Editor’s note: This is the third in a series on the Fibonacci sequence. You’ll get the most out of it if you read parts one and two first.
“To see a world in a grain of sand. And Heaven in a wild flower. Hold infinity in the palm of your hand. And eternity in an hour.” —William Blake, “Auguries of Innocence”
If the language of the universe is math, then there’s one poem engraved across scientific fields as disparate as botany, ornithology, and meteorology: the Golden Spiral.
To understand the Golden Spiral, we have to start with the Fibonacci numbers. I explained in previous articles that when we arrange the Fibonacci numbers arithmetically, we obtain the Golden Ratio. What happens when we arrange them geometrically instead?
Let’s get to building: We’ll start by taking squares with side lengths the size of the Fibonacci numbers (1, 1, 2, 3, 5, 8, 13…). We will rotate the squares counterclockwise around each other, building each square off of a side formed by the combination of the previous squares. We can follow the process of the Fibonacci sequence in the increasing sizes of the squares.
This “Golden Rectangle” (so named because the ratio between the length and the width of the rectangle approaches The Golden Ratio with each iteration) is a geometric manifestation of the Fibonacci numbers.
Text of address by Billy Graham at the Honor America Day religious service held in Washington July 4, 1970
The Bible says in First Peter 2:17: “Honor all men. Fear God. Honor the king.” And the king referred to was the Roman emperor. Since our nation is a republic and not a monarchy, this Scripture could read, “Honor the nation.”Today, in the capital of the United States, thousands of us have come together to honor America on her 194th birthday.We stand here today within the shadow of three great monuments.That great shaft over there honors George Washington, who led the revolution that obtained our freedom.Not far away is the memorial to Thomas Jefferson, father of the Declaration of Independence, which proclaimed the rights of free men and began the greatest experiment in freedom the world has ever known.Behind us is the memorial honoring Abraham Lincoln, who helped preserve the unity of this country by his courage, faith, and perseverance—and who gave black men hope that they, too, would become first-class citizens.We can listen to no better voices than these men who gave us the dream that has become America. These men represent thousands who worked, prayed, suffered, and died to give us this nation.We are not here today only to honor America; we are come as citizens to renew our dedication and allegiance to the principles and institutions that made her great. Lately our institutions have been under attack: the Supreme Court, the Congress, the presidency, the flag, the home, the educational system, and even the church—but we are here to say with loud voices that in spite of their faults and failures we believe in these institutions!Let the world know today that the vast majority of us still proudly sing: “My country, ’tis of thee, sweet land of liberty.” America needs to sing again! American needs to celebrate again! America needs to wave the flag again! This flag belongs to all Americans—black and white, rich and poor, liberal and conservative, Republican and Democrat.I think there is too much discouragement, despair, and negativism in the nation today. On every hand critics tell us what is wrong with America, where we have failed, and why we are hated. We have listened and watched while a relatively small extremist element, both to the left and to the right in our society, has knocked our courts, desecrated our flag, disrupted our educational system, laughed at our religious heritage, and threatened to burn down our cities—and is now threatening to assassinate our leaders.The overwhelming majority of concerned Americans—white and black, hawks and doves, parents and students, Republicans and Democrats—who hate violence have stood by and viewed all of this with mounting alarm and concern. Today we call upon all Americans to stop this polarization before it is too late—and let’s proudly gather around the flag and all that it stands for.Many people have asked me why I, as a citizen of heaven and a Christian minister, join in honoring any secular state. Jesus said, “Render unto Caesar the things that are Caesar’s.” The Apostle Paul proudly boasted of being a Roman citizen. The Bible says “Honor the nation.” As a Christian, or as a Jew, or as an atheist, each of us has a responsibility to an America that has always stood for liberty, protection, and opportunity.There are many reasons why we honor America today.First, we honor America because she has opened her heart and her doors to the distressed and the persecuted of the world. Millions have crossed our threshold into the fresh air of freedom. I believe that the Bible teaches that God blesses a nation which carries out the words of Jesus, “For I was an hungered, and ye gave me meat: I was thirsty, and ye gave me drink: I was a stranger, and ye took me in.”Secondly, we honor America because she has been the most generous nation in history. We have shared our wealth and our faith with a world in need. When a disaster occurs any place in the world, America is there with help. In famine, in earthquakes, in floods, in distresses of every kind, we pour out billions of dollars every year, even if we have to borrow the money and go in debt.Thirdly, we honor America because she has never hidden her problems and faults. With our freedom of the press and open communications system, we don’t sweep our sins under the rug. If poverty exists, if racial tension exists, if riots occur, the whole world knows about it. Instead of an Iron Curtain we have a picture window. “The whole world watches”—sometimes critically and sometimes with admiration, but nobody can accuse America of trying to hide her problems.Fourthly, we honor America because she is honestly recognizing and is courageously trying to solve her social problems. In order to fulfill the ultimate dream, much remains to be done—but as even our critics abroad are saying, “America is trying.” The men who signed the Declaration of Independence were moved by a magnificent dream. This dream amazed the world 194 years ago. And this dream is rooted in a book we call the Bible. It proclaims freedoms that most people of the world thought were impossible. We are still striving to achieve for all men equally those freedoms bought at such a high price. From the beginning, the dream of freedom and equal opportunity has been a beacon to oppressed peoples all over the world.Let those who claim they want to improve the nation by destroying it join all of us in a new unity and a new dedication by peaceful means to make these dreams come true.Fifthly, we honor America because she defends the right of her citizens to dissent. Dissent is impossible in many countries of the world, whereas constructive dissent is the hallmark of our freedom in America. But when dissent takes violent forms and has no moral purpose, it is no longer dissent but anarchy. We will listen respectfully to those who dissent in accordance with constitutional principles, but we strongly reject violence and the erosion of any of our liberties under the guise of a dissent that promises everything but delivers only chaos. As General Eisenhower once wrote: “We must never confuse honest dissent with disloyal subversion.”Sixthly, we honor America because there is woven into the warp and woof of our nation faith in God. The ethical and moral principles of the Judeo-Christian faith and the God of that tradition are found throughout the Declaration of Independence. Most presidents of the United States have declared their faith in God and have encouraged us to read the Bible. I am encouraged to believe that Americans at this hour are striving to retain their spiritual identity despite the inroads of materialism and the rising tide of permissiveness.On the front page of a Chicago newspaper some time ago there appeared a picture of Betsy Ross sewing the first American flag. Over the picture was the caption, “Time to check our stitches.” Let’s check the stitches of racism that still persist in our country. Let’s check the stitches of poverty that bind some of our countrymen. Let’s check the stitches of foreign policy to be sure that our objectives and goals are in keeping with the America dream. Let’s check the stitches of pollution brought on by technology. Let’s check the stitches of a moral permissiveness that could lead us to decadence. Let’s even check the stitches of freedom to see if our freedom in America has become license. A liberal British writer recently said. “You Americans have become too free until you are no longer free.”What a wonderful thing it would be if we could check these stitches before we celebrate our 200th birthday only six years from now! It could be done.As we look out on modern America today, we must admit there is a vast vacuum of the soul, a void into which millions of Americans are pouring alcohol, drugs, illicit sex, and religious occultism. Our youth are perishing in an orgy of quest. They want the older generation to tell it like it is. They have questions they are asking—the ultimate questions of life: Why am I here? Where did I come from? What is the meaning of my life? The Bible is the only book in the world that fully answers these ultimate questions.I find that there is a certain sense of fear and expected cosmic judgment that hangs over our country at this hour as the divine timepiece ticks away. Thomas Jefferson felt this so deeply that he once wrote: “I tremble for my country when I reflect that God is just, that his justice cannot sleep forever.”The Bible teaches that God will judge any nation that turns its back on him—especially a nation like America, which has been given privileges and opportunities more than any other nation in history.At President Eisenhower’s first inauguration he put his finger on a verse of Scripture, and here is what it says: “If my people, which are called by my name, shall humble themselves, and pray, and seek my face, and turn from their wicked ways; then will I hear from heaven, and will forgive their sin, and will heal their land.”If we don’t check those stitches and check them quickly, the American dream could turn into a nightmare. There are groups of university students throughout America that are beginning to use a sign, the clenched fist with the index finger pointed upward, and they cry, “One way!” They are saying to the world that there is only one way for America, one way for individuals—God’s way. That is the way. Jesus said: “I am the way.”We have raised our hands and our voices in protest over many issues and many causes during the past few years. Why not raise them now to God?During a recent crusade that I was holding at a university, a student came forward when I gave the invitation to receive Christ. As we stood there, he was holding both hands up—one had the two-finger sign and the other had the one-finger only-one-way sign. Within a few minutes his arms were tired and his hands began to wobble. A black clergyman sitting on the platform went down and stood behind him, and held his arms up. What a picture of brother helping brother find God and getting involved in the world’s suffering.Today I call upon all Americans to raise your voices in prayer and dedication to God and to recommit yourselves to the ideals and dreams upon which our country was founded. Let’s dedicate ourselves today to a renewal of faith in God. Let’s dedicate ourselves to building rather than burning.What our forefathers began, we must work to fulfill. Their goal must be our goal. Their vision must be our vision. It is the vision of one nation under God, where men can live together as brothers in freedom.I’m asking all Americans today, especially our young people, to pursue this vision under God, to work for freedom and for peace. It will not be easy. The journey will be hard. The day will be long. And the obstacles will be many.But I remember today a word spoken by Sir Winston Churchill, whose courage and faith and persistence carried his nation through the darkest days of World War II. The headmaster of Harrow, the famous prep school that Churchill had attended as a boy, asked Mr. Churchill to address the students. The headmaster told the young people to bring their pencils and their notebooks to record what Britain’s greatest man of the century would say. The moment they waited for came. The old man stood to his feet and spoke these words: “Never give in! Never give in! Never! Never!”I say to you today, “Pursue the vision, reach the goal, fulfill the American dream—and as you move to do it, never give in! Never give in! Never! Never! Never! Never!”
Now go back to the interior squares (1 unit by 1 unit) and connect the vertices of the squares using successive quarter-circles with radii that have corresponding Fibonacci numbers. The interconnected series of quarter-circles is called the “Golden Spiral.” The Golden Spiral is another geometric manifestation of the Fibonacci numbers, and it’s found in nature just as frequently as the Fibonacci numbers are found in botany.
As I’ve explained in my previous article, the Fibonacci numbers occur in botany when “spiraling” occurs. For example, in a tree that doesn’t have all of its branches pointing out in the same direction, the Fibonacci arrangement allows for maximum exposure to the sun in order to maximize the efficacy of photosynthesis.
The Golden Spiral, however, is an actual spiral—not just a way to count rotations around a fixed center point. The same efficiency found in botany is clearly applied in other areas of nature as well. But it’s applied in ways (and for reasons) that we don’t entirely comprehend. It’s there. But why?
What do a bald eagle, nautilus shell, and a sunflower have in common? It sounds like the setup to a bad “dad joke.” But it’s clear that they’re all built using the same common language, following the path of the Fibonacci numbers as they twist their way towards efficiency.
The curve of eagles’ beaks could easily be much blunter or sharper. But they’re not. They generally follow the Golden Spiral. A nautilus shell could be more tightly wrapped. But it isn’t. It’s a Golden Spiral. Just like with the Fibonacci numbers, there are examples in nature of spiraling effects that don’t follow this Golden rule. But also like Fibonacci numbers, those examples are much harder to find than Golden Spirals. A certain type of spiral is much more prevalent than any other type.
Not just DNA
Even if we don’t fully grasp all the reasons behind the Golden Spiral’s appearance in diverse forms of life, we can at least acknowledge the role that genetics and heredity play. If Mommy Eagle and Daddy Eagle have beaks and heads that follow a Golden Spiral, then we would expect Baby Eagle to have the same. But what about instances where heredity doesn’t apply?
Hurricanes are formed by a confluence of climatological features that replicate themselves frequently at certain times of the year depending on their location on the globe. The details aren’t significant if one is willing to acknowledge that hurricanes don’t have “parents” in the traditional sense. So why should hurricanes share any traits whatsoever?
But they do. As hurricanes rotate, the enormous cloud bands form Golden Spirals. The same spiraling efficiency that allows sunflowers, daisies, and pineapples to survive also allows hurricanes to be formed. But in this case one cannot claim “survival of the fittest” as the hurricane’s motivation. While a hurricane is a natural occurrence, it is not a living being. There are no inherent chemicals dictating the way in which they form.
And spiral galaxies form themselves the same way.
Taste and see
The ancients had another name for the Golden Ratio—they also called it the “Divine Ratio.” Their theology may have been all over the place, but they saw this relationship as evidence of something supernatural. It doesn’t have to be; it’s just as easily evidence of natural selection operating at peak coherence. Those of us who believe in a Creator God, however, can marvel at the complexity shown in this ubiquitous relationship. We have a model of the galaxies provided for us on the head of each sunflower. The nautilus shell tells us of the hurricanes that it witnesses on a yearly basis. Like the Earth’s steady rotation around the sun and the steadiness of a human heartbeat, God often communicates to his children by repeating himself. As humans filled with doubt, how important is it that he repeats himself, over and over again, not only in Scripture but also in his natural revelation?
And how important is it for us to repeat ourselves as well, especially in our praise to him? We long for the day when we can repeat back to him our devotion; one day we will join the chorus, day and night singing “Holy, Holy, Holy.”
