Reasoning mathematically about the Trinity, part II

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In part I of Reasoning about the Trinity, we talked a bit about the doctrine of the Trinity and the philosophical concepts sustaining  its contents.  We also looked as common analogies used to help people partially understand the Trinity, and ended with a brief exposition of the Sierpinski triangle as a possible more "mathematical" analogy with less weakness, at least, than typical, empirically based analogies. In part II, we approach the Sierpinski Triangle (from now on, I will shorthand this as just the Triangle) using some mathematical tools. 

Let's begin! The Triangle is an example of what we call fractals.  While there are multiple mathematical definitions of fractals, the gist is that these are objects which can be partitioned into mirror images of themselves; in short, they have self-similarity. The Triangle itself is made up of three copies of itself  (look at the picture in the corner, and see how the cyan, yellow, and magenta portions are themselves Triangles), and each such copy is made of three copies of itself, and so on.  

An interesting thing about fractals like the Triangle is that they can be described as minimal fixed set of Iterated Function Systems (IFS).  Not all sets commonly called fractals can be generated by IFS's but there are many such fractals (for example, the Mandelbrot set), including several familiar ones, which can be described in this way.  IFS's are finite sets of contractive functions on some metric space (for example, the real numbers or the Cartesian plane).  For the mathematical novice:  by contractive, I mean that the functions take any two points a given distance, and return two points that are even closer together.  For an IFS with functions $f_1, f_2,..., f_n$, by fixed set,  I mean a closed set $A$ such that if I take all the points $ y$ such that $y = f_i(x)$ for some $x$, I end up with $A$ again.  It turns out that the Triangle is the smallest fixed set for functions $f_1, f_2, f_3$ over real number pairs with 

 

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

One thing to note is that each function "contributes" to a portion of the fractal. Suppose that the Triangle above is in a coordinate plane, and the bottom left point of the cyan Triangle is the origin $(0,0)$. If you take the points in the Triangle and pass them through $f_1$, you shrink everything down to the bottom left corner, which takes the whole triangle to the cyan Triangle. $f_2$ shrinks the Triangle and moves it to the right, mapping it onto the magenta Triangle, and then $f_3$ shrinks it and  moves it diagonally upwards and rightwards where the yellow Triangle is.  If we call the set of points in the Triangle $T$, it's easy to see that $f_1(T) \cup f_2(T) \cup f_3(T) = T$, making the set $T$ a fixed point for the IFS $\{f_1, f_2, f_3\}$.

This is by no means the only such fractal that can be described this way.  For instance, If you have functions $g_1,g_2,g_3,g_4$ with 

\[ g_1(x,y) = (0.85x +0.04y,-0.04x + 0.85y +1.6) \]

\[ g_2(x,y) = (0.2x - 0.26y, 0.23x +0.22y + 1.6) \]

\[g_3(x,y) = (-0.15x +0.28y, 0.26x +0.24y +0.44) \]

\[g_4(x,y) = (0, 0.16y ) \] 

You get what is known as the Barnsley fern. The individual functions also contribute to the whole fern, and the pictures are color coded in such a way that corresponds to each function in the system.  Assuming the origin is a bit below the bottom stem of the fern, $g_1$ gives the main structure of the fern. You shrink it just a bit, rotate it, and move it up, mapping it to the cyan portion.  Meanwhile, $g_2$ and $g_3$ shrink it significantly with some rotation and distortion, and shift it off, generating the yellow and magenta leaves of the fern.  Finally, $g_4$ flattens it vertically into a line, shrinks the line and shifts it upwards, mapping the fern onto the green stem.  Again, the fern is a fixed point of the IFS $\{ g_1, g_2, g_3, g_4 \}$.

Let's show one more.  We can actually generate some interesting fractals with IFS's containing only two functions.  For instance, let 

\[ f_1(x,y) = (0.2875x +  1.575, 0.5875y + 1.6) \]

\[f_2(x,y) = (0.603x +0.5591y +0.3880, -0.5864x +0.6324y -0.1549) \]

You get this cool fractal:

 (Yes, these are computed using some software and much experimentation). Using what you know of distortions (rotations, skews, translations, etc.) and high school geometry and algebra, try to figure out which portion $f_1$ and $f_2$ go to. 

In fact, we can use this approach to figure out, for a a given IFS, what the associated fractal is.  Start with the origin point (0,0) (or some other point, if you want),  let $G_1 = \{ (0,0) \}$, and apply each of the functions to it to get some set

\[G_1 = \bigcup_i f_i(G_0). \] 

Then do the same thing again, except instead of just with (0,0), you do it to your newly generated set $G_1$, and let $G_2 = \bigcup_i f_i(G_1)$.  Keep repeating this, and the closure of the limit is your fractal.  The repetition also should help you see why these fractals have self similarity properties.  As you can see, each of the functions takes the fractal and maps it to a smaller and possibly distorted version of itself.  Though the individual functions are not each other and only form a part of the function system, their manifestations, so to speak, in the IFS, result in complete copies of the fractal.

Thanks for reading! Part III will describe how these tools can help us to visualize certain doctrines seemingly in tension with each other at the same time.