A hexagon can be imagined as 6 equilateral triangles all shoved together.

and that saves on one's frazzled nerves that result from the stress caused by calculating square roots of ordinary numbers then coming up with impossible fractions that have little place in the more charming and often less precise elfinesque world of pop-up cards.  

 A pop-up mechanism using a hexagon, one of my favorites, relies on the parallel edges of the hexagon running equidistant along the central fold of the card. In order to place the lines the proper distance, the height of any one of the triangles must be known, not the length of their sides. The hexagon will be attached to the card by only two parallel sides, the other four sides remain free, they compress when the card is folded. In order to cut the slits into the backing of the card to accept tabs attached to the sides of the hexagon, or simply to know where to glue the tabs, two of the six triangles within the hexagon can be imagined as a bow tie with its center placed on the spine. 

Conceptualizing a bow tie from two of a hexagon's six triangles allows you to dispense with measuring and use your God-given intuition to place lines, make cuts, and work with certainty that things will fit.

Hexagons are particularly well-suited for pop-ups. <--100% of opinion that can be backed up with more argument than anyone can reasonably be expected to endure.

If you were to draw a flower within a circle you could fold the circle flat and insert it into a card and glue its entire surface to the inside of a card along the crease. If you drew the same number of petals but didn't quite fill the circle, but then attached the ends together to force a circle, it would form a funnel but could still be folded flat and still be attached to the inside of a card, albeit at only two points or along two lines parallel with the card's central fold. This is what's happening here, a hexagon of flower petals is drawn within an octagon then the sides attached as if nothing at all was wrong. This forms a funnel which can be smashed flat inside a card. When the card is opened, the attached portions force the funnel to open along the established creases, the creases conforming with flower petals.

If you don't get it right it could still work as long as the two side remain parallel with the spine.  It doesn't really matter if the remaining sides are precisely the same size, however the resulting polygon will go wonky, which might actually be a desirable effect.

It should be emphasized, this is for a symmetrical flower, or cone, placed on the center fold.  It's intended to appear like a perfect little flower with the most precise way of standing upright and directly folding flat.  But there might be occasions when asymmetry is desireable and all this exact parallelism and equality of sides could make a thing seem stilted.  Some of the best effects are the result of asymmetry and from breaking a general rule.