Going around in circles !!

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Hypatia001

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How many different ways can 4 circles be drawn without overlapping or touching ??

How many ways can they be drawn if they are able to touch or overlap?

Is there a formula for either of these problems ?
 
You really expect us to believe that you do not know how many ways you can draw 4 circles with them touching or overlapping?
This is a help site not a site where we do homework for students.
Please follow the guidelines which you read, tell us what you tried and where you are stuck so we know what type of help you need. Definitely include a drawing with your next post.
 
Thank you for your reply.

I came up with both 8 and 9 for the non touching circles. But am really stuck on the touching / overlapping part of it.

Thats why i asked for help !!! I dont want an answer but some guidance on how to approach the problem.... hoping thats what this forum is for ?
 
Hypatia001 said:
How many different ways can 4 circles be drawn without overlapping or touching ??

How many ways can they be drawn if they are able to touch or overlap?

Is there a formula for either of these problems ?
Click to expand...
Hypatia001 said:
I came up with both 8 and 9 for the non touching circles. But am really stuck on the touching / overlapping part of it.
Click to expand...
How did you get those numbers? Why not 42?
 
I can only assume that your circles are on a plane (2-dimensional) as opposed to 3-dimensional space.

A plane is like the paper you write on but has no end, it is infinitely long and infinitely wide.

How many ways can you draw 4 circles so the circles do not touch?

You say that you are stuck on the touching / overlapping part--Can you please explain exactly where you are stuck so that we know how to help you?.

Can you please post the exact problem from your textbook or maybe even post an image of the problem?
 
Subhotosh Khan said:
How did you get those numbers? Why not 42?
Click to expand...
Hi.

I actually drew them out.

The question asks for different ways without the circles touching or overlapping... and there are no labels or sizes. So i assumed that the cirxles could all be of different sizes.

What i am really after is the maths behid it so i can apply it to any number of circles given the same rules to follow. I do not want the answer but the reason for the answer.
 
You can draw an infinite number of circles that do not touch and you can do that on any size paper that you like. Think about concentric circles (circles which have the same center).
 
I'd do it for 2 circles first. Then three. See if you notice a pattern.
 
Hypatia001 said:
Hi.

I actually drew them out.

The question asks for different ways without the circles touching or overlapping... and there are no labels or sizes. So i assumed that the cirxles could all be of different sizes.

What i am really after is the maths behid it so i can apply it to any number of circles given the same rules to follow
Click to expand...
Hypatia001 said:
How many different ways can 4 circles be drawn without overlapping or touching ??

How many ways can they be drawn if they are able to touch or overlap?
Click to expand...
Only rule you stated is that there are 4 circles and

they are drawn "without the circles touching or overlapping"

If those are the only rules then the number ways could be ∞

and I do not know of any "mathematical" rule for that!
 
Hypatia001 said:
The question asks for different ways without the circles touching or overlapping... and there are no labels or sizes. So i assumed that the cirxles could all be of different sizes.

What i am really after is the maths behind it so i can apply it to any number of circles given the same rules to follow. I do not want the answer but the reason for the answer.
Click to expand...
Part of the trouble here is that it is not clear what "different ways" are. That depends on context, possibly stated in the problem itself, if you haven't quoted it fully, or perhaps implied by what topic you are studying. That's why we ask for such information:

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My guess is that it has to do with which are inside which others. But nothing you've said makes that clear. Show us your picture, at least!
 
This is the exact question...

how many essentially distinct ways can four circles be drawn without overlapping or touching?
 
Pick a point which will be the center of all 4 circles you will draw.
Pick a length, say 1 inch. Draw a circle of radius 1 inch.
Take half of 1 inch which is 1/2 inches. Draw a circle whose radius is 1/2 inch.
Take half of 1/2 inch which is 1/4 inch. Draw a circle whose radius is 1/4 inch.
Take half of 1/4 inch which is 1/8 inch. Draw a circle whose radius is 1/ inch.
There are your 4 circles.
My rule was to start with a circle of radius 1 and then draw 3 more circles where the radii was 1/2 the previous circle. I can change that number 1/2 to any positive number I like (except 1) and get 4 circles that do not touch. Now how many positive numbers different from 1 are there?

Alternative draw a circle whose radius is 1 inch. Go enough to the right of the center for the 1st circle so you can use that point as the center of the 2nd circle of radius 1 so it does not touch the 1st circle. Keep drawing circles to the right of the previous one so that it does not touch any other circle drawn. Now change the radius of the circle to any other positive value you want.

Just understand that with an infinite size piece of paper (usually called a plane) there is always more space for another circle that does not intersect any another circle! Let the 1st 4 be your 1st example of 4 circles that do not intersect. Let the 2nd 4 be your 2nd example of 4 circles that do not intersect. ...
 
Hypatia001 said:
This is the exact question...

how many essentially distinct ways can four circles be drawn without overlapping or touching?
Click to expand...


Numberphile on YouTube has a discussion with diagrams that your question
is contained as a subset. Their topic is "How many ways can circles overlap?"
Their counts include circles not intersecting at all, circles crossing circles,
circles within circles, and combinations of those. They do not allow/count
circles touching at one point or three or more circles intersecting at one point.

Hypatia, for n = 3 circles, they show 4 ways where none of the circles are
intersecting each other. I know you are working on the case of four circles.
 
Hypatia001 said:
This is the exact question...

how many essentially distinct ways can four circles be drawn without overlapping or touching?
Click to expand...
And what is the context? What course is this for? What topic are you in?

It appears that this is not taking the circles to be distinguishable (e.g. four different colors or names), and I presume it is on a plane, not a torus. Exact locations clearly don't matter, so this is to be read topologically. But we really need to be sure.

Now, I'll tell you that I find 9 ways, by representing a configuration of non-intersecting circles as a tree. Does that sound reasonable in your context?
 
lookagain said:
Numberphile on YouTube has a discussion with diagrams that your question
is contained as a subset. Their topic is "How many ways can circles overlap?"
Their counts include circles not intersecting at all, circles crossing circles,
circles within circles, and combinations of those. They do not allow/count
circles touching at one point or three or more circles intersecting at one point.

Hypatia, for n = 3 circles, they show 4 ways where none of the circles are
intersecting each other. I know you are working on the case of four circles.
Click to expand...
Thank you so much
 
T
Dr.Peterson said:
And what is the context? What course is this for? What topic are you in?

It appears that this is not taking the circles to be distinguishable (e.g. four different colors or names), and I presume it is on a plane, not a torus. Exact locations clearly don't matter, so this is to be read topologically. But we really need to be sure.

Now, I'll tell you that I find 9 ways, by representing a configuration of non-intersecting circles as a tree. Does that sound reasonable in your context?
Click to expand...
It does ... but i want to understand how it works mathematically foolowing these rukes on a single plane.
 
Hypatia001 said:
It does ... but i want to understand how it works mathematically following these rules on a single plane.
Click to expand...

So please answer my questions! Show us your picture! Tell us what have you learned that you might be expected to use! This is very important when you ask for help -- even if you don't think these things will help us. Believe me, they will.

In particular, I mentioned trees in hope that you might tell me specifically what, if anything, you have learned about them, and also that you might try thinking that way. I don't know that it's the best way, but it's one among various ways to organize your thinking. And if it happens to be something you've learned recently, it might be a very good idea. Or maybe something else is ...

But the video strongly suggests that you are not expected to solve the problem allowing for touching or overlapping. Is that, or isn't it, part of your assignment?
 
That agrees with my picture; all you need is to convince yourself that you haven't missed anything, which depends on having some orderly way to list them. That's how I used trees (graph theory), just as a way to think about the cases.

There is no "formula" for the count, and I'm not sure you can even use any more basic combinatorial concepts to come up with a number. They certainly did nothing of the sort on the video.
 
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