# 4D Geometric Objects

## Introduction

There are various ways of constructing 4D objects. The following sections describe some of these constructions and objects that may be built in this way. Some objects can be constructed in several ways, and so may appear under multiple categories.

## The Extruded Objects

Objects in this category 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

Objects in this category are constructed by the *tapering* of 3D
objects. This is to take a 3D object and translate it along the W-axis, but
also to shrink 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, one of the CRF polychora.

The pentachoron 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.

Mathematically speaking, these objects are formed by taking the Cartesian product of 2D polygons or circles.

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 prism on their polygonal faces and bending them in 4D to form a ring, stacking*n*copies of the*m*-gonal prism 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.