Infinity Series I: Finite Area with Infinite Perimeter

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…