The 33 =  27 face centers of the cube around the unit sphere
The coordinates are assigned to the points based on the usual right-handed coordinate system.
They are then interpreted as balanced ternary numbers from −13 to 13.
The least significant trit is on the left (little-endian), so the enumeration is consistent across dimensions.
(The l.s.t. always corresponds to the first dimension — usually called x.)
In the table on the right the gray boxes to the right of the triples indicate the Hamming weight of each number ( A005812).
The weight (number of non-zero entries) corresponds to what kind of point the coordinate describes:
0 1 2 3
origin face center(line vertex) edge center(square vertex) cube vertex
While the figures (−1, 0, 1) are written as (−, 0, +), the sign of the integers is only expressed by their color.
The positive octant of this cube is the unit cube. Its vertices have the numbers 0, 1, 3, 4, 9, 10, 12, 13 ( A005836).

Right-handed coordinate system

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Own work
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