For the sphere cap model of leaves, using a sphere of radius 10, without any overlap of leaves, nor overlap of hinge pins:

For 5 leaves, the latitude angle of the circle where the hinge pins plus support are located, is about 32.2312 degrees, from the equator.
This leave about 0.5473 units (e.g. cm) for each of the 5 hinge pins plus support.
This is about the minimum angle so that blades will not overlap the hinge pins of adjoining leaves.

For 6 leaves, there is not enough latitude circumference, until 60 degrees from the equator is reached.
At this angle there is zero space for the hinge pins.

At 61 degrees, there is only 0.0923 cm of room for 6 pins, which is extremely little.
The opening in the top of the sphere is small.

- Brian