Trefoils, Borromean Rings, and Athanasius

By Bobby Neal Winters

You can look up in church windows and see them.  They are the Trefoil Knot and the Borromean rings.  I learned about them first as mathematical objects rather than religious ones because I grew up in a religious tradition that did not truck with such abstract notions.  

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Trefoil Knot	Borromean Rings

But as abstract as they are, the Trefoil Knot and the Borromean Rings are more concrete than the religious concepts they were trying to capture.  I learned about them as I was working on my doctorate in the area of low-dimensional geometric topology.  Specifically, I was beginning the study of knot theory.  My two main sources on knot theory in those days were an article “A Quick Trip through Knot Theory” by Ralph Fox and a wonderful book called Knots and Links by Dale Rolfsen. These were my Old and New Testaments, but not necessarily in that order.

Once while I was working on my doctorate, I had a conversation with one of my cousins with regard to what I was studying him.  I told him about knot theory.  His intelligent middle school aged daughter was listening and interrupted with an indignant tone.

“That isn’t math!” she said.

It leaves that impression with a lot of people.

Knot theory, like her great great grandmother Geometry, attempts to describe the visual world with words. She sees and then she describes.

Let us take the Trefoil as an example.  From the topological point of view, it is a circle. Put your finger on it and trace around it.  It comes back around on itself and starts over.  You can keep going around and around forever just like a circle.  There is a difference, though.  When you draw a standard circle, the lines never do, but in drawing a Trefoil there will be places where the lines cross.

When knot theorists draw the trefoil, we avoid letting the lines touch.  We imagine the Trefoil as being in space and the crossing as being a place where the trefoil passes behind itself.  We draw the part that passes behind as broken.

The Trefoil Knot is the simplest (non-trivial) knot.  Knot theorists can draw it many different ways, but the way one sees it in the church window is probably the best.

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The Trefoil Knot

It exhibits a three-fold  symmetry.  Note that there are four finite regions that the knot bounds: One is in the exact center and the other three are distributed around it. If you rotate the picture around the center of the middle piece by one third of a rotation, each of the other three pieces will lay exactly on one of the others.  They are completely equal in size and shape.

This three-fold symmetry is something the Trefoil has in common with the Borromean Rings.

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The Borromean Rings

The Borromean Rings are an example of a link.  Links and knots are inextricably interrelated.  (This is to avoid saying they are linked. While I don’t object to puns in general, this one is too cute.)   Links are groups of knots that are...uh..linked to each other.  The Borromean Link is special in that each of its pieces (components they are called) is a simple unknotted circle and if any of the circles is removed the rest falls apart, i.e. becomes unlinked.

The question one might ask, because of how I began this article, is why are these religious symbols.  Why do they appear in church windows? They are, of course, symbols of the Trinity.

I once asked a dear friend of mine to explain the Trinity.  I asked him because: one, he has a master’s degree in theology so he is learned in these things, and, two, he is Catholic and they just know this stuff.  He talked to me for ten minutes.  Every sentence he said made sense; they piled one upon another in coherent paragraphs.  At the end of the discourse he said, “And if you understood what I just said, then you weren’t paying attention.”

The Trinity is not something those explain the faith would’ve made-up simply help sell it better.  One can easily infer oneself to misunderstands such as Jesus is his own father.  One can find the concept in the Athanasian Creed or one can refer to the diagram below:

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The Relationships among the Persons of the Trinity

The Trefoil and the Borromean Rings provide a comparatively concrete example of how something can have three parts and still be one thing. They provide a way to defuse the easy quasi-numerical arguments against Trinitarian faith.

The strength of mathematics lies in its exactness.  There is a quote by Roger Bacon that I know because I play Civilization IV, not because I read. It is: "If in other sciences we should arrive at certainty without doubt and truth without error, it behooves us to place the foundations of knowledge in mathematics."  This certainty comes because in mathematics we are able to create an artificial world that is an approximation in some way of the real one.  We may then extrapolate from the artificial one in order to approximate the real one. Nature leaves a footprint; scientists take a cast of it; mathematicians draw a sketch of the cast.

Mathematics is Man’s extension of his language in an attempt to capture the language of Nature.  I almost said “Reality” instead of “Nature,” but that would carry the implicit assumption that Nature and Reality are the same.  I find that assumption to be farther and farther from obvious as I grow older.  Maybe it’s Alzheimer’s.

As a person of faith, I believe things exist beyond those I can touch.  I suppose this is part of being a mathematician.  We create worlds that at first seem to exist only in our own heads, but, as we interact with other of our kind, we discover they have created those very same worlds in their heads as well.  

Christian theologians found themselves heirs to millennia of tradition and scripture.  From that they distilled the concept of the Trinity. We might at this point recall the reaction my cousin’s daughter had to Knot Theory: That is not Mathematics.

While theology is certainly not mathematics, we might have even more sympathy for the theologian than for the knot theorist.  While the knot theorist is attempting to use language to capture the visual world, i.e. what we can see, the theologian is trying to use language to capture what we can’t see. Optimistic might be one adjective to describe such people; insane might be another.

But I don’t want to dismiss them.

I find something good in staring at a stained glass window being reminded that there are things I don’t understand and never will and that no one else will either, but that we still desire to strive toward understanding.