Silvery Fractal Dragons
Today we're going to switch things up with some math. We're going to look at how to build fractal curves with silver bullion.
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A dragon curve at zero iterations, just one lonely dragon.
Dragon Curve
A dragon curve is a type of fractal. Fractals are geometric structures with very specific properties. One of the most prominent ones is self-similarity. No matter which part of the object you look at, it will always look similar if not identical. Fractals also don't intersect themselves and are usually infinite structures.The dragon curve is one of those fractals and can be constructed easily with a sheet of paper (at least partially). Some of you might have seen a dragon curve already in the Jurassic Park book. Each chapter shows a dragon curve of increasing order. This publicity made the dragon curve semi-popular.
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A dragon curve after one iteration, just two merry dragons.
The Paper Method
The easiest way to make a dragon curve is by taking a piece of paper and folding it again and again. But always be careful to fold in the same direction. Afterwards, unfold and place all angles at 90°. Now you've reached a dragon curve of n-th order, whereas n is the number of times you've folded the sheet. If you use this method with a DIN 4 paper you usually only reach order four or five. The paper gets too thick to fold really fast.
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A dragon curve after two iteration, they multiply like rabbits.
Solid Problem
We could, in theory, do the same thing with silver. But I don't have any silver foil, and I don't want to sacrifice my stack to make some, so we have to improvise a bit. Luckily we can use an iterative string-based process which tells us how to draw a dragon curve.We're going to write a "construction plan" until the nth order by following those two steps. We're going to abbreviate draw right at a 90° angle with R and draw left at 90° with L. Now onto the instructions:
- We add an R to the end of the character sequence. If there is no sequence, we write R.
- In the second step we take the original sequence without the added R and invert the middle letter. Afterwards, we concatenate the altered sequence after the added R. If the original sequence is empty, there's nothing to add. Repeat with the nth sequence to get the n+1th sequence.
Through this way, we obtain the sequences for order one, two, three etc. Respectively. R, RRL and RRLRRLL.
Let's do a quick example for the third order sequence. To get the third order sequence, we take the character sequence from the second order, RRL and add R at the end.Now we have RRLR. Next, we take the first part, which is RRL and invert the center R to L. Now we concatenate the new RLL sequence after the added R. We end up with RRLRRLL as sequence.
If you want to draw this, you need to draw a line first. It doesn't matter which orientation the line has; I decided to use a vertical line. Afterwards, you draw lines in 90° angles from the previous line end. You follow the sequence and continue drawing until you've reached the end of the sequence.
To build a real dragon curve you'd have to do this process an infinite amount of times, which isn't possible. But it is in theory, and that brings us to a very peculiar property of this curve: it has an infinite circumference. If you wanted to, you could also use Lindenmayer Systems to draw it, also a string rewriting system, but more computationally sound. The result is the same.
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A dragon curve after three iteration, still manageable.
Silver Time
Now we transfer this approach to silver bullion. We don't want to draw with silver onto something. So I decide that each bullion coin represents an edge of the dragon curve. The issue with that approach is that you need lots of space and silver. Because with each iteration the number of edges increases exponentially. For order one we need just 21=2 coins, but for order six this increases already to 26=64. And you don't see the real structure of the dragon curve until order 13. That would be 8,192 coins or 125,255.68$ of silver. As this approach isn't feasible, I decided to build only a sixth-order dragon curve. For this you need the string:RRLRRLLRRRLLRLLRRRLRRLLLRRLLRLLRRRLRRLLRRRLLRLLLRRLRRLLLRRLLRLL.
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Four iterations in I've run out of dragons. Now I need to use my whole one ounce stack to continue.
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The Final Result
With the help of 25 maples and 20 Noah's ark coins, I managed to finish the sixth-order dragon curve. 65 ounces of silver.Not sure where I went wrong, but somewhere there must be a mistake in this curve. But it looks like a sixth-order dragon curve, so I'm happy.Thanks to Wikipedia I can show you a nice animated 15th-order dragon curve. With 15 iterations you might see why it's called dragon curve. It looks a bit like the profile of a dragon. Now you know how do make a dragon curve. What you do with this knowledge is up to you. Is this the first time you've heard about this, or have you seen a dragon curve before?
I think it's really fascinating how something so complex can be created by simply folding a piece of paper.
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Animation of a unfolding dragon curve. Made by Jahobr
Elly standing inside the dragon curve. She's trying to find dragons.
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Its Important to have Fractional silver to make change.
I've just got a few fun fractional Skulls to make change of full One oz Skulls in Pirate commerce.
That's true, fractional silver is cool too. It might have helped with this curve as well because they are cheaper and smaller.