a time, the free Human Design chart calculator says whether your cross holds all day; with one, it gives the span of birth times over which your cross and every other fact hold.
Reading your cross
The tradition reads a cross through its four gates. The conscious Sun's gate is read as the theme you express most directly, the conscious Earth as what grounds it, and the two Design gates as the part of the theme that runs below awareness, often clearer to others than to you. The guide to the 64 gates describes each of them. The cross is not the place to start, though: the system puts type, strategy and inner authority first, and reads the cross as the theme they serve over a lifetime. The Human Design guide explains the whole chart in that order.