To get started I used the plan from Woodturner Pro to dimension the maple and the cherry. The thickness is 3/4" and each strip of wood represents the material to be used for a specific ring. As you can see the lengths and widths vary for each ring.
At the compound miter saw a stop block is clamped so that the vertical spacers can be cut. White oak is used in this instance and as you can see there is a 3/16'' crosscut section. The grain direction of the vase construction is all horizontal. The reason for this is to maintain stability of the glue joints while minimizing wood movement.