Sometimes when you're sitting in your higher Mathematics class you feel like there isn't really any connection between what you're learning and the real world out there. Especially when you look at the board filled with those seemingly meaningless symbols. What we often forget is that Mathematics is all around us. The natural world is filled with patterns: from the structure of a spider's web, the meandering rivers, the shape of honeycombs, and even the shape of a snowflake. All these can be described by Mathematics.
One of the interesting phenomena we see in nature is the existence of fractals. These are objects or quantities that exhibit self-similarity on all scales. When you zoom in on a fractal structure you will see a similar pattern each time, making it seem never-ending. Examples of fractals are the Koch snowflake, the Mandelbrot set, and the Serpinski sieve [1]. There are also many objects in nature that are fractals such as the repeating patterns in peacock feathers, in ferns and in trees.
One way to describe a fractal is by quantifying its fractal dimension which gives an idea on the complexity of a curve's geometry [1]. Unlike the spatial dimensions that we are used to, the fractal dimension is a non-integer.
In this post we will determine the fractal dimension of a fractal using the box-counting method. In this technique we set a box size and count the non-zero cells in the box while sliding it across the image of the fractal. We do this for several box sizes and plot the box count versus the inverse of the box size in logarithmic scale and find the slope of the trend line to get the fractal dimension $\epsilon$.
Here, we try to determine the fractal dimension of a neuron as shown in the figure below.
Figure 1. Image of a neuron and its binarized version (Image taken from [2])
From Figure 2, the fractal dimension of the neuron is equal to 1.5898.
Figure 2. Plot of the box count versus box size in logarithmic scale
References:
1. Fractals. Retrieved from http://mathworld.wolfram.com/Fractal.html on June 5, 2016
2. Image retrieved from https://www.nikoninstruments.com/About-Nikon/News-Room/US-News/Neuroscientist-and-Founder-of-the-Nikon-Microscopy-Center-of-Excellence-in-Budapest-Featured-in-Tech-Heroes-Campaign-on-CNN.com
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