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 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.

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.


Last updated 16 Feb 2016.

Powered by Apache Runs on Debian GNU/Linux Viewable on any browser Valid CSS Valid XHTML 1.1! Proud to be Microsoft-free