Circle of Circles

Wade

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Mar 7, 2013
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10
Hi,

I'm trying to arrange a set of 9 circles into the largest circle I can.

Think of a necklace of circles. I'm trying to find the diameter of the largest circle that can be made from a set of circles.

The diameters of the individual circles are as follows: 18, 15,14,10,9,8,7,4, and 1.

I'd like to know if there is some kind of formula I can use to calculate this.. even though the number of individual circles, and their dimensions may vary.

I'm trying to make a sculpture, and having this information would be great.

Thanks

Wade
 
If you are thinking that drawing seven tangent circles, with their centers on a straight line, of length the sum of the diameters of the circles, then "bending" the line to make of circle with circumference that same length, that won't work. The circles will no longer be tangent at the same point and will overlap.
 
Hi,

I'm trying to arrange a set of 9 circles into the largest circle I can.

Think of a necklace of circles. I'm trying to find the diameter of the largest circle that can be made from a set of circles.

The diameters of the individual circles are as follows: 18, 15,14,10,9,8,7,4, and 1.

I'd like to know if there is some kind of formula I can use to calculate this.. even though the number of individual circles, and their dimensions may vary.

I'm trying to make a sculpture, and having this information would be great.

Thanks

Wade

Wade,

I don't find the situation clear. Are the diameters of the smaller circles to lie on the circumference
of the largest circle? Or, are all of smaller circles inside of the largest circle and internally
tangent to the largest circle?

And, I am asking these questions regardless of whether or not any of the smaller circles
are allowed to intersect with each other. For an sculpture, for instance, you could have
intersecting rings (akin to circles) or not.
 
Circles around a circle

Hi,

Thanks for the replies.
I'm trying to make a circle of circles.

I'm trying to start with the individual circles, and calculate the size of the largest circle they can make together.

Like this.

circle of circles.jpg

I hope this helps.

Thanks
 
NOTE to Subhotosh: would you please "space" my previous post. I've started to use Windows8 (another major bad decision I've made!) and the **** "return key" refuses to work; what's odd is it decides to do this only at this site. I'll appreciate any suggestion to fix this...may as well "space" this one while you're at it!!
The return key does not work for me either, and I am using Windows7.
 
Thanks Denis.

This really helps and is moving everything in the right direction.

One fine point is that I'm trying to draw one continuous inner circle that connects the outer smaller circles together.
As the drawing shows, connecting the centers of the outer circles to the boundary of the inner circle, won't allow me to generate one continuous circle that is unbroken, like I'd like to have.
This image shows the problem I'm talking about.

innner outer circles.jpg

The point where the two outer circles meet, (inside the light blue square) is not on the red inner circle.
So if I were trying to draw one continuous and unbroken inner circle connecting the outer circles, I'm hoping this image shows why the solution you proposed would not produce the inner circle I'm trying to get to. A small difference I know, but significant to me.

I hope I'm making myself clear, but fear I'm not.

TIA

Wade




Hmmm...Wade, that's a bit different than what we (me, anyway) thought:
I thought you wanted the design such that the outside circles were
ON the circumference of the larger inside circle: so a "nice circle"
went around the wearer's neck! You did call this a necklace, right?
Problem analysis (IC = Inner Circle; R = IC's Radius; IC's center = M):
- make centers of the circles go around the IC in increasing radius size
(will give same results since we're trying to calculate R)
- draw chords joining the circles' centers
(this will create an N-GON, N = number of circles)
(length of any chord will be the sum of radius of 2 consecutive circles)
- join M to each of circles' centers
(this will create N isosceles triangles, equal sides = R)
(the other side will equal the corresponding chord)
EXAMPLE.
5 circles, radius 1, 1.5, 1.75, 2, 2.25
(I used those because they "kinda" fit (not exactly) an IC with R = 3)
So the chords (or side lengths of the 5-GON) are:
1+1.5 = 2.5, 1.5+1.75 = 3.25, 1.75+2 = 3.75, 2+2.25 = 4.25, 2.25+1 = 3.25
Calculating R will involve calculating the 5-GON's area in terms of R.
I'll let someone else (like Jeff who's raring to go!) show you how!!
 
Thanks Denis.


View attachment 2680

The point where the two outer circles meet, (inside the light blue square) is not on the red inner circle.

Hi Wade,

I'll just throw in my tuppence worth at this stage. The tangent (the point where the two circles touch) is on a straight line by definition

circles1_zpsb760aab2.jpg


Since you cannot have a curve on a straight line, this particular aspect of what you seek is not possible.

Here are 5 circles of equal radius, the pentagon is made up by joining the centres and passes through the tangent points.

pentagon_zpsb6795b52.jpg


As for a way of calculating the largest circle, I'm still working on it :idea:

btw JeffM, I'm on Win 7 and return

is

working



fine :)
 
Last edited:
Another image

Hi, the circle I'm interested in, is the one that connects the others continuously. So, given the pentagram example, there is a circle that connects them all. See the dark circle in the illustration below. So, that's the one I'm trying to calculate. Given a regular polygon, or a series of circles the same size, I don't think the problem is difficult at all. But when the circles are all different in size, it makes it more complicated , because the outer circles center points will not lie in a circle, only their tangent points will. I'll post another pic shortly.
 

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Tangential circle

Hi

Here is what I'm talking about, in a simplified form with four circles of different radii.
 

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Another arrangement, in answer to Denis's question about there being only one solution. Here are the same four circles as in the previous pic showing that there placement can vary, which can vary the radi of the inner circle that connects the tangents.
 

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Approximate solution

Hi,

so, back to the original question, given circles of the following diameter dimensions, 1,4,7,8,9,10,14,15,and 18
the radius of the circle (inner circle) that connects the tangents is approx 12.68. I'm getting this solution just by experiment,geometrically drawing and varying circles. I do think Denis is correct in that there is only one solution to the problem. But I would like to know if there is a way to derive this answer of 12.68 (approx)
from the dimensions of the original circle set. And How approx is my solution? 12.68 is me working it out by drawing, so it is an approximation. Is 12.7 or 12.65 really closer to a calculated solution?
 
image.jpg


Hi,

so, back to the original question, given circles of the following diameter dimensions, 1,4,7,8,9,10,14,15,and 18
the radius of the circle (inner circle) that connects the tangents is approx 12.68. I'm getting this solution just by experiment,geometrically drawing and varying circles. I do think Denis is correct in that there is only one solution to the problem. But I would like to know if there is a way to derive this answer of 12.68 (approx)
from the dimensions of the original circle set. And How approx is my solution? 12.68 is me working it out by drawing, so it is an approximation. Is 12.7 or 12.65 really closer to a calculated solution?
 
Hi Wade,

Having failed to solve this problem I hunted further afield.

What you are trying to achieve can only be done by trial and error (as you have been doing).

A mathematical solution only exists for 4 circles around a central circle.

For 5 or more outer circles (not all same radii), to calculate the (green) radius of the central circle you need to know the area of the (red) polygon below. To calculate the area of the polygon you need to know the radius. So I'm afraid you're left going round in circles :p

You can find more information by searching for tangential polygons and tangential quadrilaterals.

circles2_zps10527fe8.jpg
 
Thamks for everything.

Thanks everyone for participating in this thread.

Ive got more info and know that The problem is a challenging one. Trial and error seems to work pretty well. I'll post some shots of the sculpture after its done. Also, I thought you might like to know I thought of this problem, after investigating squared squares, and rectangles. If you are unfamiliar with this idea, check out the web site http://squaring.net

thanks again all.
 
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