Hello.
I happened to go through the problems presented in the previous issues .The problem titled "CIRCLE AD INFINITUM" was really interesting!!! My interpretation of the problem is slightly different.
According to the solution suggested by Justin Sinz and Fabian Hanneforth , three circles are added at each stage , which of course is true for the given diagram . But if an equilateral triangle is considered at each stage, i.e., at the second stage , there are 3 equilateral triangles; each has, along with one incircle, three smaller circles-each in one triangle and each of these triangles in turn?!!! I hope you get my point!
What I am trying to say is this:

In the first stage:

1. There is 1 circle of radius 1cm

1. There are three equilateral triangles each of height 1cm
2. Three circles inscribed in 3 triangles of height 1cm---1 at each vertex
3. Each has a radius 1/3 cm
4. Totally there are 3 circles of radius 1/3 cm.

1. There are three equilateral triangles , each of height 1/3 cm
2. Three such equilateral triangles are there - IN the three triangles created in the second stage (triangles of height 1cm).
3. Each of the triangles has a cirle of radius 1/ 9 cm inscibed in it.
4. Now, there are 9 circles of radius 1/ 9 cm.

$S=2\pi [1+1+1+...]$!!

Hello Annamalai,

Firstly (being slightly pedantic) your question isn't a different interpretation of the Circles ad infinitum question but is actually a completely different question.

You are exactly right to say that you have 1 circle of radius 1 and 3 of radius 1/3 and 9 of radius 1/9, etc. So in total the sum of the circumfrences is actually infinite.

Can you work out the sum of the areas of the circles?

Is this finite or infinite? Can you tell the answer to this question without actually working out the sum?

Does it bother you that we have an infinitely long collection of lines inside a triangle which has finite area?

AlexB.

I just get the feeling that my question is stupid after all!! The fact that there are infinite no. of lines inside a finite area just didnt occur to me! Anyway thanks a lot for having the patience to read my question!
The area is finite all right! All those infinite number of circles have to get into ONE triangle!I hadn't least bothered about the area; I was so very worried about the sum of the circumferences tending to infinity!I think the last sentence of your "answer" explains the infinite circ.
The total area...turns out to be a GP with a common ratio 1 / 3 which settles down to
(pi) x [1 / (1-r)] = 3(pi) / 2.
Thanks a lot for clearing my doubt.
Annamalai Meena.

That's okay... I'm glad that you have a clearer picture now. Shapes that you get by infinite processes (like the one you describe above) have been studied a lot in recent times. The one you describe above is very similar to something called the Sierpinski Gasket. Other similar ones are called Dragon curves and Koch curves. If you get any popular book on fractals out you'll probably find these shapes in there. They all have very interesting properties.

AlexB.

Try looking at these links if you want to know more about fractals like the Sierpinski Gasket, Dragon curves and Koch curves.

• How to draw the Sierpinski gasket
• A freeware fractal generating program

Another shape with finite area and infinite perimeter can be formed by starting with a equilateral triangle, adding another similar triangle to the central third of each side. This gives a star shape. Adding a triangle to the cental third of every side is a process that can be repeated infinitely giving infinite perimeter for a finite area.