A brief definition of functions, illustrated by nature
In order to avoid the use of the not rigorously defined words ‘rule’ and 'associates’, the intuitive explanation of functions is completed with a formal definition that relies on the notion of the cartesian product. The cartesian product of two sets X and Y is the set of all ordered pairs, written (x, y), where x is an element of X and y is an element of Y. The x and the y are called the components of the ordered pair. The cartesian product of X and Y is denoted by X × Y.
A function f from X to Y is a subset of the cartesian product X × Y subject to the following condition: every element of X is the first component of one and only one ordered pair in the subset. In other words, for every x in X there is exactly one element y such that the ordered pair (x, y) is contained in the subset defining the function f. This formal definition is a precise rendition of the idea that to each x is associated an element y of Y, namely the uniquely specified element y with the property just mentioned.
Have you ever asked how math relates to our life, and received a somewhat mysterious answer involving references to seashells or flower petals, and then been expected to consider it all good and clarified? My favorite mysterious example to use is the dimensions of a piece of paper. It’s extremely satisfying to give such an answer and watch the surprise and confusion and sometimes even awe play across an audience’s face, but, as answers to a genuine question go, it’s not particularly helpful.
So what do seashells and flower petals and pieces of paper (and tree branches and pine cones and bees and rabbits and the human face and an infinite number of other surprising examples that seem satisfyingly disconnected) have in common? As it turns out, they’re all connected by a string of numbers: the Fibonacci sequence.
You’ve probably heard of it. It’s one of the most over-used examples of math’s connection to nature ever, because there’s a satisfying mystery about it–for everyone. Not even mathematicians totally understand the Fibonacci sequence, although we certainly use it enough.
The Fibonacci sequence is arrived at by adding the two previous terms of the sequence together to create the next term. One can either start the sequence at 0,1 or 1,1–I prefer 0,1, for absolutely no better reason than that I do.
Okay, so, it doesn’t look like much. At first. But then, later on, it looks like a lot. Almost too much. How can this one string of numbers connect to so many things around us? (The wonderfully fun incredibly cool answer: we don’t actually know.)
But, now, to explain:
The Fibonacci sequence was supposedly discovered (I’ve heard multiple stories on the matter, but I like this one) in connection to breeding rabbits. Immortal rabbits, actually, because, in true mathematician form, we’ve extended our real life situation into the realm of the mind and ideas. So our rabbits never die.
Now, as much as I love my immortal rabbits, I find the clearer (and more well-substantiated) story to be of bees–particularly because they’re not required to be immortal. The Fibonacci sequences traces the growth of a population of rabbits (hence the immortality–otherwise we’d keep losing chunks of our population). But the story of bees traces the history of a population, so it matters very little to us if the bees are still alive or not. (Although it’s still an idealized population, because we rely heavily on a consistent bee breeding pattern.) Let me explain:
(I’ve stolen these formal rules of our idealized bee population from Wiki, as I’m not as familiar with this example as I am with the rabbits.)
If an egg is laid by an unmated female, it hatches a male (drone) bee.
If, however, an egg was fertilized by a male, it hatches a female.
Therefore, a male bee will have one parent.
And a female bee will have two parents.
Let us now trace back the history of a male bee’s family:
We’re dealing with 1 male bee.
He has 1 parent (rule three).
He has 2 grandparents (rule four)
He has 3 great-grandparents (rules thee and four)
He has 5 great-great-grandparents (two female grandparents–>two sets of two parents, one male grandparent–>one set of one parents)
He has 8 great-great-great grandparents
He has 13 great-great-great-great grandparents
Now, I know the logic gets a little hard to follow there at the end, but you see the pattern, right? (If not, go back and look at the numbers I wrote again.)
So: we have these magical numbers that show up when bees breed. Yay. Now where does the rest of it come in?
As it turns out, spiraling patterns like leaves off a stem or branches off tree or the uncurling of a fern, are formed by two consecutive Fibonacci numbers.
(Thanks Wiki for the image.) Notice that there are 13 aqua spirals and 21 blue spirals–consecutive Fibonacci numbers. (The above is a picture of the head of a Yellow Chamomile flower, if you’re wondering.)
Okay, that’s all for now, but I’ll probably have one or two posts discussing the other occurrences of the Fibonacci numbers, including the golden ratio and Pascal’s triangle.
With many cheerful facts about the square of the hypotenuse,
from Brock, Eugenia. “Why Are Fibonacci Numbers Important in Nature? : Math Problems & Trigonometry.” YouTube. Ed. Stephen Brock. EHow Education, 23 Feb. 2013. Web. 22 Oct. 2014. <https://www.youtube.com/watch?v=-GVQ2-3Nv2s>.