Summary

Vases with varying shapes, generated directly from a Python script.

Very quick to print with single perimeter thickness, and surprisingly rigid thanks to all the folds and curved surfaces.

Update: Removed the non-commercial part of the license.

Instructions

Slice and print with one perimeter, no top layers, no infill.

I use PLA with 0.2 mm layer thickness, and a 0.4 mm nozzle. With 40 mm perimeter speed (and active cooling) a 10 cm high vase takes just over one hour to print.

Slic3r complains about a hole, because the top surface is missing from the uploaded stl:s. However the result is fine.
The reason is that this way I get an object which is displayed with thin walls, but Slic3r views it as a solid object so you can still slice it with fine control over the wall thickness (number of perimeters in the slicer settings).
The downside is of course that it is no longer a solid object, but Slic3r does not seem to mind (much).

The stl files are generated by the included python scripts:
Install Python 2.7.
Unzip vasegenerator.zip.
Run vasegenerator.py

To generate solid solids, with top surfaces, just change the parameter "top" to True in the function "gen".

It would be great to see more designs based on this, so I'll give a few hints to get you started:

The "gen" function takes as input a number of steps (the resolution) and a polygon generator function.
The polygon generator function will be called once per step with a value which increases from 0 to 1. It must return a polygon with the same number of segments each time.
There is one polygon generator function for each vase. Typically a polygon is created using some primitives described below, then a z-component is added to make the vase rise, and possibly some rotation arounds the z-axis.

I have used a few "turtle graphics" primitives to generate the polygons, but you can generate polygons any way you like.
Here is an overview:
koch, circle, square, gear: Generates "turtle" output.
bevel: rounds corners by inserting short segments.
turtle: Converts "turtle" output to a polygon.
morph: Blends different polygons or turtle outputs.

NB: One or two of the vases fold into themselves (intersecting facets), but slic3r handles it without complaint.

Scripted vases