Can you live in a Fractional Dimension? – II

To quote Anthony Liccione, “A mind is not weighed by its magnitude, but by the dimensions of its thoughts”. Of course, it’s not the same dimension I am going to talk about here but the spirit of the statement is a desirable dimension (pun intended). Hope you have read the first half of this post. No? never mind just keep reading ūüôā . I will do my best to answer the question posed in the simplest form. To start of, think about a line segment like this

Straight_LineIf you double the length of this line then you will get 2 copies of the line segment. Okay, consider a square as shown in the figure. sqAgain, double the sides of the square. Now, you get 4 copies of the original square. I know, by now you would have extended this logic to a cube and figured out that 8 copies come out by doubling the length of each side of a cube. Okay, summary:

Line: 2 = 2^1 (2 to the power of 1)

Square: 4 = 2^2

Cube: 8 = 2^3

It is very easy to notice that the last numbers (exponents) are the number of dimensions in which these figures exist. In fact this is one of the definitions of dimension called Hausdorff dimension. Quite a cool definition actually. Of course, the definition I just gave is not totally correct in mathematical sense but for explanation, good enough. Have you heard about this amazing convoluted triangle called the Sierpinski gasket shown below?¬†As you can see there is a regular and conspicuous pattern exhibited by the¬†Sierpinski gasket, recurring all the way to infinity. Some of you would recognize this instantly as a fractal which, of course, is a bigger collection of objects having the same properties –¬†regular and conspicuous pattern¬†recurring all the way to infinity. There is a reason these structures are called fractals and I am going to explain you just that. Look back at the structure below.

Now, let’s double each side of this triangle.

If you observe closely you get 3 copies of the original triangle not 4. The implication of this is captured in this equation:

2^d = 3
Where d is the dimension in which the¬†Sierpinski gasket live. ¬†A quick evaluation puts d at 1.5849… which is a fraction……. fraction….. frac….. fractals?! It’s true the name fractals was coined because these closed figures exist in a fractional dimension. After ‘how’, some may ask ‘why’ fractional dimension? Because, Self-Similarity. Period. Remember Cantor Set from the previous post, yes even that falls under the category of fractals. There is an inherent redundancy (Self-Similarity) in fractal formation and even a rudimentary knowledge in set theory will tell you that sets don’t contain repeated elements which is exactly the reason that leads to non-integer dimensions. There are some beautiful fractals which, I think, I am obliged to share here.

 

Are there any fractals in nature? Oh, yes. The famous 1967 paper¬†“How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension” puts¬†the dimension of Britain’s coastline at 1.25. Intriguing? What about this one: ¬†each branch of a cauliflower¬†carries around 13 branches, each 3 times smaller. This renders the dimension of a cauliflower to 2.33.

So, do you live in a fractional dimension? At least your brain surface does. The Hausdorff dimension of the surface of the brain is at 2.79 and that of the lung surface is 2.97. If such an essential organs can, then why not the whole body surface? Yes, it is possible but there is a key ingredient which is missing, remember, Self-Similarity. So one can safely claim that the dimension of our body is 3 (integer) in the conventional sense. Too bad, I know.

In summary, you can but you don’t live in a fractional dimension but if you desire to, ask this guy.


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