Mathematics Is Beautiful!!!
A bold statement.
Putting aside any negative feelings towards mathematics you may have, you may be wondering how the statement, mathematics is beautiful, can be made. Isn't beauty purely subjective in that what is beautiful to me may or may not be beautiful to you?
Perhaps.
From our perspective beauty may be subjective but I submit for consideration that from God's perspective, mathematics is beautiful! Indeed, all is beautiful! Consider the Story of Creation in Genesis, Chapter 1:
3 The God said" Let there be light, and there was light.
4 God saw that the light was good
10 God called the dry land "earth," and the basin of water he called "sea." God saw that it was good.
16 God made the two great lights, the greater one to govern the day, and the lesser one to govern the night, and the stars.
17 God set them in the dome of the sky, to illuminate the earth,
18 to govern the day and the night, and to separate the light from the darkness. God saw that it was good.
21 God created the great sea monsters and all kinds of crawling living creatures with which the water teems, and all kinds of winged birds. God saw that it was good.
25 God made every kind of wild animal, every kind of tame animal, and every kind of thing that crawls on the ground. God saw that it was good.
27 God created mankind in his imagel in the image of God he created them; male and female he created them.
...
31 God looked at everything he had made and found it very good.
(Note: All scripture passages taken from the New American Bible, revised edition 2010. Emphasis added was mine)
God created everything and God was quite pleased and FOUND IT VERY GOOD. Everything created by God and given to us as gifts is beautiful to God! And since mathematics is a gift given to us, mathematics is beautiful! Our ability to recognize the beauty in mathematics, I would think, then, is pleasing to God. Indeed, whenever we see something beautiful in nature (think of a beautiful sunset) our minds (should) turn to God for the recognition of beauty gives glory and honor to the creator of the beauty (God) and lifts our hearts and minds to God. The beauty in mathematics is (or should be) no different!
But what exactly is beautiful about mathematics? Well, a proof is most certainly beautiful! The combining of thoughts, symbols and logic to show an absolute truth in an elegant way is quite beautiful! An equation can be quite beautiful (think of the quadratic formula!). A series of equations that represent fundamental laws of nature are quite beautiful (think Newton's Equations of Motion). The list can go on, but the point is that mathematics is beautiful. By definition, it has to be.
With that in mind, whenever I derive a mathematical truth for my students, I always conclude with the statement, "Isn't that wonderful?!" Or, "Isn't that beautiful?!" A rhetorical question perhaps, but yet, it is true! Mathematics is beautiful!
The beauty of mathematics, however, goes beyond elegant proofs or equations. It is even more apparent when a relationship between two seemingly unrelated ideas, each beautiful in-and-of-themselves, have a relationship that is totally unexpected. Among many examples, the Fibonacci sequence and the Golden Ratio will be used to illustrate this concept. Only cursory information about each is presented here for the focus is on the beauty of the unexpected relationship between the two. Please refer to the series of articles on the mathematics of each and their relationships I wrote. The articles can be found here at Fibonacci Sequence / Golden Ratio. Be sure to check out the "Fun Facts" section in each. Fibonachos anyone???
First consider 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 and also information about Fibonacci himself.) As a mathematical sequence it is indeed quite beautiful and appears in unexpected places.
Next, consider the Golden Ratio, the special number (phi = 1.618) obtained when the ratio of the longer length of a line segment to the short length segment equals the ratio of the total line length to the longer segment (a diagram of the golden ratio is included at the end of this article; also refer to the article Description, Mathematics and Other Fun Facts of the Golden Ratio I wrote for more details). Along with the golden ratio there is the golden rectange, a rectangle whose side lengths are in proportion to the golden ratio (a diagram of the golden rectangle is included at the end of this article). The golden rectangle is (arguably) thought to be the most aesthetically pleasing of rectangles and has been used in architectural designs throughout history and even to this day. The ancient Greeks are said to have known of the golden ratio and the Golden Rectangle and apparently used each in their architectual designs. The Parthenon (picture included at the end of this article), for example, seems to (approximately) fit within a golden rectangle. There is so much more which can be said about the golden ratio (indeed, books have been written about it ) and its amazing properties, such as it being considered the most irrational number of the irrational numbers. For these reasons (and many others) the golden ratio is often referred to as the Divine Proportion.
(As an aside and just for fun, I created two informal surveys, Survey #1 and Survey #2, to test the idea that the golden rectangles are the most aesthetically pleasing. Should you want to participate, and I hope you do, please take the surveys and future posts will report the results.)
The interesting thing, however, relative to this discussion is the unexpected relationship between the Fibonacci sequence and the golden ratio. One example is that the limit of the ratio of two consecutive Fibonacci sequence terms is the golden ratio!!! Another is that the Fibonacci sequence and the golden ratio occurs abundantly in nature (see The "Nature" of Mathematics for a look at how these two beautiful mathematical concepts are prevalent in nature). It is certainly quite interesting that two seemingly disjoint, yet beautiful, ideas are so closely related!
It gets better. Not only is there a relationship between the Fibonacci sequence and the golden ratio, but consider that the Fibonacci sequence can also be found in, of all places, Pascal's Triangle (another mathematical beauty)! Amazing! Simply amazing!
(For a complete description of the relationships of the Fibonacci sequence refer to the Relationships of the Fibonacci Sequence to the Golden Ratio, Pascal's Triangle and other Fibonacci Sequence Relationships article I wrote).
Are these relationships mere coincidences? Some might say, yes. But I think not!!! I have faith that they are not! Are they absolutely beautiful?? I think yes!!! How about you? Something to ponder!
Returning to the opening statement, yes(!) mathematics is beautiful! Of course it is, for is a gift from God. Yet we don't have to be mathematicians with a Ph.D. to fully appreciate the beauty of mathematics! Just knowing that these are revealed truths, beautiful truths, is enough. At least it is for me.
What is also beautiful is that we don't have to be geniuses to apply these truths. By learning mathematics sufficient enough to apply that knowledge in whatever our vocations might be can only help to build up the body of Christ. Now, that is beautiful! Just like a beautiful equation, using our talents, whatever they are, to serve God warrants the statement "Isn't that beautiful?!"
So, yes, it is true that mathematics is beautiful and the recognition of such gives glory and honor to God! It is logical, then, that one can:
See God in Mathematics!
(There is so much more to write about the beauty of mathematics and as time goes on will do so. This introduction, however, should hopefully give some insight and be food for thought.)
Golden Ratio
Golden Rectangle
Parthenon
(The above picture acquired from, to the best of my knowledge, the public domain.)