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Friday, August 5th 2016


Phi in Nature, part 1

  • STATE OF EXISTENCE: a good tired

This entry comes to you from deep in the American South, where we are visiting brother Ralph and his wife Lisa for a few days. This has been a totally spur of the moment trip, the kind of thing you can do once you’re retired and the nest is empty. June is continuing to struggle a bit health-wise, and we’re looking forward to her appointment with Dr. B on August 17. Speaking of that date, you may also recall that it’s the day on which we celebrate the birthdays of William Carey, Davy Crockett, and some other really cool people.

Ralph and me

We’re having a great time together here. In case you didn’t see the post on FB, on Thursday night Ralph and I sang and played together again for the first time in quite a few years. There was a time when neither one of us thought we would ever perform again as individuals, let alone as a duo. So, last night's StreetJelly.com was an incredible blessing for both of us. We had a total riot, and I think that many folks in the audience had a good time, too. There’s a video of us rehearsing on my Facebook timeline.

Now I'd like to come back to the topic that I may have treated too cavalierly earlier, namely the appearance of the golden ratio and ϕ in nature.

Phi in Nature

Do I need to define “nature”? I would think not, but it may be helpful if I emphasize a few of its characteristics. Somehow I want to maintain the idea of “nature” as that part of the created order that has not been formed by human beings. Of course human beings are themselves a part of nature in important ways. So, maybe rather than saying “formed,” a better word might be “subdued” or perhaps “governed.” A sunflower is a part of nature; a picture of a sunflower, strictly speaking, is not. Nature includes our bodies (to a large extent anyway) as well animals, plants, germs, diseases, rocks, astronomical objects, subatomic particles, and lots more stuff. Nature manifests signs of being made by an intelligent personal being, though nature seen as a whole is neither personal nor intelligent. It contains personal intelligent beings, but nature per se cannot think, plan, wish, imagine, invent, like or dislike. For example, contrary to the popular saying, nature does not “abhor” a vacuum. The regularities that we see in nature lead us to conclude that a vacuum will be filled with air or some other gas as soon as soon as access is provided. However, nature has no opinion on this issue—nor on any other, for that matter.

A recent article in the Smithsonian (Danny Lewis, “Why the Turtle Grew a Shell—It’s More than Safety”) introduces us to some recently discovered ancient turtle fossils. These newly available specimens shed fresh light on the reason why turtles may have developed shells. It was not for the sake of defense against other animals, as we might think, but as an aid for them to burrow underground in order to survive the inhospitably hot climate of South Africa, their abode at the time. The article closes on a cautious note:

"While more research needs to be done to determine whether the earliest turtles known to have shells were diggers themselves, it just goes to show how adaptable nature can be."

Well, I’m afraid I need to add one more item to my list above: nature cannot be “adaptable” either. The term implies foresight and intentional reactions; but the impersonal forces of nature cannot qualify as being “adaptable” in a meaningful sense. You can say that a certain life form became adapted thanks to the potentials found in nature or that some species have turned out to be more adaptable than others in hindsight. However, any kind of intentionality on the part of “nature” or the species in question is overstepping the boundaries of evolutionary biology—unless you accept that there is a Creator and Sustainer who has been supervising the entire process. If you see something “smart” in nature, it’s not nature itself that’s smart because nature itself has no mind. The smartness points us back to the one who made nature.

Having said that, there are some wonderful examples of apparent smartness in that collection of things in nature, and, unsurprisingly, some of them are classified along that line due to the presence of the golden ratio.

How can we know if some item or phenomenon exhibits ϕ?

1. For some things we can take measurements to see if there are lines that are related to each other in the golden ratio. This is an easy method for human architecture and pieces of art, though we also have a large amount of room for fudging there. It’s also possible to do so for some natural items, such as crystals. For the most part, though, it’s not a practical way to proceed.

2. Bring on the Fibonacci numbers. It appears that some things come only in numbers of the Fibonacci series. Furthermore, if we can find relationships based on the Fibonacci numbers in nature, we can avail ourselves of the fact that they do converge to ϕ, and so the series in question can be said to be in keeping with the golden ratio.

3. Measure the angles. This approach is particularly applicable to anything that appears in spirals. Remember that we said that logarithmic spirals, of which the golden spiral is a special case, distinguish themselves from Archimedean spirals by the fact that their increase in length maintains a consistent ratio, so that the spiral moves away from its point of origin at a much faster rate. In that case, any line drawn from any segment of the origin of the spiral should manifest the same angle.

log spiral

Thus, in the graph above, the angle at which any of the three lines intersect the spiral is the same.  Livio helps us calculate the value of that angle for the golden spiral.

360° ÷ ϕ ≈ 222.5°

Because this result crosses the 180° line, we should come to it from the other direction.

360° - 222.5° = 137.5°

This is the angle at which all the lines cut the spiral whenever they intersect it. If the origin of the spiral converges to the center of the graph (i.e. point 0,0), then the x and y axes will also intersect with the spiral at that angle.

Please keep in mind that a golden spiral is a logarithmic spiral, but not every logarithmic spiral is a golden one. Nevertheless,  Livio makes the case that golden spirals are very likely the ones in nature. Apparently, in whatever way living beings expend energy in assuming formations, the golden spiral based on phi is the least demanding. So, when we see a logarithmic spiral in nature, the poor chambered nautilus notwithstanding, there’s a good chance it may also be a golden spiral.

Let me try to explain the above point a little more. Livio characterizes phi as the “most irrational” among the many well-known irrational numbers. Let’s make a quick comparison between ϕ and π. In some ways, ϕ is more approachable than π; in others it isn’t.

, :







Can it be derived from an algebraic formula?

No. π is “transcendental.”


½ (1+√5) = ϕ

Can it  be approximated by a fairly simple rational fraction?

Yes. 22over7



Calculating ϕ involves referring to another irrational number, namely the square root of 5. The approximate value of one irrational number is computed by including another irrational number. Consequently, if any items are arranged in a way that makes use of phi, no two items are ever going to be sharing one or the other coordinates in space.

[A similar thing applies to time. On occasion one hears the idea that, given an infinite amount of time, it becomes, not only probable, but even certain that every event of the present or past will recur at some point in the future. But that’s not true, as can easily be demonstrated with the following mind experiment conceived by Georg Simmel (1858-1918) and reported by Walter Kaufmann (1921-1980) in his magisterial book, Nietzsche: Philosopher, Psychologist, Antichrist. (Fourth Edition, Princeton University Press, 1974, 327). It uses pi, but phi would work just as well in light of the above considerations. Think of three disks rotating at different speeds. One completes a full rotation every minute. The second one does so every 2 minutes (thus, running at half the speed). The third one has a cycle of 1/π minutes.

eternal wheels

The disks start at a point we can call zero (maybe the twelve on a clock face). That particular arrangement will never occur again. Similar events, though not formally devised by anyone, occur frequently since pi as well as other irrational numbers come up over and over again in everyday life.]

So, let’s look at a particular example now and a few others next time.

Flower petals and plants

1.  It is an observable fact that the number of petals on many species of flowers come in Fibonacci numbers. Once again, I have noticed that some of the same pictures appear website to website, and I cannot credit the person or organization that created them in the first place. Here is a photo with a set of pictures that demonstrate the “preference” of many flowers to take the numbers of their petals right out of the Fibonacci series.

Fibonacci Petals

2. Even more interesting is the phenomenon of how the petals of many flowers are arranged. Petals and leaves frequently form a logarithmic spiral around their center. Here's why this pattern is a valuable asset for a plant. 


Imagine a flower with several rings of petals, and it takes exactly four petals to make up one ring. So, we start out with a nice flower with four petals surrounding the center.


Four Petals

But we stipulated a flower with several rings of petals, and the second ring also follows the rule of four to a ring.

Eight Petals

Clearly, this isn’t going to work too well because the second layer will just cover up the first. And if we continue that pattern, the third layer will cover up the first and second.

Twelve Petals

The flower is in trouble. How might it work better?

The flower would be far more functional if the relationship among its petals was not governed by such an easy integer as four. But, if we stay with straightforward rational pattern, the same thing will happen. Increasing the numbers of leaves by decreasing the ratio with larger integers, we would just get more layers of petals or leaves pancaked together.

Twenty-four Petals

We could play out this scenario further, but you already know where I’m heading with this. In many plants, the petals are arranged according to the golden angle of ca. 137.5°, and thus form a golden spiral. Thereby exposure to sun and rain, the attraction of insects for pollination, or whatever else the plant needs to thrive is maximized. Rather than my pitiful drawing above, we can see a beautiful rose that incorporates a rather complex as well as beneficial mathematical pattern in its structure. 


Is this amazing? I think so. Does it point to the idea that a very knowledgeable and powerful being must have overseen the creation and further adaptation of flowers? I should think so.

There is a point that someone could make and unreflectively think that it counted against the above idea. It could be, as Livio hypothesizes, that the Fibonacci sequence and formations according to ϕ represent the least expenditure of molecular energy in the formation and continuation of plants. That’s a good idea, and I already mentioned it above, and—in fact—it is a great idea. It so great that we’ve only pushed the issue one step further back. It only increases the wonder of the phenomenon, and, thus, leads us to an even stronger conclusion that the principle was installed by a supreme creator.

More examples next time.

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