Introduction
The 4D objects described below are grouped into categories based on how they're constructed. Some objects can be constructed in several ways, and so may appear under multiple categories.
The Polychora
In 2D, we have polygons, which means “many sides”. A polygon is bounded by linear sides. In 3D, we have polyhedra, meaning “many faces”. A polyhedron is bounded by polygonal faces. In 4D, the analogous objects are bounded by polyhedral volumes. Hence, we use the term polychora, which means “many rooms”. Polygons, polyhedra, and polychora, as well as their higher-dimensional analogs, are known collectively as polytopes.
The polyhedra may be constructed by folding 2D polygons into 3D such that they enclose a 3D volume. Analogously, the polychora are constructed by folding 3D polyhedra (solids bounded by polygons) into 4D, so that they touch each other at their faces and enclose a finite 4D volume. Here, we consider some common convex polychora.
The Regular Polychora. These are polychora whose cells are regular polyhedra (Platonic solids).
The Pentatope or 5-cell.
The Hypercube or 8-cell.
The Uniform Polychora. These are polychora whose cells are either Platonic or Archimedean polyhedra, and whose vertices are congruent. There are 64 of these polychora in total, not counting the infinite families which include the polyhedral prisms and the duoprisms. (Duoprisms are discussed below in the duo-cycles section.) George Olshevsky's Uniform Polytopes page contains a detailed catalogue of these beautiful polychora. Among them are:
The mesotruncated pentatope, consisting of 10 identical cells in the shape of truncated tetrahedra.
The mesotruncated 24-cell, consisting of 48 identical cells in the shape of truncated cubes.
The Extruded Objects
This category of objects are constructed by the extrusion of 3D objects. This is to take a 3D object and translate it along the W-axis, and taking its trace (the 4D volume it sweeps out as it's translated along the W-axis).
The cubical cylinder, or “cubinder”, as some people call it.
The spherical cylinder, or “spherinder”.
The conical cylinder.
The tetrahedral prism.
The cubic prism, or the hypercube, which is also among the Regular Polychora. One way of constructing the hypercube is by extruding a 3D cube along the W-axis.
The Tapered Objects
This category of objects are constructed by the tapering of 3D objects. This is to take a 3D object and translate it along the W-axis, but also shrinking it linearly at the same time, so that it has shrunk down to a point at the end of the translation. The 4D tapered object is the trace formed by this process.
The spherical cone.
The cylindrical cone.
The cubical pyramid.
The pentatope or 5-cell, which is also among the Regular Polychora. It can be constructed by tapering a 3D tetrahedron along the W-axis.
There is another common 4D object which may be constructed by tapering. The 16-cell can be constructed by tapering a 3D octahedron in two directions: the positive and negative directions along the W-axis. The 16-cell is among the Regular Polychora.
The Duo-cycles
These curious objects are constructed in a way possible only in 4D or higher. They are formed by joining together two perpendicular rings or cycles along the sides.
The duoprisms are constructed from two (possibly identical) types of 3D polygonal prisms. A specific member of this set is called an m,n-duoprism, where m and n specify the two types of polygonal prisms used. For example, the 3,5-duoprism is made of 5 triangular prisms and 3 pentagonal prisms. The hypercube is the same as the 4,4-duoprism.
An m,n-duoprism is made by stacking m copies of the n-gonal prisms on their polygonal faces and bending them in 4D to form a ring, stacking n copies of the m-gonal prisms and bending them in 4D to form a second ring, and then joining these two rings together along their sides to form a closed object.
The prismic cylinders may be thought of as the limiting cases of the m,n-duoprisms, where m is taken to infinity. The prismic cylinders are made of n cylinders and an n-gonal torus. The cubical cylinder is a member of this set.
The duocylinder is a peculiar object that has two perpendicular sides on which it can roll. It may be thought of as the limit of m,n-duoprisms as both m and n are taken to infinity. It is bounded by two interlocked, perpendicular circular torii.
