Infinity Series I: Finite Area with Infinite Perimeter

Original lithograph (M. C. Escher, 1961)

This is the first entry of the “Infinity Series”, which will introduce you to the peculiar world of infinity. Today, we will be talking about something with finite area but with infinite perimeter. That sounds odd and impossible but I’ll show you how it can be done.

Draw a circle and an equilateral triangle inside it, with the three vertices of the triangle touching the circle.

Since we know that the circle has a finite area, the triangle inside must have finite area as well. At this point, we still have finite perimeter on the triangle as well.

Now, divide each edge of the triangle into thirds; draw an equilateral triangle using the middle piece of the divided edges as the base, as follows:

With careful observation, you can see that the perimeter has increased. Specifically, the perimeter increased by a third: instead of one edge, now two edges of each newly-drawn equilateral are part of the shape . The area inside the shape increased too – but notice that it is still within the circle, so the area is finite.

Now, perform a similar operation by dividing each edge into thirds and draw an equilateral triangle using the middle piece as the base. It now looks like this:

Although a small change, the perimeter has increased and so has the area. But since the shape is still within the circle, the area is still finite.

As you can imagine, this operation can be performed an infinite number of times and the perimeter will keep getting bigger and bigger onto infinity. On the other hand, the area keeps getting bigger and bigger as well but will always remain smaller than the circle, hence finite. For a more mathematically-mature audience, the area converges while the perimeter diverges.

(Wikimedia Commons user António Miguel de Campos / Wikimedia Commons)

Hence if you were to make a football field in this shape, you will only need a finite amount of grass but infinite amount of white paint to draw the sidelines.

On a soccer field of this shape, the assistant referees in charge of calling the offsides will have to run an infinite distance to get from the goal to the half line (whatever that may be).

If a castle with such walls had guards monitoring on top of the walls, only an infinite amount of time will suffice the guard to make one circle and the kingdom would’ve needed an infinite amount of rocks or bricks to build the walls to begin with.

Odd?

Welcome to the world of infinity.

NOTES: This shape is called the “Koch snowflake” and is a good introduction to the world of fractal geometry.
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