Math Circles 5 -- The Wonders of the Fibonacci Sequence
Learning to see beauty with new eyes.
James Nickel’s Math Circle 5 introduces readers to the work of Leonardo of Pisa and the wonders of the Fibonacci Sequence.
1. What are you pointing to, Leonardo?
EN: Elton Trueblood suggests that if the world is created by an Infinite Mind, its beauty should not surprise us. When did the elegance of mathematical patterns—like the Fibonacci sequence—first strike you as pointing beyond chance to something deeper?
JDN: I noticed this sequence when I started teaching high school mathematics in Australia (1982). To prepare for this assignment and during my teaching, I bought many resources, all sparking interest:
Howard Eves, An Introduction to the History of Mathematics (only a few pages)
A textbook by Harold R. Jacobs, Mathematics: A Human Endeavor (one lesson)
Matila Ghyka, The Geometry of Art and Life (one chapter)
Theodore Andrea Cook, The Curves of Life (several chapters relating the sequence to spirals and growth)
Dan Pedoe, Geometry and the Visual Arts (some discussion of the Fibonacci sequence in art and architecture)
William M. Ivins, Jr. Art & Geometry: A Study in Space Intuitions (general introduction to the relationship of intuition to geometric forms)
Leroy C. Dalton, Algebra in the Real World (detailed study in several enrichment lessons)
Herbert E. Huntley, The Divine Proportion: A Study in Mathematical Beauty (general introduction with plenty of examples)
Rochelle Newman & Martha Boles, Universal Patterns—The Golden Relationship: Art, Math & Nature (a tour de force of geometric connections in art and creation)
The animated featurette Donald in MathMagic Land (1959), a Walt Disney production (the Fibonacci Sequence, the Golden Rectangle, and the Golden Ratio get plenty of colorful and captivating exposure)
My high school course in Geometry never touched on any of these fascinating relationships. None of my university undergraduate courses in mathematics exposed me to this pattern. The “History of Mathematics” course may have mentioned the Fibonacci sequence, but I did not enroll in it.
It was these resources that I was studying in the 1980s-1990s that led me to see that the Italian Leonardo Bonacci (ca. 1170–ca. 1240–50), aka Leonardo Fibonacci, was pointing to something deeper, an opulent beauty revealed in the interplay of God’s creation, the art and architecture created by humans—the imago dei, and geometric relationships.
There was a nadir of connectivity in Euclid’s Elements (ca. 300 BC) although Greek architects made use of the Golden Ratio in their buildings. For Euclid, Geometry was solely and only a matter of deductive analysis, i.e., “Euclid alone has looked on beauty bare.” [from a poem by Edna St. Vincent Millay (1892-1950)]. That deductively oriented priority, that beauty denuded, was how I received this subject in both high school and university, a segue into Question 2!
2. Learning to See Beauty
EN: The way you teach these things (Fibonacci Sequence, Golden Ratio, Pi) really makes it come alive. My own recollection of learning about the Fibonacci numbers in school was, “Oh, that’s interesting.” It was never connected to any broader implications. What do you think is the most important thing a teacher can do to help students move from simply learning math… to actually seeing the beauty and meaning behind it?”
JDN: To grasp that the most important thing a teacher can do to help students move from simply learning math to actually seeing beauty and meaning behind it, we start with two instructive observations.
First, when I visited New Zealand in 1992 as a keynote speaker at an education conference in Masterton (in the South Island), the headmaster of a primary school in Paraparaumu gave me a unit from her Primary Mathematics course (1967, 2-1). I quote from the introduction:
“When we look into the wonderful world of Nature, we look into the birthplace of all man’s mathematics—his surroundings. Mathematics, the science of numbers, shapes, and space, began when men first started to record relationships between things in the world around them. Although we do not know just when this was, we can assume that it began thousands of years ago.
In time, these ideas lost their association with Nature, and became the bases of the systems of symbols and signs with which we are so familiar.
However, the relationship of mathematics to Nature is very interesting, and in this Unit we are going to take some time to re-associate the mathematics of shapes with the Natural world.
At the same time we shall look more deeply into some important topics in order to prepare you for further work in mathematics and science.”
Second, in the preface to the revised second edition of Universal Patterns (1992, p. x.), authors Newman and Boles state:
“This world is of a single piece; yet, we invent nets to trap it for our inspection. Then we mistake our nets for the reality of the piece. In these nets we catch the fishes of the intellect but the sea of wholeness forever eludes our grasp. So, we forget our original intent and then mistake the nets for the sea.”
We start “hands-on,” eyes open, with our vision trained to look for patterns that surround us in the universe of the Father, Son, and Spirit’s making. In the early years, we engage students in recognizing geometric shapes in flowers, pinecones, sand dollars, crystalline formations, and the multifaceted spiral revelations, from the chambered nautilus to galactic swirls.
We use mathematical signs and symbols to abstract those patterns, while we always remind ourselves that these abstractions are grounded in objective realism, in the structure of the outside world, a universe held together and sustained by the Incarnate Son of the Father in the communion of the Holy Spirit.
Beauty confronts us as we penetratingly look at the multi-faceted nature of creational form. This seeing is FORM-1 beauty, and it serves to spark our intuitive capabilities.
FORM-2 beauty is our abstractions, the wonder that we can catch the Newman-Boles “fish” of creation using the “nets” of our intellectual making, the marvel that there is a creational connection between the way our minds work and the structure of the universe. Through the syntax of mathematics arising from FORM-2 beauty, FORM-3 beauty is the discovery of an abundance of interconnections, not only within the internal structure of mathematics but also with other branches of creation. The power of mathematics lies in these connections. That the irrational number represented by the Greek letter , the ratio of the circumference to the diameter of a circle, appears in the normal curve of probability theory, the analysis of spring and bob motion, Einstein’s field equations for relativity, and the Schrödinger equation that governs the wave function of a non-relativistic quantum-mechanical system, just to name a few, should stun and awe the beholder. This power becomes revelatory of stunning beauty.
Beauty is the revelation of creational design, the unveiling of the shared wisdom of the creative play (i.e., “Let’s do this!”) of Father, Son, and Holy Spirit (Genesis 1, John 1).
Beauty is the interconnection of creation’s proximate form with its function, a mapping that declares, for the eye trained to see it, the ultimate beauty of the interpenetrating interpersonal glory of the Creator and Sustainer of all things, a segue into Question 3!
3. The Ethos of Kepler
EN: You close your essay with Kepler’s almost hymn-like praise of God, where scientific discovery becomes an act of worship. Do you see the study of mathematics—especially patterns like the Fibonacci sequence—as something that can lead a person not just to knowledge, but to awe or even worship?”
JDN: In December 1981, while packing and organizing our things for my young family’s move to Australia, we were living at my parent’s home in Dinuba, California. For the first eleven months of 1981, I was both working at a funeral home in Ventura, California, and picking up whatever I could afford related to mathematics at local used bookshops.
Through my research, I came across a recommendation to read the publications of mathematics historian Morris Kline (1908-1992). Dinuba had a relatively new library, which was built in 1975 after its Dale Carnegie Foundation-funded building (built in 1916 with $8000 in Carnegie funds) was demolished, so I went to it to see if Kline was in the analog card catalog. I found his 1959 publication Mathematics and the Physical World, checked it out and began a two-month intensive study, copiously filling a notebook. In it (p. 119), Kline quoted Johannes Kepler (1571-1630):
“The wisdom of the Lord is infinite; so also are His glory and His power. Ye heavens, sing His praises! Sun, moon, and planets glorify Him in your ineffable language! Celestial harmonies, all ye who comprehend His marvelous works praise Him. And thou, my soul, praise thy Creator! It is by Him and in Him that all exists. That which we know best is comprised in Him, as well as in our vain science. To Him be praise, honor, and glory throughout eternity.” (Kline’s reading of his translation of Harmonies of the World [1619])
The responsory phrase “I was flabbergasted” was an understatement. My formal education had not taught me that!
I kept Kepler in mind after arriving in Australia in March 1982. Early in 1984, I managed a trip to the University of Adelaide Barr Smith Library to do two weeks of research. Among other goals, I wanted to read the context of Kepler’s quote. As I read him, I was able in part to feel with him as he was “being carried away and possessed by an unutterable rapture over the divine spectacle of the heavenly harmony.” (Max Caspar, Kepler, p. 267.)
Kepler’s psalm of praise to God was one among several. The context was his work on his planetary laws; an interpenetration of intuitive insight with mathematical equations derived from astronomical data obtained from an observatory built by Tycho Brahe (1546-1601). Kepler’s equations were the Newman-Boles “nets.” Intuitive insight came from indwelling the data, the reality of the Newman-Boles “sea of wholeness.” To Kepler’s credit, he did not mistake the nets for the reality of that ocean. That reality was the sustaining power of God, or as Colossians 1:15 states, the incarnate Son of the Father holding together all things of creation, visible and invisible. His psalm of praise flowed from the nets he created to the ocean source, i.e., the Author of creation’s wonder and beauty—Father, Son, and Holy Spirit.
Kepler wrote in his calendar for the year 1604:
“I may say with truth that whenever I consider in my thoughts the beautiful order, how one thing issues out of and is derived from another, then it is as though I had read a divine text, written into the world itself, not with letters but rather with essential objects, saying: Man, stretch thy reason hither, so that thou mayest comprehend these things.” (Max Caspar, Kepler, p. 11.)
Indwelling, i.e., stretching one’s reason hither, into the Fibonacci Sequence as it interpenetrates an ocean-full of creational objects, will generate the same wonder of praise.
Read: The Fibonacci Sequence in God’s Creation.
Further investigative links:
Recursion and the Fibonacci Sequence
https://biblicalchristianworldview.net/Powerpoint/keplerRavishingDelight.pps
https://biblicalchristianworldview.net/documents/IncommensuratesFibonacci.pdf