The Cubical Cylinder

The cubical cylinder, or cubinder for short, is one of the ways to generalize the 3D cylinder to 4D. This object can be constructed by extruding a 3D cylinder along the W-axis by a unit distance. The following is a diagram of the cubinder in parallel projection:

The following is a perspective projection of the cubinder:

The inner cylinder is actually the same size as the outer cylinder, but appears smaller in perspective because it is farther away. Also, the upper and lower frustums connecting the inner cylinder to the outer cylinder are actually themselves regular cylinders; but they appear as frustums because they are being viewed at from an angle.

And the next diagram shows another perspective projection, this time with the cubinder rotated 45 degrees in the ZW plane.

Here, it is more apparent that the cubinder consists of four 3D cylinders joined at their circular ends. What is not so apparent, however, is that their round surfaces are joined by a square torus. This is the torus generated by rotating a square in the ZW plane (displaced in the Y direction by one unit) in the XY plane. One could also construct it by extruding the round part of the cylinder along the W-axis.

These diagrams do not have hidden surfaces removed, so that the whole structure of the cubinder is more apparent. However, a 4D being looking at the cubinder would not see the entire structure simultaneously. The inner cylinder in the second diagram would not be visible, for example, and neither would the two inner cylinders in the second diagram.

There are 4 circular surfaces on the cubinder, on which it may roll. All 4 surfaces, however, can only rotate parallel to the same plane, and hence the cubinder can only cover the space of a line by rolling. (This may sound counterintuitive to our 3D-centered understanding of rotation, because the 4 cylinders are in 2 perpendicular pairs, and so have perpendicular central axes. In 3D, two perpendicular central axis define two different planes of rotation. However, in 4D, rotations don't happen around an axis, but around a plane. The perpendicular central axes of the cubinder are actually both perpendicular to the plane of rotation. As one can see from the diagrams above, all circular sections of the cubinder lie parallel to the same plane, and hence can only rotate in that plane.)

The reason for the cubinder's name is that one of its possible projections is the 3D cube:

The blue edges in the cube show where the 4 circular ends of the cylinders are. They have collapsed into a single line in the diagram because they are perpendicular to the direction of projection.