March 04, 2017 Saturday
Bedtime Story
Fractal Increase or Fractal Scaling
Now what is a fractal?
To mathematicians a fractal is bit different than to most other
average bipedal apes.
A fractal is a set which when analyzed or looked upon at any scale
will exhibit the same repeating pattern.
Another term that is used in the mathematics of fractals is
self-similarity.
Self-similarity essentially means that at the set or an object is
exactly or near-exactly the same at every scale.
It also means that a self-similar object is same or nearly the
same as part of itself.
A common example often used to explain this is any coastline.
The idea of fractals had its beginnings in the concept of
recursion.
This is all great.
But that is a fractal increase or increment?
Well…this is yet another reason why fractals are so fascinating.
It is the way in which they scale in contrast to the other
geometric figures.
I had no idea what is scaling so I read it up.
Scaling is a kind of linear transformation (some guys use the term
mapping and some folks use the word function) that either enlarges or shrinks
an object by a scale factor that is same in all the directions.
When you enlarge a photograph or zoom in an image on your computer,
the algorithm carries out uniform scaling.
Take for example any polygon.
If you double the edge length of a polygon, you will increase its
area by four.
A simple square will help you clear this idea.
In this case, a linear transformation has taken place of a scale
of 2.
In a case of sphere, if its radius is doubled, then its volume
increases eight times.
In this case, the scale of increase is 8.
The scale was measured in each of the above cases by the formula:
(new side/old side)dimension on which the object resides
In case of a square it was 22 and in case of sphere it
was 23.
In the case of a fractal, it is bit different.
When one-dimensional sides of a fractal are doubled, the scale of
increase takes place by a power that is not an integer.
I will stop at that for now, as going deeper into it will take me
into the realms of topological space and Lebesgue covering dimension.
Stay tuned to the voice of an average story storytelling
chimpanzee or login at http://panarrans.blogspot.in/
Good night mon ami and my fellow cousin ape.
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Another great educator and a teacher that I am aware of is
Professor Subhashish Chattopadhyay in Bangalore, India.
While I narrate stories, Professor Subhashish an electronic
engineer and a former professor at BARC, does and teaches real mathematics and
physics.
He started the participation of Indian students at the
International Physics Olympiad.
Do visit him here:
All his books can be downloaded for free through this link:
For edutainment and English education of your children, I
recommend this large collection of Halloween Songs for Kids:
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