# Tetrahedra

I made two tetrahedra, both of which demonstrate the strategy behind setting up triple integrals. One tetrahedron came from our calculus textbook, another came from Professor Keller. The former is defined by the equations $$x + 2y + z – 2, x = 2y, z = 0$$, and $$x = 0$$ (Example 5 of 12.5 in Stewart’s Essential Calculus), and the latter by $$y=-6, z=0, z=x+4$$, and $$2x+y+z=4$$. When I first tried to print the Tetrahedron from the textbook, the equation on the bottom face did not appear. Later, Dave Pfaff told us that it was because the object was inside out in Cinema 4D. I fixed the issue by reversing the normals on the object.

I then tried to print the object in addition to a set of round coordinate axes, but one of the axes was either knocked out of position by the left extruder (the unused one) or it curled up because it cooled (or maybe both). Later, I tried to print the tetrahedron with Prof Keller’s tetrahedron, but they curled up at the ends because the cooling of the plastic is exacerbated when the length increases. Next, I used a raft to print the two shapes, but the equations looked awful. So then I used some ABS juice on the half of the build-plate that contained the sharp vertex of Professor Keller’s tetrahedron and the print came out well, except for a messy-looking number “4” and some stringy filament on one of the faces (see blue shape).

Finally, I printed Professor Denne’s tetrahedron along with another set of coordinate axes (see black objects). Since there was some juice left over on the build-plate, the only end that curled up was at the origin of the coordinate axes, and the rest of the objects looked great.  These tetrahedra can be found on Thingiverse here and here.