Sections of the Four-Cube
This displays the section of the 4D cube by a hyperplane orthogonal to the normal vector described by the hyperspherical coordinate angles , , and and at a displacement from the origin along the normal. A projection of the four-cube from a view point along the normal is also shown, as well as of the , , , axes.
The unit normal vector is controlled by the angles , , and . The first two are just like the same angles in regular 3D spherical coordinates: , with , is the angle to the positive axis (or more precisely, the -- hyperplane), and , with , is the angle to the positive axis (or rather the - plane). The angle , with , is the angle formed by the normal and the positive axis and is analogous to in that way. If , then and have no effect, just as when , the angle has no effect. When the normal is changed, it appears that the four-cube rotates. But it is not moving; it is your view point that is moving. Of course, the two kinds of motion look the same.
Animate the distance and view the series of sections orthogonal to the normal from one end to the other. The coordinates of the vertices of the four-cube are , so the vertices are two units from the origin. The displacement is the displacement of the hyperplane, and thus the section consists of all points of the four-cube, the projections of whose coordinate vectors onto the normal are equal to times the normal.
Each cell (bounding 3D cube) of the four-cube is drawn with the same color as the face of its section by the hyperplane. By playing with the opacities, you can see the relationship between the cube-cube's cells and the intersection.
The projection displayed is from a point 3 to infinity units away from the origin along the line through the origin parallel to the normal (infinity gives a parallel projection). The four-cube and section are projected onto the hyperplane through the origin and orthogonal to the normal. A 3D coordinate system for this hyperplane is attached to the normal and rotates around with it as the normal is changed. This coordinate system will "precess": as the normal moves around (you need to change at least two angles) and moves back (along a different trajectory) the section will appear to have changed position. But it has not. It is the coordinate system used to project the section onto the screen that has changed.
Some important angles: , , , .
Snapshot 1: cell first (normal parallel to center of a cube-cell): cube
Snapshot 2: face first (normal parallel to center of a 2D face): rectangular box with a square face
Snapshot 3: edge first (normal parallel to midpoint of edge): equilateral-triangular prism
Snapshot 4: vertex first (normal parallel to midpoint of vertex): regular tetrahedron
Snapshot 5: hexagonal prism
Snapshot 6: regular octahedron