Saddle

Saddles

12.1 Planes and surfaces
12.2 Graphs and level curves
12.6 Directional derivatives and the gradient
12.8 Maximum/minimum problems

Note: Section numbers refers to Calculus, Early Transcendentals (2nd edition) by Briggs, Cochran, and Gillett.

Two saddles and a monkey saddle

Saddle surface

Saddle with grid lines and saddle with level sets

Here are two difference versions of a quartic surface: a standard mathematical saddle surface. This surface is the graph of the function 

In the red and blue version, the red curves second color depicts the level sets of the surface.  In the blue and white version, the blue curves depict the fixed x and fixed y grid lines. This point (x,y,z) = (0,0,1) (the middle point) is a critical point, meaning all partial derivatives are zero, but the point is neither a maximum nor a minimum. 

Monkey Saddle

The following print illustrates the graph of the function

This is a cubic surface. Like the standard saddle, it still has a critical point at (x,y) = (0,0), (the center), but it is even flatter than the standard saddle. It is called a monkey saddle since a monkey could sit on this saddle with room for both legs and its tail. 

Technical Details for Printing

These objects were designed in Mathematica. These technical notes were determined by trial and error (or more accurately error and error and error ad nauseam)

• The parametrization lines are created using the Tube command in Mathematica.
• This was printed with neither raft nor supports. 
• The blue base piece is needed in order to make the print work without falling over, the thickness of the white surface is 2, and the thickness of the blue parametrization is 2.5.
• In order to make the blue base pieces, I made spheres in Mathematica and used the hole options in Tinkercad to cut off the bottom half. Do not just leave the sphere intact and assume that the printer will ignore the part below the build plate. This actually seems to stop the printer for a long time and set there thinking about printing the bottom half of the sphere.
• Do not try to print something too steep - it seems that if it's about the same height as length and width you'll be better off. A steep piece can break off the base under its own weight. On the other hand, too wide will give the dreaded overhangs of more than 45 degrees.
• The side walls that are printed on the dual print can topple over. This seems to be a build plate adhesion problem. No consistent solution to this problem.

Quartic Surface with Level Sets

The saddle surface with level sets is made much the same as above, but the tubes are used for fixed height levels. 

Technical Details for Printing the monkey saddle

The standard monkey saddle function does not include the 1/2, but this made for too steep a figure, which started to break off at the corners.