I like some of the vases I've found here but being a tinkerer, I wanted to make changes. I quickly found myself dealing with math that I haven't touched in over 20 years. It was a great learning experience and open ended with possibilities. This first post will be some of my early designs. I will post others later once I'm happy with them.

I've printed these in ABS, HIPS and PLA. I'd recommend a 1mm shell at 0.25mm resolution. These are hollow objects so you don't have to mess with infill or removing the top layer.

I'm including the OpenSCAD files so others like me can play with the designs. Note that the basic vase prints just fine at a lower granularity (higher numbers). You will see the vertical and horizontal parameters (vgrain and hgrain) at the top of the file. The more complex vases need to set vgrain and hgrain to 1. Be aware that they take forever to render. The designs do seem to scale well. The straight fluted vase in the picture was scales to 2 before printing.

I've commented the OpenSCAD files but here are some generalities. Each vase is drawn by layers. Each layer is broken into wedges. The size is controlled by vgrain and hgrain. There is a mathimatical formula to represent the layer. For the basic vases it is just the trig formula for a circle. Additionally there is a mathimatical formula that represents the vertical profile of the vase. In the basic vase file I've included four profiles. All of them are based on the cosine function and its harmonics. I've included a file called vase_profile.scad which allows you to quickly see what a profile might look like. It is how I came up with the profiles I'm using.

Be aware that if you change the formula for generating fluting or make the diameter too small you may run into errors. What happens is the nice arc shapes actually generate loops that loop back on themselves. For example the stuff you might see from playing with a Spirograph as a kid. The loop backs cause the left points of the wedge to be less than the right points of the wedge and create an invalid shape. They seem to work in the quick preview but when you render, they will throw errors. Enjoy.