To print a 3D object you need a description of that object in an .stl file. One of the easiest ways to make a simple .stl file is to construct an object in Tinkercad, an in-browser CAD drawing program that you can use for free (for one design) or a monthly fee (to save multiple designs).
Here are two prints of a cone that is cut to reveal a parabolic conic section. You can't slice things in half in Tinkercad, but you can start with a cone and then use another object (like a slanted rectangular solid) to make a "hole" in it. I did this for the little piece and then moved the rectangular solid to cut out the other half for the larger piece. They don't fit together perfectly - or stand up very well together - but it is cool to see the parabola. The green model was printed on a Replicator 2 with PLA filament material and the white model was printed on a Replicator 1 with ABS filament material.
Other ways to get .stl files for printing include visiting Shapeways or Thingiverse and downloading an existing design, using the demo files that come with your printer, or even using Minecraft to build something and export to an .stl file (more on this later). You can also use software like Mathematica or MATLAB to generate code for .stl files (more on this later too).
In the picture below, clockwise from the upper left:
- the one sheet hyperboloid was modeled in Tinkercad, printed on a Replicator 1
- the white rounded cube was made in Tinkercad (to examine infill properties and resolution), printed on a Replicator 2
- the green cube was exported from Minecraft (although it is too small to see the details in this model, it is a die with video game designs on the faces), printed on a Replicator 2
- the dogbone-shaped object is a dual dodecahedron from Thingiverse, printed on a Dimension Elite
- the Klein bottle was created from MATLAB code and the hole in it was put in with Tinkercad, printed on a Dimension Elite (it is surprisingly thin)
- the nut and bolt is from a Replicator 2 demo file (it actually screws together very smoothly) and printed on a Replicator 2
- the chain is from a Replicator 2 demo file (it was printed linked together, with the pieces sort of standing up diagonally on the build platform so they didn't touch) and printed on a Replicator 2