Sandbox
Rotation through a dimension you cannot point at
Drag to turn the figure in ordinary space. Hold shift and drag to turn it through w. Set any plane spinning with the sliders, then freeze it and study the still.
drag · rotate in space — shift + drag · rotate through w
Figure
The 4-cube. Its three-dimensional cross-sections are ordinary cubes — no single cube is the tesseract.
Rotation planes
Slide a three-dimensional slice through the figure. The slice is real — but it is never the object.
Six planes
In three dimensions there are three planes of rotation. In four there are six: xy, xz, yz — and xw, yw, zw. The last three rotate the figure through the dimension you cannot point at.
Why it warps
Nothing is bending. You are watching a perspective shadow of a rigid object, cast from four dimensions down to two. The distortion is the price of the projection.
The slice
Turn on the cross-section and a single three-dimensional slice is picked out in blue. It is entirely real. It changes as you drag. The figure does not.
A conscious life stands to its moments exactly as this figure stands to its slices.