Crafting Fractal Snowflakes, and Exploring the Incarnation of Christ. A Christmas Math/Craftivity.

 
At Christmastime we celebrate God becoming man, and walking with us.

And the Word became flesh and dwelt among us, and we have seen His glory...John 1:14a

The incarnation, infinite God in a finite form, can be difficult for children to grasp.  Honestly, it can be difficult for adults to wrap their minds around.  In fact, it's a concept theologians have been grappling with since the beginning of the Church. And yet, it's a concept we can see illustrated in nature, if we look closely enough, and mapped out for us in mathematics textbooks - cool textbooks, anyway - in the back chapters dealing with fractal geometry.

Fractals provide beautiful illustrations of infinity in a finite space.  They also lend themselves nicely to Christmas crafting, as so many of them are shaped perfectly to become (with proper coloring, and a touch of imagination) Christmas decorations.  I'm hoping to explore several this week, with the children, as we craft, bake, decorate our house, and contemplate the wonder of the Christmas story.

We started today, with a very simple math lesson/Christmas craftivity - building a Koch snowflake, one iteration (or step) at a time, out of marshmallows and toothpicks.

I starting by laying out a couple of plastic tablecloths on the living room floor.  Then, I asked the younger children to make three lines, each with 36 toothpicks connected by marshmallows.

They connected the three equal rows together to form an equilateral triangle. 

We stopped for a minute to review what an equilateral triangle is - a triangle with three equal sides, and three equal angles - measuring the angles with a protractor to make sure they were equal, and that they did in fact add up to 180°, as we expected. 

Then, we reviewed the concept of perimeter - the distance around the outside of an object - and measured the perimeter of our triangle in toothpicks, recording our findings on our dry erase board.

With that done, we then divided the sides (each of which was 36 toothpicks long) into three sections (12 toothpicks each), and built three new equilateral triangles (similar - the same shape, but different size - to our first triangle) over the center third of each side.

Once we had noted, that even though smaller, our 12 sided triangles were still equilateral, with angles all measuring 60°, we removed the base of each of the small triangles, and discovered that now, instead of a triangle with a perimeter of 108 toothpicks...

...we had a star, with a perimeter of 144 toothpicks.  At this point, the children thought we were running out of room on our tablecloth, and would have to stop expanding our shape.  So, they were surprised, when I instructed them to make additional equilateral triangles over the middle third of each newly created side. 

The sides of our star measured 12 toothpicks each, so our new triangles had sides of 4 toothpicks, and changed our star into a snowflake - an amazing snowflake, that somehow did not move any closer to the edges of the tablecloth than the star, which had had a perimeter of 144 toothpicks, even though it now had a perimeter of 192 toothpicks.

We stopped there, as to go any further, we would have had to have broken the toothpicks into sections.  But, it was easy enough to see, as far as we had gone, that our shape could continue increasing in perimeter - theoretically on into infinity, by continuing to divide the sides into thirds, and adding triangles onto them, without ever moving any closer to the edge of tablecloth.  Just for good measure though, we also watched a video about the Koch snowflake from Khan Academy, to prove it to ourselves.

Koch snowflakes, finite infinity - just about as hard to grasp as God incarnate, but awesome all the same.