Patterns, Fibonacci and Our Awesome Universe

While watching an episode of Rod Serling's Twilight Zone, "Night Call" (Episode 139 -- February 7, 1964) there is a scene where the old woman has been assisted into a wheelchair. The camera angle is from the viewpoint of an adult looking down to the woman. Across her lap is a knit afghan with a zigzag pattern similar to the kind my grandmother used to make, and for just a moment the camera lingers on the pattern.

One of the thoughts I had in that moment: what would an afghan look like if instead of being knit into a pattern, the colors and knitting were totally random? Isn't it the pattern or design that gives the afghan its interest?

In 2008 I wrote about color as a facet of making or appreciating art. Design and pattern could be added to the list of things which can make a drawing or painting interesting.

Nature is full of patterns, from atomic structure to the design of galaxies... from the incredible Fibonacci sequence to the rhythm of waves... from the phenomenon of day and night to the miracle of a heartbeat...

In the Renaissance, artists did not see beauty as accidental. They believed it could be discovered, studied, and deliberately constructed. Pattern—whether in proportion, repetition, or geometry—was one of their primary tools for doing so.

Figures like Leonardo da Vinci and Leon Battista Alberti were not only artists but thinkers who sought underlying order in the world. They were deeply influenced by classical ideas of harmony, especially the belief that mathematics revealed the structure of reality. This conviction shaped both painting and architecture.

One of the most intriguing patterns associated with Renaissance art is the Fibonacci sequence and its related “Golden Ratio.” While not always applied consciously in a strict numerical sense, its proportions—roughly 1:1.618—appear repeatedly in compositions. Painters arranged figures, horizons, and focal points along these invisible lines, creating balance that feels natural rather than forced. The eye is gently guided, not commanded.

In architecture, this pursuit of proportion becomes even more explicit. Alberti and others designed buildings where height, width, and spatial divisions followed harmonious ratios. The result is not merely structural integrity but a sense of calm coherence—spaces that feel “right” even if the observer cannot explain why.

What makes this especially compelling is that these patterns mirror those found in nature: the spiral of a shell, the arrangement of leaves, the unfolding of galaxies. Renaissance artists believed they were not imposing order but participating in it—echoing a design already embedded in creation.

In this sense, pattern is not decoration. It is revelation. It is the quiet framework beneath the visible surface, the hidden scaffolding that gives a work of art its unity and power. And once you begin to see it, it is hard to unsee—the world itself starts to look like a carefully composed canvas.

Elton Trueblood once suggested that if the world is the product of an Infinite Mind, its beauty should not surprise us. That idea becomes more than philosophical speculation when you begin to trace patterns—not just in art or music, but in mathematics itself.

Educator/mathematician James Nickel describes how for him this realization did not come through formal schooling, but almost by accident. While preparing to teach high school mathematics in the early 1980s, he began exploring resources that connected numbers to the real world—books on geometry in art, the growth patterns of nature, and the Fibonacci sequence. What struck him was not merely the elegance of the numbers, but their recurrence—in shells, architecture, spirals, and proportions that seemed to echo across creation.

What had been presented in school as isolated facts suddenly revealed itself as something cohesive, even luminous. Fibonacci—Leonardo of Pisa—was no longer just a historical figure attached to a number sequence, but a doorway into what Nickel called “an opulent beauty,” a harmony linking human creativity with the structure of the natural world, and ultimately pointing the Creator God of All, all nature declaring His glory.

Equally striking was what had been missed. Traditional math education, with its emphasis on deduction and abstraction, often stripped away this sense of wonder. Geometry became a system of proofs rather than a language describing reality. The “pattern” was still there, but the meaning had been muted.

And so the rediscovery of pattern becomes, in a sense, a recovery of vision—learning again to see what was always present.

Check out this interview with author/mathematician James Nickel on The Wonders of the Fibonacci Sequence.

Poetry Corner: The Geometry of Innocence

The Geometry of Innocence

The geometry of innocence,
A perfect shape so pure and bright,
A circle, free of angles and bounds,
A symbol of honesty and light.

But as we grow and learn and age,

The edges start to dull and fray,

The lines become less clear,

And innocence slips away.

We're taught to fear and mistrust,

To guard ourselves and lock the door,

The circle becomes a cage,

Trapped forevermore.

But oh, to hold onto that pure,

Uncorrupted, innocent form,

To live in truth and honesty,

Would be a life truly reborn.

So let us strive to keep that flame,

To hold onto the geometry of our youth,

For in innocence, we find a peace,

That we can never lose or uncouth.

* * * * 

The origin of this post is a line from Dylan's "Tombstone Blues." The poem and image here have both been generated by an AI. The poem was created by ChatGPT. I chose not to edit this, to let it be. The illustration was generated by DreamAI. 

Normally I'd share my thoughts at this point but for now I am interested in yours. Feedback welcome in the comments. 

For some reason the phrase is mentally stimulating. I've reflected on it many times over the years. Here's a blog post I wrote about the Geometry of Innocence in March 2012. 

Meantime, life goes on. Enjoy the sun.

* * * *

The Alhambra: An Introduction for North American Readers

I recently received an email regarding a new exhibition, titled Symmetry's Portal, by Lisbon-born artist Margardia Sardinha. The show was exhibited at Centro Cultural Jaime Lobo e Silva in Ericeira and opening this past week at Lisbon's Ismaili Centre, which belongs to the Aga Khan Foundation.

The press release begins with this enticing description:

Symmetry 's Portal is an exhibition organized by the artist Margarida Sardinha that has, over a lengthy period of time, been extensively focused on the concepts of symmetry and optical illusion. The exhibition expands geometrically a wide photographic survey of the Alhambra in Granada documented by the artist. This photographic assemblage is used by Margarida Sardinha to deconstruct symmetry and to generate illusory semblances. Symmetry 's Portal is a body of work of optical illusions where the symmetric and random are diluted in three-dimensional works originated from two-dimensional planes.

In order to properly write about M. Sardinha's current work I found myself challenged by my ignorance regarding the central feature of this work, the Alhambra in Granada on the southern plains of Spain. Since learning of the show last week I downloaded several books about the Alhambra to my Kindle and have been surprised yet again by another wonder of the world.

First surprise: Washington Irving, author of the Legend of Sleepy Hollow and Rip Van Winkle, wrote a book about his journey to the Alhambra in the mid-nineteenth century. Irving, who later completed a five-volume biography of George Washington, had been appointed ambassador to Spain from 1842-45. During this time he took the opportunity to travel through Don Quixote territories to this most unusual palace and fortress complex which had first been established by the Moors when they conquered Spain. Irving, being an author, wrote in great detail and an entertaining manner about the trip to the Alhambra and the marvels he encountered there.

The Alhambra began as an outpost in about the 9th century, was later rebuilt and added onto in the 11th century by the Moorish emir Mohammed ben Al-Ahmar of the Emirate of Granada, who built its current palace and walls. It was converted into a royal palace in 1333 by Yusuf I, Sultan of Granada.

But its history doesn't end there. After the Muslims were removed from Spain, Christian rulers moved in to portions of the Alhambra and added some of their own embellishments. In fact, it was here that in 1492 Christopher Columbus made his presentation to King Ferdinand and Queen Isabel of Spain to obtain funding for that crazy-wild notion of going to India by a direct route West across the Atlantic.

The structure over time fell into disrepair until a much later time it was recognized for the jewel that it was, with restorations initiated in the 19th century.

This history is not, however, the focus of Symmetry's Portal. Rather, it is the remarkable tiling patterns with which this palatial wonder has been decorated. I learned more about this feature of the Alhambra by reading John Jaworski's volume A Mathematician's Guide to the Alhambra.

For those unaware, Islam forbids representation of the human form, hence the "kaleidoscope of colorful tilings, carvings and reliefs" in Islamc architecture. According to Jaworski, the phenomenon that is featured in the Alhambra regarding these designs is that there are examples of all 17 possible tiling patterns here. The designs of the Alhambra thus become a catalog of mathematical forms.

17 is an interesting number. First, as a prime number, it is indivisible. Second, among other things, it is the only known prime that is equal to the sum of digits of its cube (17 to the third power = 4913 and 4 + 9 + 1 + 3 = 17).

To understand  what is going on in the patterns of the Alhambra it is helpful to understand tiling, symmetry and tessellation, the latter being "the tiling of a plane using one or more geometric shapes, called tiles, with no overlaps and no gaps." Many of us have a measure of familiarity with the concept through the etchings of M.C. Escher, who was himself inspired by the Alhambra.* Not all tiling patterns have symmetry, and according to those who have studied this, there can be no more than 17 distinct ways to repeat these motifs. This standard of 17 exists only in Egyptian temples and the Alhambra.

How did these artisans in Egypt and of the Alhambra discover this? How did this all come about? As this is my first introduction to these notions, I can only say that I look forward to learning more, and aim to share more insights from M. Sardinha's penetrating work, which appears to be an outgrowth of concepts that she has been developing for many years.

"Know oh brother ... the study of sensible geometry evokes skill in all practical arts, while the study of intelligible geometry evokes skill in the intellectual arts because this science is one of the portals through which we move towards the knowledge of the essence of the soul, and this is the root of all knowledge..."
~Rasa'il Brotherhood of Purity, translated by S.H. Nasr

TO BE CONTINUED

* Henri Matisse and David Hockney are additional 20th century artists who found inspiration in the designs of the Alhambra

Photo Credits
Top Right: "Ceiling in Alhambra" by Liam987 - Own work. Licensed under CC BY-SA 3.0 via Wikimedia Commons.
Lower Photo: "Vista de la Alhambra" by bernjan - Flickr. Licensed under CC BY 2.0 via Wikimedia Commons.

Interview with the Author of Mathematics: Is God Silent?

I met James D. Nickel at Bethany Fellowship in 1976, a Missionary Training Center in Bloomington, MN. We were both amongst the older set of freshman who came in that year, have been through college and a bit more seasoned by life, which added value and perspective to our experience.

In those days we called him Jim. At 6'6" with brillo pad hair he was someone you looked up to. I did, especially, as the shortest member of our Bethany Men's Quartet, which we originally called The Glory Boys, along with 6'5" Jim Towner and my 6'2" room mate Ken. With Jim Nickel's great bass voice (I chimed in at high tenor) we'd sing in the stairwells (the acoustics were terrific), in the dining hall, and eventually at a number of churches around the Twin Cities. Those were fun memories.

A major feature of the school's training was a research paper called the Senior Project. I remember reading a draft of Jim's paper and being impressed with his lucidity on this topic. Eventually the seeds of that research led to his commitment to complete a richly insightful book called Mathematics: Is God Silent?

Rather than break this interview into three segments, I will violate the principle of keeping blog entries brief and dissect the interview in half, the rest to follow tomorrow. Please read it all because James offers a wealth of insight here.

Ennyman: When did you first take an interest in mathematics?
JDN: High school, the result of the influence of my Algebra/Geometry teacher Wilbert Reimer (1965-1967). Unlike most high school math teachers, he was a mathematician and he knew what he was doing. He was tall, thin, reserved (when something excited him, he merely “raised his eyebrows”), and possessed a quiet, yet sparkling wit. He would give extra credit questions on his math tests like, “Who is buried in Grant’s tomb?” and “When was the War of 1812 fought?”

In college, I decided to major in math (simply because of all the subjects, I liked it the most). My freshman Calculus instructor was Larry Walker, a former World War II B17 bombardier. I remember him connecting math to the parabolic flight of bombs as they are released from the bomb bay … unforgettable images!

As the course material advanced beyond differential and integral calculus, it seemed like I had entered into an n dimensional domain of transcendent abstract analysis, aimed not at the Elysian fields of delight, but at the specter of the null and the void. On graduation day, I made an internal vow, “I will never open another math book again as long as I live.” I had the opportunity then to go into graduate school but why study the void? I chose a more practical and financial rewarding route … computer programming (where I spent nearly 25 years of my working life).


After college and entering the work force (1973), my interest in mathematics waned. I was seeing it connected to something tangible, though, because of my work for the United States Navy (I had to write code dealing with three-dimensional coordinate systems tracking F14A Tomcat test flights).

In 1976, I traded work (and its mathematical connections) for missionary training. I never thought I would ever be involved with mathematics again, but, as part of my training (1978-1979), I found myself teaching a year of high school math in Kailua-Kona, Hawaii (with Youth With A Mission). In this tropical paradise, my interest in mathematics was rekindled after meeting, in April of 1979, Dr. Glenn R. Martin (1935-2004), professor of history at Indiana Wesleyan University (then known as Marion College). In a series of profound lectures, he affirmed that all knowledge is understood in terms of presuppositions and that mathematics needed to be studied (and, more importantly, reinterpreted) on that basis. These lectures formed a catalyst that brought mathematics back under my purview, but this time on a foundational and thereby highly motivating level! I began a long and pleasurable journey of rethinking the subject, a study that eventually culminated in the publication (first printing in 1989) of Mathematics: Is God Silent?

Ennyman: What were the most surprising things you discovered as you studied the lives of great mathematicians while working on your first book, Mathematics: Is God Silent?
JDN: I had never known anything about the lives of the great mathematicians, especially the key ones … Galileo, Kepler, Newton, etc. In the early months of 1982, before leaving the States to work as a Christian schoolteacher in Australia, I read Morris Kline’s Mathematics and the Physical World (1959). In it (p. 119), he had this quote by Kepler from The Harmony of the World (1619), “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.” Astonished, I thought, “I never learned this in school!” Thus, during my years in Australia (1982-1987), I researched the lives of many famous mathematicians and scientists. I spent countless (and highly enjoyable) hours at the library of the University of Adelaide combing through stacks of books, both off the shelves and from its special loan section. The most surprising discovery that I made was reading Kepler. He would write page after page of math equations and then, overwhelmed by what he was seeing in them, pause to write a psalm of praise to God! Then, he would continue with his equations and, a few pages later, stop and compose another paean!

Another fascinating mathematician I learned about is Leonhard Euler (1707-1783), a man whose gift of systematics enabled him to explore and expose a multiplicity of “unities in diversities” within the structure of mathematics. Euler, the mathematician, was also a gifted teacher (you cannot always equate the two). He wrote one of the first textbooks on the Calculus and every succeeding Calculus textbook (approaching “infinity” as a limit!) owes Euler a substantial debt. Everyone should read his fascinating “Letters to a German Princess” (1768-1772). As a committed Christian, Euler often received the vitriolic “wit-wrath” of Voltaire and other French atheists. One of them, Pierre-Simon Laplace (1749-1827), encapsulated Euler’s influence on mathematics, “Read Euler, read Euler, he is the master [i.e., teacher] of us all.”

Kline and a host of other mathematics historians are naturalistic in their presuppositions. To them, the devotion of Kepler (and others like him) to God form an anomaly. Science historian Stanley L. Jaki (1924-2009) amplifies the reason why, “When historians are baffled by ‘the religious convictions that formerly motivated some of the finest research,’ it is because they have never experienced what it means to look at the world as the product of a personal, rational Creator” (The Road of Science and the Ways to God, p. 47).

Not all of the scientists and mathematicians of the 16th to 18th centuries were thorough going and consistent Christians, but all of them, Christian and non-Christian, were doing mathematics in a framework or cultural ambiance built upon Biblical Christian truth: (1) God is a Wise and Rational Creator and (2) His creation reflects His wisdom and rationality. Hence, we can expect to find, via mathematics and science, laws that are “commentaries” on the “language fabric” of creation.

Ennyman: Why is it that so many people seem to be intimidated by math, especially the higher levels like trig and calculus?
JDN: Primarily, the intimidation comes from two sources: (1) the symbology and (2) the disconnect with wonder. Mathematics is a unique language and, in order to appreciate its wonder and power, you must first master its grammar. Many math teachers (and textbooks) do not know how to teach this grammar in a way that engenders mastery. Professor Warren W. Esty of Montana State University has written a wonderful text that attempts to fix the problem: The Language of Mathematics.

In the 1960s, Morris Kline said, in effect, that math teachers know little of how to connect mathematics to the physical world (i.e., science). Since then, nothing much has changed (but there are some splendid exceptions … see the textbooks written by Harold R. Jacobs as an example). My daughter’s College Calculus teacher just taught the “mechanics” of the subject. He never connected any of the concepts to history or physics (where the methods of the Calculus really “shine”). More students would get excited about math if a teacher taught it in a classroom with “walls open to the outside world.” Topics of biography, history, and elementary physics are portals to wonder. For one example of how to do this, see my essay on the rainbow.

CONTINUED

As one who both loved and burned out on mathematics during my school years, I find these thoughts inspirational. Come back tomorrow for the rest of this valuable interview.