02 / SIX PLANES, ONE SPACE
Rotations in four dimensions
A rotation is defined by a plane. In four dimensions, there are six coordinate planes: XY, XZ, XW, YZ, YW and ZW.
Turn X toward W.
An active XW rotation transforms a point using two familiar sine and cosine equations, leaving Y and Z unchanged:
w′ = x sin θ + w cos θ
At 90°, the positive X direction becomes positive W. Aetherfold rotates the basis of the visible slice. A direction hidden a moment ago can become a direction you can walk.
Edge projection for illustration; not a solid slice.
Order matters.
Turning in XW and then ZW generally produces a different result from doing those same turns in the reverse order. Both involve W. By contrast, rotations in disjoint coordinate planes, such as XY and ZW, commute.
The campaign introduces XW and ZW turns. The Observatory lets you explore all six coordinate planes, including rotations involving the vertical axis.