The "Nature" of Mathematics
The "nature" of mathematics. A clever (or at least, cute) way of saying that mathematics, and all its beauty, is found in nature, and all its beauty.
In The Beauty of Mathematics it was seen that, yes, mathematics is beautiful. The example highlighted was the Fibonacci Sequence and its unexpected relationship to the Golden Ratio and Pascal's triangle. What is even more unexpected is how these beautiful mathematical truths are manifested in nature, in the beautiful world itself.
Recall the Fibonacci sequence which is a sequence of terms where each term - after the initial conditions - are the sum of the two previous terms. The sequence is {1,1,2,3,5,8,13,21,34,...} (refer to the article Description, Mathematics And Other Fun Facts of the Fibonacci Sequence I wrote for more details on the Fibonacci Sequence). Though beautiful from a mathematical perspective, the Fibonacci sequence might seem quite innocuous. To that I say, consider the columbine flower:
It is quite a beautiful flower. But notice the number of petals: 5. Interesting! The number 5 is a number in the Fibonacci sequence! Okay, so what, no big deal. Well, consider the black-eyed Susan flower.
Count the number of petals.
There are 13 petals in the black-eyed Susan flower. The number 13 is a number in the Fibonacci sequence! Interesting! I know my interest has been piqued. Consider the good old daisy.
Did you count the number of petals?
Though perhaps hard to count, there are 21 petals. The number 21 is a number in the Fibonacci sequence!
Are you seeing a pattern here? The number of petals on the above flowers are found in the Fibonacci sequence! Whoa! What's going on here?
Upon closer inspection you will notice that the petals begin to overlap. They are growing in a spiral! A botanist I am not, but this spiral pattern is so that new petals do not block sunlight from reaching the others and, perhaps most importantly, so that the amount of water (in the form of rain or dew) that gets routed to the roots is maximized.
Without getting too technical, the spiral growth is such that a new petal is rotated some fractional amount relative to the one before it. If the fractional growth was a simple fraction then the resulting petal arrangement would, as more petals are grown, most likely end up with a linear pattern. Petals would potentially completely overlay the ones below it and/or there would be large gaps between the petals. This can be problematic to the survival of the flower.
To avoid this, the fractional rotation of a new petal should be as far from a simple fraction as possible. This would imply an irrational number. And an irrational number which can not be easily approximated by a rational number would be even better (the value of pi, for example, is often approximated as 22/7). As was described in the article Description, Mathematics And Other Fun Facts of the Golden Ratio, the golden ratio is perhaps the most difficult of irrational numbers to approximate by a rational number. It is often referred to as the most irrational of the irrational numbers! The rotation from one petal to the next tends to be a fraction that corresponds to two successive Fibonacci numbers. As the growth continues the spiral of petals grows and the fractional growth approaches the golden ratio!!! Why is this? Well, as described in the article Relationships of the Fibonacci Sequence to the Golden Ratio, Pascal's Triangle and other Fibonacci Sequence Relationships, the limit of the ratio of two successive Fibonacci numbers is the golden ratio!
Here is an example of petal growth corresponding to the spiral pattern described. Notice there are 2 petals, then 3, then 5, etc. as the spiral pattern expands producing a (relatively) perfect petal arrangement - no direct overlaps so that light and water can be efficiently distributed. A perfect survival mechanism!! Nature provides! Or should I say, God provides!
It is for this reason that it is very common to observe petals adhering to the Fibonacci numbers (though it should be noted that other plant survival mechanisms do exist). The beauty of the Fibonacci sequence and its relationsip to other beautiful mathematical entities such as the golden ratio is now made evident in the beauty of the natural world! Amazing! Simply Amazing! (at least to me it is)
It gets better, if that's possible. If you were to take a closer look at the sunflower head where the seeds are (again, a botanist I am not), you would observe something quite interesting.
Can you see it? The growth pattern of the seeds are in two distinct spirals going in opposite directions. Each new seed is grown after making a fractional turn from the previous one. For basically the same reasons as described for the petal growth, it is the optimum growth pattern. That is, the rototation from one to the next tends to be a fraction that corresponds to two successive Fibonacci numbers. As the growth continues the spirals grow and the fractional growth approaches the golden ratio!!! (For, again, the limit of the ratio of two successive Fibonacci numbers is the golden ratio.) Wow!
There are many more examples of the Fibonacci sequence, the golden ratio and spirals (Fibonacci, golden or logarithmic) in nature, perhaps too many to reference. I would be remiss, however, if I did not at least point out the pineapple (nothing like a delicious Hawaiian pineapple!). The scales on a pineapple (usually) have three sets of spirals with different numbers on each of the spirals. What is is amazing is that the number of scales on each of its spiral are Fibonacci numbers. Here is an example of a pineapple with 5, 8 and 13.
Not to take away anything from the flower or the fruit world (or should that be the vegetable world? Is a pineapple a fruit or vegetable?? There seems to be some controversy on this?! Egads!!) but one of my favourite examples of the Fibonacci sequence and golden ratio in "nature" is the human body (or biology) itself!
The biology of the human? Yes, human biology, which is not often discussed within this context. I will not attempt to summarize the main points. I will let the paper, "Fibonacci Series, Golden Proportions, and the Human Biology", speak for itself. I will, however, provide the following quotes from the paper to pique your interest to hopefully at least peruse it. The first quote is from the abstract, the second is from the conclusion:
"Despite its wide-spread occurrence and existence, the Fibonacci series and the Rule of Golden Proportions has not been widely documented in the human body. This paper serves to review the observed documentation of the Fibonacci sequence in the human body."
"Fractal geometry lays the foundation to understanding the complexity of the shapes in nature. In the exploration of the origins of life through mathematics, the occurrence of the Golden Ratio (and the) Fibonacci Series ... are observed in several aspects of life on planet earth and within the cosmos."
Amazing!
The opening premise was that mathematical truths, beautiful in-and-of-themselves, can be, and often are, found in the beauty of nature, something I find truly amazing!! It is relatively easy to see God in the beauty and wonders of nature (think of sunsets, sunrises, flowers, or even galaxies). I will concede that for some, it may be harder to see God in mathematics (though hopefully I gave some food for thought in The Beauty of Mathematics) BUT when one , however remotely, can see (or is shown) the connection between the two then the next time a flower is gazed upon (or even a pineapple) one can't help but appreciate the wonder of God even more! Said another way, one might love mathematics for mathematics sake or one might be especially in tune with nature and yet I believe that there is even more appreciation of both, and therefore God, that can be gleaned when considered together.
If one can see God in the beauty of nature and there is the realization of how nature relies on the beauty of mathematics, which is a gift from God, it is logical, then, that one can:
See God in Mathematics!
(There is so much more to write about the "nature" of mathematics and as time goes on will do so. This introduction, however, should hopefully give some insight and be food for thought.)
(The above pictures were from, to the best of my knowledge, the public domain.)