Calculus Topology Uncategorized

Koch Snowflakes is a weird Circle(Also Disk)

Koch Snowflakes is this beautiful shape that I fell in love with in my school years, I didn’t know much math then( I don’t know much now either!),but the catchphrase is this shape has “infinite parameter but with a finite area”

If this phrase doesn’t make you uncomfortable you are very stoic person (hats off!), I got to know this fact for the first time from a  Bengali child novel called “আমি তপু” (I am Topu) by Muhammad Zafar Iqbal, and probabily the only novel in Bangladeshi literature ever to romanticized math(but no one really remembers it somehow!)

I’m not going tell the story here but this kid named Topu discovers Koch snowflakes by himself while sleeping in the kitchen alone while talking to a rat!

And honestly when I first read it I didn’t appreciate it much, but nevertheless it is one of the fantastic beasts that math has to offer!


             Poster child of real analysis

Koch snowflakes should be the poster child of analysis and should be in the cover of atleast some analysis books because it does capture all the powerful and fascinating ideas we have about real numbers.

Firstly the the shape or set it self is the limit of shapes, and I’m stressing this point out again that neither of the individual shapes are the Koch snowflakes, they are all just stages, The main shape is what all those stages approaches.

secondly and most importantly it carries a diverging and a converging sequence in a absolute stunning way like Devi whom manifests as “infinite perimeter” and “finite area”.

Helge Von Koch and his is not the father of microbiology!

I noticed that I haven’t describe what a Koch snowflakes is! (because I think it’s just very clear from the pictures).It is because I assume that people reading this already familiar with it and want a trivia like post rather than a introductory one, and if you want a introduction what better than to read straight from the Scientific American and I may quickly copy paste from the wiki because it’s not that big.

‘The Koch snowflake can be constructed by starting with an equilateral triangle, then recursively altering each line segment as follows:

  1. divide the line segment into three segments of equal length.
  2. draw an equilateral triangle that has the middle segment from step 1 as its base and points outward.
  3. remove the line segment that is the base of the triangle from step 2

Now observe that each stage comes off with its own perimeter so we get a sequence of perimeters(which is just a real number), and this sequence does not converge, it just grows and grows without any bound. In the same way each stage comes off with its own area and we get a sequence of areas(also sequence of real numbers) and this sequence converge nicely and gives us a fixed area. Thus it capture the two basic kind of sequence simultaneously in itself.


                    Circle vs Disk

I also noticed that I haven’t yet uttered the word ‘Fractal’ yet where I know some people know it as a the go-to fractal, but I am not going to talk about it now rather than I want talk about the differences between Circle and Disk.

In Euclidean Geometry we use Circle synonymously as Disk, that’s why Circle has both perimeter and area formula, but the area is actually belong to the Disk if you think about that. Circle is the set of points equidistant from a origin point(but doesn’t contain the origin point) where Disk is the set of points enclosed by the Circle(which does contain the origin).

Circle is is inherently one-dimensional because you need just an angle θ to uniquely specify a point in but embedded in two-dimensional, where Disk is inherently two because it needs two number, and angle and a distance (θ, r) to specify itself. The Disk contains The Circle as boundary and if you remove it becomes an Open Disk. That’s why area of a Circle doesn’t make much sense but perimeter of a Disk is just a perimeter of its boundary Circle. So to be clear the Area of the Koch Snowflakes is the Area bounded by the closed Koch Curve.

When a Topologist sees a Koch snowflakes he sees a Disk, very Strange Disk indeed, it’s a limiting Disk whose Boundary Circle is very wiggly but this Boundary Circle is the Same homeomorphic copy of the Kinoshita-Terasaka Knot or all the Complex number z, where |z| = 1 or just plain old and wise Euclidian Circle and the Disk bound by it is the same homeomorphic copy of Ribbon Disk bound by a knot in Four-dimension or all the complex number z, where |z| ≤ 1 or just the Plain old Euclidian Disk.

Disk bound by ribbon knot
Kinoshita-Terasaka knot

                           Why bother . . . . .

It can be a legit question that why even Topologist bother every thing make out of rubber? doesn’t it takes away the beauty of every shape? but consider this, when you prove a theorem about the Topological Circle or Disk it automatically true every other Homeomorphic version of it from every other multiverse. Its like proving theorem from ground up. As an example take the crown jewel Brouwer Fixed Theorem which basically say every continuous function from a Disk to Disk has a fixed point, and it is also true for this Koch Disk! no matter how weird it looks.


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