Watching a cell divide is mesmerizing.

But it gets even more spellbinding when you realize there's a mathematical pattern underneath it. Over many divisions, the generations of cells reproduce mathematician Blaise Pascal's legendary triangle, in which the two numbers directly above add up to the one below them.

At first there is one cell: A. Then there are two cells: A and B. At the next division, A will have produced two "B" cells, and the lone B cell from the first division will have produced one cell, which we'll label "C." So there are now four cells: 1 "A", 2 "Bs", and 1 "C." The pattern continues. At the next division there will be 1 A, 3 Bs, 3 Cs, and 1 D. And so on and so forth.

Mathematical patterns manifest across all corners of life. Sometimes they're broad and sweeping, like Swiss biologist Max Kleiber's Law. It states that an organism's basal metabolic rate -- the amount of energy it consumes at rest -- is roughly equal to its mass raised to the three-quarters power. Kleiber's Law has also been found to apply roughly to lifespan in both plants and animals! In general, the larger an organism, the longer it lives. There are many exceptions to Kleiber's Law, prompting some biologists to take umbrage with its generalized nature. Nevertheless, the notion definitely has traction, receiving thousands of citations in the scientific, peer-reviewed literature.

In some places, mathematics is less observational and more integral. Fractals, exponentially branching figures, can be found in all sorts of places: "leaves, gills, lungs, guts, kidneys, chloroplasts, and mitochondria, the whole-organism branching architectures of trees, sponges, hydrozoans, and crinoids, and the treelike networks of diverse respiratory and circulatory systems," physicist Geoffrey West noted.

But why? A prominent explanation, which I previously wrote about, is that fractal-like shapes are fantastic at maximizing surface area. This extra area allows nutrients and gasses (like oxygen in your lungs) to be transported more efficiently within and throughout biological entities and structures.

For most evolved life, efficiency is everything. It is in pursuit of this perfection that some of nature's most astounding patterns have arisen. Ever count the petals of a flower or the spirals of a pinecone? Each will almost always* be a number from the Fibonacci Sequence, in which the previous two numbers add up to the next: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, etc.

At first, this is mind-boggling. Why would Nature do this? But as YouTube educator Vi Hart pointed out, the reason is beautifully simple. Plants want to maximize the amount of sunlight they receive, so logically, each petal should never completely block another out. Thus, petals are placed based upon an irrational "Golden" ratio: 1.61803, which the Fibonacci Sequence closely mimics!

"This design provides the best physical accommodation for the number of branches, while maximizing sun exposure," the University of Georgia's Nikhat Parveen described.

The Golden Ratio also appears on you! Various proportions of the human body -- on our face, fingers, and arms -- roughly average out to 1.61803. How quaint: humanity is inherently irrational!

*Correction 1/19: An earlier version of this article made it seem like the Fibonacci numbers were a constant in plants. They are not constant, just extremely common.

(Images: Flickr/Didier.bier, NikonU, Youbing Yin, UGA)