Sections of the 16-Cell
The 16-cell is a four-dimensional cross polytope, an analog of the octahedron. It is bounded by 16 tetrahedra (cells). This Demonstration displays the section of the 16-cell 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 16-cell from a viewpoint along the normal is also shown.
The unit normal vector is controlled by the angles , , and . The first two are just like the same angles in regular 3D spherical coordinates: is the angle , with the positive axis (or more precisely, the hyperplane) and , , is the angle with the positive axis (or rather the plane). The angle 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 4-cube rotates. But it is not moving; it's the viewpoint 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 16-cell are ±1, 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 16-cell the projections of whose coordinate vectors onto the normal are equal to times the normal.
Each pair of opposite cells of the 16-cell is drawn with a different color. Each face of the section is drawn with the color corresponding to the cell that the hyperplane passes through. By playing with the opacities, you can see the relationship between the 16-cell's cells and the intersection.
The projection 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 16-cell 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 (change at least two angles) and moves back (along a different trajectory) the section will appear to have changed position. But it hasn't. It is the coordinate system used to project the section onto the screen that has changed.
Some important angles: , , , .
Snapshot 1: cell first (the normal is parallel to the vector from the origin to the center of a tetrahedron cell): a "clipped" tetrahedron
Snapshot 2: face first (the normal is parallel to the vector from the origin to the center of a 2D face)
Snapshot 3: edge first (the normal is parallel to the vector from the origin to the midpoint of an edge): rectangular box capped with a pyramid
Snapshot 4: vertex first (the normal is parallel to the vector from the origin to a vertex): regular octahedron
Snapshot 5: cuboctahedron
Snapshot 6: hexagonal dipyramid