If nature is supposed to be so 'ordered' mathematically, with patterns all over the place, supposedly in the ways plants grow, seeds show up in fruits, and animal populations grow, why is that instead of nice, even, and easy to see numerical patterns of whole numbers, nature seems to favor irrational numbers more than rational ones? Doesn't this go against the whole patterning thing and any mathematical understanding of it the world?
New Orleans, LA
First, it is helpful to distinguish the meaning of the words 'rational' and 'irrational' as applied to numbers in mathematics, and the meaning of these same words outside math.
In the world of Mathematics, the words 'rational' and 'irrational' are objective terms with no subjective judgment implied. That is to say that neither category of numbers is superior or more desirable than the other category. RATIONAL, which comes from the word "Ratio", simply means any number that can be expressed as a ratio of two integers. For example, the fraction one-half is a rational number because it can be expressed as 'one-over-two' or 1/2, and both 1 and 2 are integers. On the other hand, the famous "pi", from the world of circles and pi(r-squared), is not a rational number because it cannot be exactly expressed as one integer over another. The closest reasonably useful approximation is 22/7, but even this does not exactly equal pi when you do all the division.
The reason I mention that there is no 'judgment' regarding "rational" numbers or "irrational" numbers in mathematics, and that they are purely objective terms, is that outside the world of math it is often the case that these same terms are often not considered objective terms that are 'separate but equal' descriptions; rather, "irrational" often caries an implied stigma of being a person, idea, or behavior that is without reasonable explanation or one that does not follow a 'logical' series of steps for derivation or justification. And 'rational', when used as the non-mathematical converse to "irrational" is, of course, the converse of unreasoned and illogical thought or behavior or a person responsible for such. Though there are many exceptions (most notably in the arts and other creative endeavors), 'rational' behavior is typically considered a more intelligent and educated approach than 'irrational' behavior.
BACK TO THE WORLD OF MATH:
Nature is indeed extremely mathematical in so many ways that is continues to impress, excite, and perplex those who study it for these properties. It never ceases to amaze observers for its mathematical genius. And THIS is precisely why it favors irrational numbers over rational ones. Specifically, it is the blend of its mathematical wizardry with its infinite 'interest' in efficiency that makes irrational proportions and distributions to be the best (most effective) design for almost everything in nature.
It's like this:
Nature may or may not 'think', but if it doesn't, then it has some kind of need or mechanical obsession with efficiency. And that makes sense: Nature has quite a lot to do each minute of each day of each year, and it shows no sign of slowing up. Nature is infinite even though it is all 'contained' in a world that is closer to finite than infinite in its resources. So efficiency is critical.
This is where irrational numbers come in. Since, by definition, irrational numbers cannot easily be divided up into simple ratios of integers, such as 1/3 or 2/5, they have a natural leaning toward coming into being at 'off' amounts from each other, so that when their existence is repeated elsewhere or multiplied by a rational number the result yielded will be a new, unique measurement, which is even harder to repeat. WHAT???????
It's like this: Branches that grow out of the sides of a tree follow a spiral push as the tree grows taller and taller, and newer and newer branches spring out of higher and higher places. BUT, it would NOT be the most efficient system for the branches that are there (or the future ones to come) to follow a neat and simple rational fraction of the tree's circumference for each branch's starting point, because then after a certain number of spiral revolutions around the tree trunk one branch would spring out directly above one below it, and thereby cover the lower one in shade and deprive it of the sunlight which it needs for its photosynthesis, which is the process that makes a branch's leaves green and healthy. If a branch were to spring out every time the spiral had progressed one-third around the tree, for example, then the pattern would be: 1/3, 2/3, 3/3, 1 1/3 ..., and the first to appear over another would be the fourth one, (the 1 1/3 one), blocking the light from (and gradually killing) the first one, which was the one located directly underneath, or at the point we can think of as the 'address' 1/3. For any part of a tree to kill any of its own growth is inefficient, and early trees that did that when trees first began to show up in the world did not do as well as the more efficient ones, and nature, being the shrewd manager she is, selected the right progeny for each job and gave the others smaller jobs, or jobs that were temporary so that each lifetime was as efficient as all living creatures need to be.
It may sound weird, but it certainly seems to be true.
And, to me at least, when I look at a tree or a branch or a leaf -- or just about anything else that nature has created, I see perfection. During a quiet moment, take the opportunity . . . and check it out!
Hope this helps,