Mathematica is a commercial mathematical software known for its user friendly interface, language, formatting, and graphics. Mathematica is developed by Wolfram Research, founded in 1987 by S. Wolfram.

The above images were all generated by Mathematica for the 1992 User’s Guide for Macintosh. It is likely that these images are copyrighted by Wolfram research.

These images are, in pairs, wireframe and single colour styles of the same underlying object.

The first pair is a knotted torus, the knot has no given name, but has braid word (1122-1-1-2-21-21-2) [I use additive notation for lack of TeX].

The second pair is a complex variety of some sort.

The third pair is the Mandelbrot set.

The fourth pair is the famous figure 8 immersion of the Klein Bottle. It’s famous because the two mobius bands are readily seen.

The last pair is simply “conchoids”, i.e. spiraling conics. That is, in cylindric coordinates, the surface is a spiral for every height and is a conic (in this case a circle) for every angle.

What is amazing, is that the input code that generates all of these is shorter than what I have written in this post. This is because these are actually very simple objects, yet they quite clearly contain an abundance of inherent mathematical beauty.