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Wednesday, August 24th 2016


Phi and the Golden Ratio: Conclusion


We just went through a tornado warning about as close to us as I ever want to get. Thankfully the bad part of the storm decided to move about a mile north of us.


If you would like to get caught up on all of the previous installments of this series, please do so by going to the website on which I have been collecting it: PHI—Let’s Get It Right. I’m putting in corrections, and pretty soon navigation aids, there, rather than backtracking on the original blog entries. Some of the sections will be totally rearranged so that the sequence makes sense and some duplications can be eliminated.

This has been a long mega-series, and no one is as surprised by its length as your fatigued bloggist. The initial motivation was a pretty modest one. As I stated right at the outset of this series, my point has been to introduce some cautions into how much use a Christian apologist may make legitimately of the golden ratio, the Fibonacci numbers, and phi (ϕ) in an argument for the reality of God.  It has not been my goal to give a full description of the history of phi or of all of its occurrences, let alone the Fibonacci series. Still, I wound up adding a whole lot more material than I had originally intended since I realized in the process that the need for analysis was greater than I had anticipated, and that any number of items would not make sense without providing at least some background.

Having occupied myself with these things for several months now, I’m thinking that the phenomena connected to phi or the golden ratio are best combined with other features of the world for the sake of an argument for God’s existence. Still, the fact that phi shows up in so many different and apparently unrelated places surely is remarkable. As you have read, I am even more amazed by its intrinsic properties as they emerge in geometry and algebraic equations than in their occurrence in nature. Consequently, as a believer in God, I do see his handiwork as an all-knowing and all-powerful creator in this aspect of the world he made. When I say “world,” I’m referring to both the physical world of matter and the world of numbers.

However, it is in the much broader context of mathematical and physical reality that the evidential side of the subject matter becomes strong.  Phi is a number; in fact, it is a unique and truly remarkable number. So is pi, though in different respects than phi, and so is e, a number whose characteristics would take too long to explain here. Let us not forget 0 (the “additive identity constant”) with her many characteristics, or 1 (the “multiplicative constant”) to whom we did not pay much attention in this discussion. There is i, the square root of -1. There are so many kinds of numbers, as represented by their sets from ℕ through ℤ, ℚ, ℝ, to ℂ, as well as special sub-categories, such as ℙ. When we come to terms with the complexity of the universe, while simultaneously recognizing how mathematically fine-tuned it is, it becomes impossible for me to say, “It just happened."


The evidence for God is there. Unfortunately, some people confuse the creator with his creation and think of the created order as God. However, nothing can be the cause of its own existence and, if there are indications of intentional regularity in the universe, they reflect on the one who made it. When I think of various non-theistic theories of the origin of the universe, I can’t help but wonder how anyone can be satisfied with them. For example, take Michio Kaku’s claim that our present universe is the result of a collision of two previous universes. This idea not only strikes me personally as bizarre (which by itself does not count as evidence against it), but leaves all of us hanging with the question of what caused the existence of those previous universes. Another set of colliding universes? And another one before them? Are we back to turtles supported by turtles?

Most important for this series, though, has been my intention to encourage Christians to test the evidence before using it. I have been quite critical of the unbridled way in which many people have attempted to “find” the golden ratio in various parts of nature, architecture, and art. For many of them, the need to find phi in virtually every beautiful building or painting, seems to be driven by a mystical outlook that directly links beauty and the “golden number.” As far as I can tell, a direct causal relationship between phi and the perception of beauty has neither been completely undermined nor proven. However, there is definitely no necessary relationship between beautiful objects and numerical patterns embodied in them. In many cases where the Fibonacci numbers appear, e.g., among flower petals, their purpose appears to be practical rather than aesthetic. This assessment does not take away from its wonder or from its contribution to the evidence for a Creator, but it does extend our understanding of phi beyond creating visual appeal.

Phi is a number. More specifically, it is a number that arose out of the relationship between various geometric lines found in the pentagon. It belongs to geometry, analysis, and number theory. It is not something that we merely run across in our day-to-day existence.

Consider the example of the bones of a finger, which we mentioned earlier under the heading of “Phi in Nature, part 3.” Why are we interested in their relative lengths? If, indeed, they do measure 2, 3, 5, and 8 units in length, that discovery per se doesn’t tell us much. 8 divided by 5 equals exactly 8/5 a nicely rational fraction of two integers, which can also be expressed as 1 3/5 or 1.6 with no further remainder. There doesn’t seem to be anything special about it in isolation. If we were ignorant of any further implications, such as the Fibonacci numbers or phi, we might write down the results of our measurements, but we would probably not excitedly post them on a website to share with the world.

The reason why we are intrigued by the lengths of the finger bones is because we already know that these numbers have a larger significance: they are a part of the Fibonacci series, which converges to phi. We fuss over these numbers because we anticipate what’s coming. When we do nothing but measure the proportion between the proximate phalanx and the metacarpal, there is a huge difference between the result we get,1.6, and the number we would eventually reach if we could go on forever with the measurements, namely ϕ. We do not see ϕ exemplified on the x-ray, and our limited data would not allow us to compute ϕ. However, we have learned previously that the Fibonacci series will converge to phi (1.618033…).  Thus, we find the Fibonacci numbers in nature and give them significance based on the Fibonacci series, but the series is actually a mathematical entity. It is the outworking of a recursive equation,

Fn = Fn-1 + Fn-2

with start-up values F1=1 and F2=1,

not a principle emerging from the observation of human digits or fecund lagomorphs.

So, what am I saying? Nothing that I haven’t said many, many times, but ever more frequently over the last few years. Apologetics is not about memorizing answers and arguments. We can leave that approach to the internet atheists and their irrepressible need for rabbit trailing. It’s about finding answers to relevant questions as a part of the total project of demonstrating the truth of Christianity, both in evangelism and in dealing with our own doubts. And in that context, memorized answers (“If they say this, then you say that”) without a personal understanding of both the question and the answer, are of very little help, if any.

I’m afraid that phi is a case in point. It’s easy to list it as one of the marvelous aspects of the world that points to the Creator. But I need to ask, to what extent can you substantiate the make-up and meaning of the golden ratio? Can you discriminate between what is science and what is pseudoscience? What is nature and what is numerology? What is truth and what is fantasy or deception? I’m hoping that this series has helped create a little more understanding and, perhaps, even a little bit of interest to delve further into this topic or the role of math in apologetics.

As you have seen, I have built bridges by means of hyperlinks into some of the blog entries and even more into the combined site so that you can jump over some of the meanest-looking equations and calculations. It’s a great feeling when you learn to work through an equation and actually get the result you’re supposed to, but doing the math in all of its fine points if not obligatory. Nonetheless, I do want to say that anyone making use of the remarkable number phi should have some basic understanding of:

1. the difference between phi and the Fibonacci series;

2. the fact that phi is not derived from the Fibonacci series, but that the Fibonacci series converges to phi;

3. the nature of the proportion (whole to large segment = large segment to small segment) that constitutes the golden ratio;

4. the fact that phi originated in geometry;

5. the basic nature of a golden triangle, rectangle, and spiral;

6. some of the genuine occurrences of the Fibonacci numbers in nature, art, and architecture;

7. the lamentable fact that people do fudge the data in their apparent eagerness to find the golden ratio everywhere;

8. our obligation as Christians to be honest and forthright in our learning and speaking. It’s not right to use bad information, even if the person to who you are talking believes it. (Nor, for that matter, is it okay to make up something on the spur of the moment if you don’t know the answer, not even if you think you just experienced a moment of revelatory inspiration [or inspired revelation].)

9. my concern that in order to become a good Christian apologist, one should study fields of knowledge as a whole rather than just picking supposedly good apologetic tidbits out of them.

10. the happiness that results from studying and learning!


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