Editing 16-cell (section)

=== Coordinates ===
{| class="wikitable floatright"
!colspan=2|Disjoint squares
|-
|
{| class="wikitable" style="white-space:nowrap;"
!colspan=2|''xy'' plane
|-
|( 0, 1, 0, 0)||( 0, 0,-1, 0)
|-
|( 0, 0, 1, 0)||( 0,-1, 0, 0)
|}
|-
|
{| class="wikitable" style="white-space:nowrap;"
!colspan=2|''wz'' plane
|-
|( 1, 0, 0, 0)||( 0, 0, 0,-1)
|-
|( 0, 0, 0, 1)||(-1, 0, 0, 0)
|}
|}The 16-cell is the 4-dimensional [[cross polytope|cross polytope (4-orthoplex)]], which means its vertices lie in opposite pairs on the 4 axes of a (w, x, y, z) Cartesian coordinate system.

The eight vertices are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by edges except opposite pairs. The edge length is {{radic|2}}.

The vertex coordinates form 6 [[orthogonal]] central squares lying in the 6 coordinate planes. Squares in ''opposite'' planes that do not share an axis (e.g. in the ''xy'' and ''wz'' planes) are completely disjoint (they do not intersect at any vertices). These planes are [[completely orthogonal]].{{Efn|In 4 dimensional space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is opposite or ''[[completely orthogonal]]'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}}

The 16-cell constitutes an [[Orthonormal basis|orthonormal ''basis'']] for the choice of a 4-dimensional reference frame, because its vertices exactly define the four orthogonal axes.