Step 1: What you need
Any saw that can make round cuts (bandsaw, jigsaw or even a scroll saw).
A lathe (or a sander, but this will take very long!).
compass (for drawing circles)
Paper and pencil:-)
Step 2: Design phase
After sketching it up rouhly i decided for a minimal wall thickness (~10mm in my case) and drew the lines accordingly. Now I basically got a cross-section of the vase.
From this cross-section you can easily meassure the diameters/radiuses of all the layers.
Draw the the circles (or half circles) onto the board using a large compass (or a string + pencil, if you don't have one).
As I had only long and narrow boards i cut out half-circles and glued those together. If you are using a jigsaw and you have got wider boards you can of course cut out the whole circles.
The circles don't have to be ultra precise, bettter make them a little bit too big than too small. You can remove material quite easily on the lathe
Step 3: Gluing and turning
Apply glue, put the rings back onto each other and clamp everything together with large clamps (or put heavy weights on top of the vase). You should add a larger board on top of the vase, so you can clamp it properly.
IMPORTANT: When you are using glued together half rings, don't align the glue points on top of each other! you might have a weak spot during turning and the whole thing can fall (or better "fly") apart!
After gluing all of the pieces together, put it in the lathe.
I added another small baseplate to the bottom, so i could attach it to my chuck. I also glued a baseplate ON TOP the vase to stabilize at further with the tailstock/spindle (my circles were not cut perfectly, therefore it was were imbalanced).
I first roughed it down to be really round and to get rid of the steps between the layers using a roughing gouge, then I used a fingernail gouge to clean everything up.
Then I turned off the top baseplate (at the tailstock), so that I could turn the inside on the top part (only as far, as you can easily see from the top)
I didn’t turn it on the inside because you barely can see inside the vase and i lack the proper tools (and skills...).
After that i sanded it with fine grid sandpaper (up to 400) and parted it off.
Unfortuntately I don't have any pics of this steps. But there're a lot of videos on youtube on turning hollow forms, check those out.
Your gouges shold be really sharp for cutting plywood (ok, this holds in general for wood turning).
If you don't have a lathe you can also sand it down using a belt sander. I've done this in another project, but it takes really a lot of time (see here)
Step 4: Finish
My mom is using mostly dry flowers, so I didn’t seal the inside. I think you could use some water proof lacquer on the inside by simply pouring it in and turning the vase, so that it spreads equally.