The painting of Saint John the Baptist by Leonardo da Vinci seems to be one of his final paintings. Leonardo depicts John the Baptist with a similar enigmatic smile as the famous Mona Lisa. Yet, there is no background and the hues are dark enough that the viewer struggles to identify the Baptist’s garments. Leonardo poses his subject in such a way that he points upward while looking downward. In many respects, the Baptist pulls us into a contemplation of a spiritual enigma. How does the spiritual relate to the physical?
One of the elements that pulls us into this enigma is the positioning of John the Baptist’s head and arm. The face and hand are placed on the same level. The nose and index finger have a linear quality to them. In fact, there are many lines that can be traced on the figure, most of which cross each other, forming a series of “V” shapes. One can superimpose the golden ratio on the figure to identify a stunning set of relationships between the parts of the Baptist’s body.
From the placement of the head and face on the canvas to the angle of the nose, one can see an intricate design that informs decisions Leonardo made in this painting. It is possible that the golden ratio might not have been so obviously sketched out, as this late in his career Leonardo may have intuited such design principles. But we know that he studied deeply the geometric proportions of the human figure. Take for instance his sketch of the Vitruvian Man. It comes as no surprise, then, to find such exacting placement of the chin, the part in the hair, the location of the wrist and the curvature of the smile.
Geometry: A Classical Pursuit
What stands behind Leonardo’s fascination with the golden ratio is an ages long discourse into the fundamental principles of nature and the universe at a geometric level. Classical principles of design can be see in ancient temple ruins, sculptures and mosaics. As these principles were discovered, geometers felt they were unveiling the secrets of the universe. It makes sense that these ancient peoples would construct their most sacred spaces utilizing geometric principles. We see this not only in the Greek Parthenon, but also in the US Capital building, the Taj Mahal, and Notre Dame cathedral in Paris.
Geometry is all around us, especially if we stop to notice it in nature. Take for instance the spiral of shell of a snail. The proportions of each concentric spiral follow the golden spiral which is made from the golden ratio. A related concept, the Fibonacci sequence of numbers, contributes to our understanding of relationships that occur in nature. You arrive at the Fibonacci sequence through simple addition. Add 1+1 to arrive at 2. Take the sum of the final two numbers, 1+2, and you arrive at 3. Take the sum of the final two numbers, 2+3, and you arrive at 5. Continue this algorithm repeatedly and you create the following sequence: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, etc. The ratios of these numbers approximates the golden ration, or phi (φ). 5:3 equals 1.666… 8:5 equals 1.6. 13:8 equals 1.625 and so on. Similar to pi (π), the golden ration has an infinite decimal expansion. So, just as we use 3.14 as an approximation for π, we can use 1.618 as an approximation for φ. Based on this we can see that the Fibonacci ratios bring us close and closer to the golden ratio. These numbers – five, eight, thirteen – appear throughout nature. Take flower petals as a very accessible object with which to measure the Fibonacci sequence. We have flowers with three petals (iris, lily), five petals (wild roses), eight petals (delphiniums), thirteen petals (marigolds, daisies), twenty-one petals (black-eyes susans), and so forth.
The Wonder of Geometric Thinking
We’ve taken this deep dive into the world of geometry (and really only one small fraction of it) to introduce us to the wonder of geometric thinking. Geometry is a field of study within mathematics. Included among the quadrivium, it is one of the most ancient arts with Greek philosophers such as Pythagoras and Euclid remaining prominent figures even down to today. Geometry can be boiled down to three master ideas: shapes, solids and measurements. Seemingly over simplistic, these master ideas contain vast arrays of concepts. For instance, shapes contains the Euclidian concepts of points, lines and planes. Solids include cones and cubes. Measurements involves such things as surface areas and volumes. Geometry can range from the very abstract – “a point is that which no part” (σημεῖόν ἐστιν, οὗ μέρος οὐθέν) – to the very concrete – the surface area of a cylinder is the label on a soup can. It encompasses subatomic particles as well as every object conceivable in the universe.
Geometry is not merely a class in the mathematical sequence. Students learn to apply geometric thinking, which has applications both in mathematics but also in all parts of life. Barbara Oakley in her book A Mind for Numbers gives the advice, “One of the most important things we can do when we are trying to learn math and science is to bring the abstract ideas to life in our minds” (205). To accomplish this, we need to train our minds to visualize and imagine mathematical concepts. Students will often pose the questions, “How does this relate to real life?” or “When will I use this in real life?” These often arise when encountering abstract concepts. And the temptation is to relate the concepts to a specific real-life application. But this sometimes reduces the potential for exciting the mind to creative exploration. Because at one level, the real life application occurs all around us and even within us in unexpected and obscure ways. Only by training the mind to think mathematically can we even begin to articulate certain realities and processes in life.
To take an example from real life. Picture a child or even yourself riding a bicycle. This is a very physical experience involving balance and motion and energy expenditure. Underlying the movement of the pedals, the structure of the chain connecting two gears, the friction between the tire and the pavement, the motion forward against the wind in your face. Each one of these has a mathematical expression or geometrical underpinning. I recently showed my geometry students how the chain connecting two bicycle gears involves two common external tangents to circles, and we investigated the properties of the circles and tangents to understand the mechanics of how a bicycle works.
Geometric thinking is fundamentally about precision and creativity. In addition to mathematical thinking skills, geometry has components of logic, philosophy, art and writing. Clark and Jain concur when they write in The Liberal Arts Tradition, “Besides Euclid’s focus on deductive proof from first principles, his lessons also contain an element of visual artistry and delight” (61). I regularly tell my students that geometry class is about seeing clearly what it is we see.
The Many Integrations of Geometry
We accomplish geometric thinking through proofs and constructions as Clark and Jain point out. I have found that students need to be carefully walked through the process of doing proofs, to see the underlying logic of what a proof is. They often begin by looking at the figure and saying, “Why do we have to prove it, you can already see it in the figure?” True, but we need to see beyond what is there to identify the materials with which we can actually prove something to be true. Take for instance a simple parallelism proof. You often have to learn to see lines connecting two parallel lines as transversals. You have to learn to identify how the angles correspond to one another. You have to apply postulates and theorems to take the materials at hand to conclude that, yes, these two lines are indeed parallel. (Note: I actually find that students understand proofs better using triangle congruence proofs than parallelism proofs.) So within geometry students learn to apply the tools of logic to ideas that seem self-evident.
Constructions can be so fun. Everyone takes out their compasses and straight edges. We learn how to form parallel and perpendicular lines. We learn how to construct regular polygons. We even learn how to draw three-dimensional objects in two-dimensional space using one-, two- and three-point perspective. The act of construction can unlock artistic ability and insight. Soon students can see that representing objects in our world on paper takes thoughtful arrangement and special care to accurately depict items on the page.
Alongside the logical and artistic aspects of geometric thinking, students get a good dose of philosophy. Consider Euclid’s notion of a point as having no height, depth or width. It is a zero-dimensional object. He literally says “a point has no part.” This is the basis for all other geometrical notions. How do we identify a point if it has no dimensions to it? When we represent a point with a dot are we being truthful? I ask my students whether there’s any place in the universe where we could find an actual point according to Euclid’s definition. The only quality a point has is location. Are there any objects in the universe that have location, but no other dimensions? Here students will often posit angels, which is brilliant. Notice how one simple mathematical concept touches on a world of philosophical and even theological discussion.
Another geometrical thinking skill pertains to expression of mathematical ideas in writing. Having students journal in math is a great way to train students in expository writing. I will have students write out the steps of an algorithm, or they must explain the parts of the slope-intercept form of a line. I recently introduced trigonometric functions, and the exercise they did later that week was the narrate based on the prompt, “Tell me everything you know about SOH-CAH-TOA.” Some students filled a page with explanations of sine, cosine and tangent along with diagrams of right triangles and expressions of the trigonometric ratios. Writing in the STEM fields has highly technical qualities to it. Learning how to explain mathematical concepts in writing begins that journey by demonstrating with words, figures and formulas how to express one’s ideas.
Learning to Wonder in Any Subject
Now, this has been a deep dive into a singular subject. My hope is that even if you do not teach geometry, or even math, that there are inspirational concepts throughout this article that you can apply to your own subject areas. Here I will distill three ideas that you can apply in a similar manner to anything you teach.
First, consider what the master ideas are for any subject you are teaching. For instance, in history one of the master ideas is continuity and change. At any moment in the study of history we can use this master idea to pose questions to students. “How did things remain the same after the revolution?” “What changed for people after the revolution?” I could even ask students to consider how a past era is similar and different to our current day. Literature has another set of master ideas, one of which is perspective. Whenever we read a text, we are encountering an author’s perspective. We can use this master idea to guide discussion toward the author’s message and meaning no matter what book we are reading. I have found that across the curriculum there are several master ideas associated with each area of knowledge. Doing some exercise to consider what these are can be of genuine benefit as a teacher. They usually amount to a small handful of ideas that you can always go back to with your class.
Second, notice how geometry is self-integrated. It is not as though philosophy and art have to be integrated artificially. They are already intricately linked. It simply takes a teacher who can see the internal connections to bring to life the inherent properties of a subject. With literature, one doesn’t need to figure out some artificial connection to history, the text already has within it a connection to history. Whether it is the setting of the book or the author’s own historical context, history surrounds the book at every turn. One can also access grammar through the same text. One can cultivate a sense of writing style from that very same text that can be imitated in writing class. Integration is not something we add onto the curriculum, it occurs within the curriculum. Often we can allow the students to make these connections, to see how different subjects relate. But we as teachers can also do some creative investigations into how we can identify pathways to make the most of each lesson.
Third, track the specific thinking skills within each subject. For instance, in history we can identify chronological reasoning. Can students place people and events in a proper order? This might involve recollection of specific dates, but one can go beyond dates to consider what might have led from one event to another, or why one event would come after another. It might seem obvious to us that the Civil War would come after the Revolutionary War, but can students trace the steps from one event to the other? In science, we could highlight classification. In grammar, students think about the form, use and meaning of different parts of speech. In theology, we grapple with justification for our beliefs. All of these different thinking skills when articulated enable us to determine how students are doing in a given class. Once I know what skills we’re cultivating, I can then articulate where a student can grow.
There is much at which to wonder. Through geometry the wonders of the world of shapes and spaces are manifested in great works of art and God’s great work of creation. Let us all be on the search for the many ways we can guide our students into worlds of wonder in all the ways that they learn each day.
Unlock the wonder of any subject through narration. Read Jason Barney’s latest book A Short History of Narration, available now through Amazon.
