Circles in geometric progression, tangency problem II?

gianlino, see link for a simulated copy of the graphic you provided in the previous question. This is the reason why we shouldn't let our stealth planes or other technology fall into Chinese hands---reverse engineering. "This problem...when solved...becomes simple".

Give me some more time to study this and add more comments later. What I find totally remarkable about this is that there are only exactly 6 degrees of freedom present in a system of 6 equations, and yet there is a very distinctive and non-trivial solution. At least this one!

Edit: Yes, there's obviously a dual family of circles, as each is tangent to 6 others. No, I haven't worked out the "algebra" behind this one. Why is it surprising to me? Because if you have just 6 variables to solve for in a family of 6 equations, there's only going to be a limited number of solutions (just one if the equations are linear). Most of them are going to be uninteresting trivial cases in cases where there's too much symmetry present. I should ask you, why should I even expect anything other than a trivial solution can exist? Moreover, if a non-trivial solution does exist, there's probably a lot more to be said about those non-trivial solutions. Probably related to other self-similar arrangements, such as the one just posted by Vikram P, see 2nd link, which I think is what you referred to in your first posted question, when you said that the circle tangent to 6 others can't itself be a part of the sequence.

How many other non-trivial solutions are there for this one? We know that there's two, the dual families of circles. Sunflower seeds and Fibonacci numbers come into mind, I wonder if there's a connection here? Well, it'd be fun if there was.

Edit 2: gianlino, as Vikram P pointed out in his posted question, this matter of self-similar sequence of tangent circles is related to the Riemann sphere, which makes this all the more interesting. I'm sure all of this, including Coxter's Loxodrromic sequence, are ultimately related. Probably worth a book by itself. But what I love about this is that this is a great example of how surprising and complex solutions can arise in situations where you wouldn't have expected it. As much as I like doing math problems in Y!A, I've actually spent most of my years thinking about physics, and this just resonates with me, because we see the same thing arising from "laws of physics". Anyway, I'm not done yet with this problem.

Edit 3: Well, "Loxodromic" should have given anyone a clue, as in loxodromes on a surface of a sphere. Map that onto a plane as per Riemann spheres to get your self-similar logarithmic spirals. Anyway, I've worked out much more precisely the variables to get close 6 circle tangents:

a = 0.688192

b = 1.84083

c = 114.8 degrees

r = a b^n

z = b^n Cos(c) + b^n Sin(c) i

R = 7.70656

X = 3.1039

Y = -3.47153

The gap errors are in the 10^(-6) range. See 3rd link for updated graphic. The angle is where the ratio of gap errors with the last 2 circles is close to 1, so that as one is reduced, so will the other be, proportionally. The "sweet spot".

Now I'll go look for any other solutions with such sweet spots.

Edit 4: Nope, there isn't anything close to approaching a solution where 3 non-overlapping circles are inside and 3 non-overlapping circles are outside the common circle tangent to them all. That 114.8 degree angle is certainly one sweet spot, a very unusual solution that just happens to exist. I will have to look for other arrangements, but obviously the dual family to the above is one of them.

Edit 5: See 4th link for graphic of the dual family of circles, note common circle tangent to 6 others. I'm not seeing any other solutions. There just might be another pair of dual families, but it will require more work and space that I have room in here for.

I'm not sure what you mean by that extra degree of freedom. What I see here are just 6 degrees of freedom, if the sequence of circles are to be self-similar. In any case, I think we've run out of time and room here, and my computer files on this are now enormous. I'll try to figure out what you mean by that, though. For just 6 degrees of freedom, you know that the number of solutions are finite, and most of them are going to be either trivial or imaginary (who says mathematicians aren't condescending, with such choice of words?).

Also, gianlino, it's this dual family of circles that have this wonderful property of being orthogonal to each other when expanded by a factor of √2. See 5th link for graphic. It is really nice. This isn't arbitrary, and I'm sure that somehow it's all tied in with everything else we've discussed, but, as I said, this would require a book or something. Thanks for bringing this to my attention!

Edit 6: gianlino, unfortunately, Y!A is now cutting me off. I can't even comment on what you just said, as much as I wanted to.