All about Bones’ Bones
Welcome to your guided tour of a Fox Network TV series, anthropology, the human body, fractal geometry of nature, mathematics, computer science, signal and image processing and hardware maintenance while catching a killer!
Let's start with the bones the killer used. In the episode, you can see that the bone that contains the fractal pattern coding is one of the real ribs of the thorax. You can't quite make out which one, but any one between T1 and T6 Additionally, you can see that the fractals are engraved on the rib head.
For those of you who are not quite as firm about the structure of your own body, here are some graphs of ribcage as well as a single rib. Note that the red marked ribs are the real ribs.
The fun part starts now! Quick question: "What is the average size of a T3 rib-head?"
Yes, it is good that you also know that the average dimensions vary between a height of 11.3 mm (discrepancy 2.5 mm) and a width of 6.0 mm (discrepancy 2.0 mm). In the following I assume that the rib of bones is a real rib T3, which has a height of 11 mm and a width of 6 mm.
Furthermore, the neck of a rib is between 2.5 and 3.0 cm long. I will not make any further assumptions here, as it is deep enough to engrave a pattern.
To illustrate what I mean, I have prepared some nice pictures for you here:
Now that we know the dimensions, we can calculate how much space is available for fancy fractal code. Simply multiply the height by the width (11 mm by 6 mm) and we get an area of 66 mm². I will stick to two dimensions, since the neck of a rib is deep enough, I don't need to calculate the volume.
Fractal Patterns – What are Fractals again?
Rest assured, I will not delve into the depths of fractal geometry. That would be quite insane (for now at least ...). In a nutshell: Fractals are patterns that describe any naturally occurring object or structure. Everything that is not man-made architecture follows fractal patterns. The structure of trees, clouds, the human blood circulation, distribution of stars in a galaxy and many more.
For those, who want to know more about fractals, here is a nice video:
To continue with our reality check: Fractals are patterns that are self-similar (yes, and for this post, I won't open the "not all fractals are perfectly self-similar" barrel). They repeat themselves, over and over again, no matter how deep you zoom in or out. The pattern is always the same.
Shortly: To construct a fractal, there is a law of education, which repeats itself every time you wish to proceed.
To get back to Bones, the fractal you see in the series is very similar to a so-called Sierpinski carpet. That's pretty amusing, since it's a universal object, which is long enough to contain lots of information and even other fractals. To construct it, just take away the square (1/3) in the middle, as it is shown here:
You can also create fractals in 3D, which is more similar to the way you create bones. If you assemble a Sierpinski carpet in three dimensions, you get a so called Menger Sponge.
If you now cut this sponge open, you will get this picture, which looks quite similar to the markings on the bones in the series.
Carving Fractals on Bone – Is it possible?
Okay, we now know what fractal patterns look like and how to estimate the size of the victim's rib.
Now it is time to think about whether it is even possible to engrave such structures into a bone. Since our killer is under house arrest, I assume he has no access to fine engraving machines or lab equipment. Therefore I searched for micro engraving tools online and found out that widths up to 0.5 mm are available.
To continue, the smallest square that can be achieved with such a tool is 0.5 mm by 0.5 mm, which gives an area of 0.25 mm². So what?
Previously we assumed that the area of the rib head is about 66 mm². If it were flat, this area could be engraved with 0.25 mm² small squares, resulting in a maximum of 264 such fields (this will be relevant in the next chapter on QR codes).
Since the pattern is a kind of Sierpinski fractal, how many iterations can be engraved on a bone considering the given dimensions?
The reduction factor of the fractal is 1/3, so we can calculate the sides of the white squares until they become smaller than 0.25 mm². I won't go into the details here, but we can achieve three levels, which gives 73 white boxes.
Thus we see that it is possible to engrave a sufficient fractal on a rib head with a micro-engraving tool that is not part of high-tech equipment.
In part three, I will talk about why this is relevant.