Mathematical Analogies: Reasoning about the Trinity: part III

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 In this final installment on the Trinity and the Sierpinski Triangle, We show how some of the principles  and techniques described in part two illustrate in some interesting ways the orthodox theology on the Trinity, particularly in cases where the doctrine holds certain ideas seemingly in tension with each other together. 

Previous posts for part I and part II are here and here (include hyperlinks).  

Let's start with the initial observation about the Triangle's self similarity.   Observe that the cyan, yellow, and magenta portions of the Triangle are themselves full Triangles, and recall that the Triangle is the minimal closed fixed set of the following Iterated Function System $\{f_1, f_2, f_3\}$ with 

\[ f_1(x,y) = \bigg(\frac{x}{2}, \frac{y}{2}\bigg) \]  \[ f_2(x,y) = \bigg(\frac{x}{2}+ \frac{1}{2}, \frac{y}{2} \bigg) \] \[ f_3(x,y) = \bigg(\frac{x}{2} + \frac{1}{4}, \frac{y}{2} + \frac{\sqrt{3}}{4} \bigg). \]

  We can initially analogize the self similarity in the yellow, magenta, and cyan Triangles to the fact that the three persons, the Father, Son, and Holy Spirit, are themselves fully God, while not being each other, and by the self similarity of the Triangle, when put together, we see that they form one Triangle as well.  The weakness of this, of course, is that we can just as easily say there are three Triangles, but we can't say there are three Gods. 

Is there a way to strengthen the analogy to at least partially avoid the problem?  One way, I think, lies in allowing ourselves to consider the multiple ways we can talk about the "threeness" of the Triangle.

  Recall that the Trinity consists in three persons in one God.  But we cannot think of the persons as parts.  Recall also that the concept of "person" answers a different question than that of nature.  Persons answer who God is, while substance or nature refer to what God is. Thinking of God according to his persons  is thus a different mode of thinking according to his substance or nature.  There is a sense in which this is also true of an Iterated Function System and its fractal.  We can study the Sierpinski triangle and its properties according to the fractal itself; it can be described, after all, as a compact set of points in the coordinate plane.  We can also think of the Triangle in terms of the three functions $f_1, f_2$, and $f_3$ which generate it.  The functions work together in coordination to generate the resulting figure. Alone, they don't mean much, but together, they define the Triangle itself.

God cannot be God except as Trinity. When we baptize, we do not say "I baptize you in the names (plural) of the Father, Son, and Holy Spirit," but rather in the "name" (singular). In Hebrew, there was a strong connection between something's essence and its name (hence the philosophical significance of God's response to Moses of "I am who am" when asked what his name was).  In light of this, Christians will see hints of the Trinity in the Old Testament as well.  Consider, for instance, the first few verses of Genesis: 

In the beginning, God created the heavens and the earth. The earth was without form and void, and darkness was over the face of the deep. And the Spirit of God was hovering over the face of the waters. And God said, “Let there be light,” and there was light.

 Traditional Christian Theology looks at these verses as presenting God as Trinity: God the Father as creating the world in his person, the Son, also known as the Word of God, who comes from the Father making creation happen (see also John 1:1), and finally the Spirit of God, hovering over the chaotic waters to bring life. 

When we focus on any one of the three persons as God, we also encounter the other two persons.  When the Father creates, the Son and the Spirit are creating.  When we see the Son, we encounter the Father and are filled with the Spirit.  The Holy Spirit who moves us, points not to itself, but to the Father and the Son.  To encounter one is to encounter all, as if in a never ending dance.  In pondering this, Christians, especially in the East, called this folding back of the Father, Son, and Spirit on each other as a perichoreisis. 

 This is much like how when we see $f_1$'s "contribution to the formation of the Sierpinski Triangle, we don't just see $f_1$; we see the Triangle again, which itself is generated by $f_2$ and $f_3$ as well.   Again, let $T$ be the set of points of the Triangle.  Then we have 

\[ f_1(T) = f_1(f_1(T) \cup f_2(T) \cup f_3(T) ) =  f_1(f_1(T)) \cup f_1( f_2(T)) \cup f_1(f_3(T) ). \]

That is, looking at $f_1$, you will also see $f_2$ and $f_3$. 

Pope (now emeritus) Benedict XVI, in talking about the Trinity, considers the persons themselves to be relations.  The Son is not Son without a Father, and the Father is not Father without the Son.  The Spirit (meaning breath) can't be Breath without one Breathing.  While people might think of the persons like the corners of a simple triangle, I wonder if Benedict likely saw the persons more akin to the sides connecting the corners.  At least, when I read him, that was the picture that came to mind.

In a sense, $f_1, f_2, $ and $f_3$ provide the "personality" of the Triangle, while the actual fractal points more to  its "substance." The functions are themselves mathematical relations, taking the triangle and mapping it to the points that reveals the particular contribution of the function, but in such a way that is in concert with the other functions.  These relations, understood as sets themselves, are also isomorphic copies of the Triangle, each distinct, but bearing the marks of the other persons in the works commonly assigned to each.  It isn't quite right to compare the persons to the self similar parts of the Triangle, therefore.  These self-similarities arise from certain special relations (i.e., functions), which in the end only make sense when seen in concert together, but each encapsulating the fullness of the Triangle. 

Welp, that's it, y'all!  Obviously we can't stretch the analogy too far, and even an enhanced one like this falls far short of the real thing.   The Trinity is quite gnarly, after all; it is hard to capture one part of our understanding of it without at the same time imaginatively letting go of an equally important part. Having said that, fractals like the Sierpinski Triangle are themselves are pretty gnarly objects.  My hope is that the gnarliness of latter could provide some fun, light hearted reflection on that of the former.