The first step on making this was the design. I will illustrate multiple steps I went trough, and how and why I did them.
I took student membership at Techshop (www.techshop.ws), so I can use the facilities. I took a SBU for wood lathe and went to work. I got some personal tools for advanced hollowing.
Step 1: Design
Segmented vase is made by putting together pieces of wood, to make a rings, stack the rings, and then turn them on the lathe. Hollowing becomes easier, but the design is constrained after you start.
The design is first created as a half section. This is the shape of the vase or bowl you have in mind. I roughly used the thickness and the width, I will need to have sufficient overlap between layers. I decided to make each rings with 12 segments (each segment is a regular trapezium, with an included angle of 30 degrees). I tested my math skills, and just to be sure, by cutting some foam core and putting them into a ring.
I made the a CAD file (this can be done on paper with a ruler and pen too) and took a 1:1 print of the segments i have to cut and number of segments i have to cut.