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## Question

1. Simple problem.
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Two unit circles touch each other and a horizontal line segment.

What is the radius r of the largest circle that fits in the space bounded by these objects?
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2. Harder problem
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What is the greatest number of circles that can have each pair of circles tangent at a single point?
How many configurations are there?
Can you find a relationship among the radii of the circles?

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I suppose we're searching for a proof that this is really the largest number of circles possible in such a configuration (and that there are no other configurations with the same number)? Or is it just that these are the best solutions anyone has ever found?

The topological proof that 4 is max isn't that hard to state.

First, [my bad] the OP meant to restrict things to the plane, and the circles from intersecting.

Well, if two circles are tangent, they can't also intersect, so that point is covered.

OK. Begin by placing three circles [WOLOG they can be unit circles] so that they touch each other.

This separates the plane external to the circles into an included area and the area beyond the circles.

A mutually tangent fourth circle can be placed in either of these areas.

But not both, because a circle in either area cannot touch one in the other without intersecting the original circles.

And we can add only one in either area. Because, once added, it also isolates the tangency points, requiring intersections by any other circle.

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The topological proof that 4 is max isn't that hard to state.

First, [my bad] the OP meant to restrict things to the plane, and the circles from intersecting.

Well, if two circles are tangent, they can't also intersect, so that point is covered.

OK. Begin by placing three circles [WOLOG they can be unit circles] so that they touch each other.

This separates the plane external to the circles into an included area and the area beyond the circles.

A mutually tangent fourth circle can be placed in either of these areas.

But not both, because a circle in either area cannot touch one in the other without intersecting the original circles.

And we can add only one in either area. Because, once added, it also isolates the tangency points, requiring intersections by any other circle.

Nice. I was working on a really long laborious proof, but this is much neater. From fiddling around with these ideas though, it does seem to me intuitively as if 4 is the maximum possible irrespective of the number of dimensions allowed (although extra dimensions do allow for many more arrangements). I'm trying to work on a proof for this - don't know if anyone else will be interested in trying their hand at it - either finding an example of 5 in 3D, or a general proof for 4 being the maximum across all dimensions.

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Sorry for the third add-on. Was working on my own proof borrowing from your approach, when I realised that we can't actually approach it this way. We're talking about a new circle being tangential to the existing circles, so there is no requirement at all that the two points where the new circle touches the existing circles lie on a tangential line to the existing circles. For example, think of two tangential circles in the 2D space. To add a third tangential circle to this group (still in 2D), the new circle does not touch the existing circles on a tangent line.

If you have two tangent circles on a plane and rotate one along the axis going through the centers of both circles some arbitrary angle (so it is now only intersects the plane at two points), would they still be considered tangent?

My proof assumes that to be tangent two circles must have the same tangent line at the point where they touch (regardless of which dimensions they pass through). If they just touch (approaching the point from different angles) I don't believe those are tangent.

I would change the first part of this sentence in the proof though...

"[For this line to be tangent with both existing circles], it would need to be the tangent line between the two existing circles."

to [since a circle on this plane could only be tangent to space S at one point and in the direction of this line]

For example, think of two tangential circles in the 2D space. To add a third tangential circle to this group (still in 2D), the new circle does not touch the existing circles on a tangent line.

But where any two circles do touch, they have the same tangent line at that point. The restriction to a line was because that was the only place that the plane containing the new circle would intersect the current 3d space (and it could have been explained better... fixed above though).

If this definition/restriction on tangency in 3d does not work for you, there's a simple example of an infinite number of circles being "tangent" to each other. In fact, they are all "tangent" to each other at two points each. Simply choose unique circles (avoiding the tangency points of the circles already chosen) of radius r from a sphere of radius r.

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My proof assumes that to be tangent two circles must have the same tangent line at the point where they touch (regardless of which dimensions they pass through). If they just touch (approaching the point from different angles) I don't believe those are tangent.

Actually, if they touch from different angles, they will in fact share a tangent line. It's just that this tangent line will not be in the same plane as either of the two circles. I believe that what you're saying is that in order to be considered tangential, they must share a tangent line that is coplanar with both circles.

Personally, I don't think this needs to be the case. Take for example two tangential circles in 2D space. These have three tangential lines - the line through the tangential point, and two other lines perpendicular to this line, touching the "top" and "bottom" of the circles. Now in 3D, you are saying that if we rotated the circles so that the common tangential line remained tangential to both circles, then that should now be considered the only tangent to both circles. The other lines touching both circles are not tangential lines as they are not coplanar to both the circles. I personally don't think this is a natural restriction.

However, I tried to Google an exact definition of tangent in 3D space, but couldn't find anything that resolved the issue, so I suppose either of our interpretations could be valid to some extent. (I would point out though that my interpretation does not lead to the "sphere" solution, as in that case, the circles touch at two points. A tangent touches at one and only one point. If it touches at two points, it's an intersection).

For your definition of tangents, yes it does appear as if you've got a working proof of how adding the 4th dimension and above would not add anything to a 3D solution.

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For your definition of tangents, ...

FWIW I've never seen a definition of tangent outside the plane.

Tangent is one of the line-like properties [along with radius, diameter, secant, chord] of the circle. What distinguishes tangent from the others [except radius, I guess] is the single contact point. Well, if you go outside the plane, there's no need to make such a distinction. You can't have a radius, diameter, chord or secant outside the plane. So tangent needs a uniqueness statement [definition] only in the plane. In 3 dimensions there is only a single type of line that touches a circle. Although there are two types of lines outside the plane that don't touch a circle.

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Nice. I was working on ... finding an example of 5 in 3D, or ...

See

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See

I couldn't directly translate the sphere solutions to a circle solution...

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So tangent needs a uniqueness statement [definition] only in the plane. In 3 dimensions there is only a single type of line that touches a circle.

Yes, I take your point about the line, but a "tangential circle" will still need a definition. In my interpretation, it was two circles that touch but do not intersect, which defines it independently of any reference to a tangent line.

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