All petals are leaves that have been modified by the plant to both protect its reproductive organs and attract pollinators to them. Most petals are symmetrical in some fashion- some are arranged radially, while others may be bilaterally arranged. Regular flowers, like a buttercup, have petals that are all relatively the same in size and shape. Irregular flowers, like orchids, are arranged on at least one plane of symmetry.
Fibonacci Sequence - 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144...
First written about in 6th century India, the Fibonacci sequence has powerful applications in nature.
The pattern appears in the family tree of bees. An unfertilized egg hatches into a male bee while an egg fertilized by a male bee hatches into a female bee. So a male bee has 1 parent, 2 grandparents, 3 great-grandparents, 5 great-great-grandparents etc.
It also appears in how humans inherit the female X chromosome.
In plants this pattern usually reflects the most efficient way for it to grow, as can be seen with the tightly packed seeds of this sunflower.
34 blue spirals veer from the center towards the right, while 55 red spirals veer to the left. To ensure that the maximum amount of seeds fills the space, each seed grows at about 137.5 degree angle from the last. That irregular angle places each seed in a way that naturally forms these spirals.
The video below is an adorable and fun explanation, while popmath.org has a great written explanation.
Vihart is a youtube blogger and super genius who makes fun videos about math. Her animated story, What was up with Pythagors? cracks me up and shows a great way to think of numbers.