After much fiddling with which radii intersect which bits of the circle clusters, and which opposite or adjacent ate what with whom, I eventually noticed that the solution to the issue of finding the dimensions in clusters of same-radius tangent circles that fit within a larger circle of known radius is to think of the polygon associated with the smaller circles as a polygon of two times as many sides as there are circle clusters.In other words, if you look at the first figure in this post there, you see five clusters of three smaller circles contained within a larger circle. The turquoise polygon linking the centers of the inner-most circles is a decagon. Near the end of the previous post I had been attempting to find the radius of the smaller circles by considering a pentagon, in this particular case, linking the centers of the circle clusters. I imagine there's a way to do it, but it is rather more direct, I found, considering a polygon with twice the number of sides as there are circle clusters.
Still, starting from the simpler...ish case of an arrangement of n clusters of 3 circles (as opposed to n clusters of n circles), I may set the defined value as the radius of the larger circle, the one that encloses every last cluster, without any arc of that larger circle passing inside the region encircled by any of the smaller circles...*pant*
I suppose it is referred to as geometrization when one defines one constant as one instead of the more actually applicable constant that could, in practice, end up being complicated, and thereby making all the rest of the mathing complicated. It is done with the field equations for Einstein's General Relativity. It is done with the unit circle so fondly sited in trigonometry text books. The radius of the smallest circle to enclose the circle clusters can be set to one, and from that you can derive to equations for the dimensions of the circles it contains. Just gotta set one side of the equation to one, and solve for whatever.
...You also may need to figure out how to connect the formulary dots from the equations for dimensions of line segments in an n-gon to the dimensions of the line segements of n n-gons contained exactly half of the radius of the smaller circles inside the smallest wholely continaing circle with a radius of r1(the inner-most n-gon's inscribed circle's radius) plus r2 + R2 of the smaller n-gons (radius of the smaller, outer-more n-gon's circumscribed circle's, plus the radius of the circle the aforementioned n-gon is inscribed within...IF the recently aforementioned n-gon has an odd number of sides), plus a/2, where a is the length of every line segment to make up the n-gons inside the largest circle...
The problem is more generalizable though. The collection of the innermost circles--and these are, every one of them, circles of the same radius--likely remain numbered two per outer cluster, and each "cluster" corresponds to a polygon of n sides.
The simpler generalization of the thing is in the R and the r--R senior and junior. If the outer polygons are corresponding to an odd number of circles, then the critical sum in their case equals R + r + a/2, a/2 having already been discussed...
Since the inner polygon and the polygons implied by the circle clusters have the same side-length, I divided the known value, the radius of the largest circle, 1, into a sum of...
r1 - inscribed radius of the inner polygon
r2 - inscribed radius of the outer polygons
R2 - circumscribed radius of the outer polygons
a/2 - half the length of a side of any of the aforementioned polygons
I then substitute in those regular polygon equations that include the side length variable, a. For r2, and R2, the angle within the trigonometric function would be pi/3, as they refer to the radii of a regular triangle. Therefore, what I substitute in for these is...
r2 = (a * cot(pi/3))/2
R2 = (a * csc(pi/3))/2
r1 is the variable I could use to generalize at least the cases where the outermost polygons we're concerned with are triangles, and all the circles the same radius, and arranged as I have arranged them, with one circle pointing outward from each. I just leave the variable, n, which equates to the number of sides within the innermost polygon (two times the number of three-circle clusters there are) within the trigonometric function when substituting r1 with it's equation that includes a. Thus...actually, I suppose it may be more consistent to have a new variable, N, which is defined as two times the number of circle clusters, explicitly. Thus...
r1 = (a * cot(pi/N))/2 , where N = 2 * [number of circle clusters]
The complete equation in this case would be...
which, solving for a works out to be...
a =
...formatting these things is a mild pain, but anyhow: testing...
According to this equation, the length of any polygon side in the figure at the beginning of the post should beee...2/(1+sqrt(3)+sqrt( 5 + 2 * sqrt(5))), approximately 0.344249819... Checking the length of a in the figure I drew in autocad I get 0.34424982, which is an even better match than I expected, really, given the tiny discrepancies I had encountered in working these things into figures in autocad. Understandable given the fact I learned to use autocad at all by fiddling with it, and the framework itself is given to various of the approximation errors of a digital system.
This equation would, of course, not work with similar clusters of circles beyond or bellow three, nor do I see a way of easily generalizing the idea of what I've been doing thus far. For example, a cluster of four circles would not fit the generalization exactly. For one thing, in the framework I used, I used r and R, the inscribed and circumscribed radii, of the triangles, and the way a polygon with an odd number of sides works those two radii line up with the center of the outer-most circle, and from there the outermost circle is as far from the largest circle's edge by the length of it's radius.
With a cluster of four circles, whether there are two of it's circles arranged along the inner polygon, or one of it's circles, the radii going up it's center would be two times it's inscribed radius, or circumscribed radius, not a sum of both. Additionally, if two of it's circles are along the inner side, the line going between it's outer-most circles from where the radii end would be more than 'a'.
The set up of the equation I came up with I will consider a special case of a more generalized set up that includes contengencies for clusters of any number of circles rotated any applicable angle about it's local center. With all the applicable values plugged into this as of yet undefined equation many values would cancel out, leaving you with what I worked out...perhaps not directly...anyhow, more later.
