The Geometry of Heaven?

One more reflection on math, and I think the frustrated math teacher in me will finally be sated.

Augustine said that the nature of God is like a circle whose centre is everywhere and whose circumfrence is nowhere. Whatever else that might mean, it does show how mathematical concepts can sometimes open up new ways to reflect on the spiritual life.

Take fractal geometry, for instance.

Fractal geometry is the study of infinitesimal space. A "fractal" is a geometric image that gets infinitely more complex as it gets infinitely smaller. Mathematical fractals are based on an equation that goes through iterations, reproducing designs at smaller and smaller scales. Every enlargement of a fractal reveals a shape of increasingly fine detail and complexity.

One of the more immediate ways to "get" fractal geometry is through the deceptively simple concept of the Koch Snowflake.

Here's how you do it: start with an equilateral triangle with a side x. The perimeter of this triangle (3x) can be increased by a factor of 4/3rds by adding an equilateral triangle with sides 1/3x to each of its three sides. Add an equilateral triangle with sides 1/3rd as big again (1/9x) to each of these 12 sides, and you increase the perimeter by another 4/3rds. Add triangles to each of these 48 sides and increase it another 4/3rds. And so on. Because every side formed by adding a new triangle can always be divided by three, the process can be repeated infinitely. Each time we increase the perimeter by a factor of 4/3rds, but the perimeter never touches itself.

If we could zoom in to even the tiniest side of snowflake, we'd see an edge that looks like this:
The length of the perimeter of the snowflake at the nth iteration is
x*3*(4/3)^n. We take the limit of the sequence thus:

(i.e. because n has no limit, the Koch Snowflake has an infinite perimeter. )

But here's the thing that makes you go hmmm. We can draw a circle around the Koch Snowflake that clearly encloses a finite area (A=π r^2). And the infinite perimeter of the Koch Snowflake will never go beyond the finite area of the circle. Actually, the area of the snowflake can be calculated specifically as:


We say, then, that the Koch Snowflake has a finite area bounded by a perimeter of infinite length.


Infinity in the finite; the finite in the infinite.

The ancient rabbi says God has put eternity into our finite hearts-- and still we cannot comprehend what he has done from beginning to end.

Now, just in case my theology prof is reading this, let me stress, I don't believe this is somehow mathematical proof for the eternity of heaven. But there's this. In The Last Battle, the children are pressing deeper and deeper into Aslan's country, finding each depth more ponderous than the one before. And Mr. Tumnus says: "It's like an onion, except that as you continue to go in, each circle is larger than the last."

Winsome words for the ways of heaven.

Sort of reminds me of the Koch Snowflake. And it strikes me that sometimes a mathematical analogy can be just as evocative as a poetic one.