COUNTING CIRCLES.
DRAFT COPY ONLY.
6/27/2009.
© Raimond A. Struble.

Professor Emeritus
Department of Mathematics
North Carolina State University at Raleigh
Raleigh, NC.
http://www.infiniteproduct.info/strucncl.htm


Send comments and correspondence to: raimondstruble@yahoo.com

Mathematics Review Subject Classification Number: 51M04.

ABSTRACT.

We introduce a simple scheme of counting the greatest number of touching circles lying between two concentric circles. This counting scheme leads to an approximate characterization of the circumstances when many touching circles close up a ring between two concentric circles. The counting of touching circles, which do not close a ring, are employed to obtain the digits of π.

A. Consider two concentric circles of diameters D and D+4. Within the ring determined by these two concentric circles, one can place unit circles of diameter 2, which touch the two concentric circles. We are interested in counting the greatest number N of such, non-intersecting, touching circles lying within the ring. (The touching circles may touch one another.)

It is clear in illustrative Figure 1,
5211.
Figure 1. D=17.5.
that the diameters of the touching circles approximate (they are less, but not by much) the arclength segments along the median concentric circle of diameter D+2. Therefore, the greatest number of non-intersecting, touching circles N, can be expected to be the greatest integer in the circumference of the median circle, divided by the diameter of the touching circles:
½(D+2)π = 9.75π = 30.63...  .
For D=4, with ½(D+2)π = 9.424776..., the construction is somewhat less demanding, graphically,
5202.
Figure 2. D=4.
and certainly validates the formula
(1) N = GII ½(D+2)π = GII(D/2+1)π.
Here, GII denotes the greatest integer in.

For D=1, with ½(D+2)π = 4.712388...,
5203.
Figure 3. D=1.
the formula remains valid, but clearly fails for D=0, which results in 3.141592... instead of N=2. The formula also fails for D smaller than 1/5.

B. The quantity (D+2)π is the circumference C of the median circle, which contains the centers of the unit circles. A graph of C/2 = (D/2+1)π versus D,
5224.
Figure 4.
indicates the application of formula (1).

Indicated in this figure are the particular cases D=4 and D=1, corresponding to Figures 2 and 3. Also, indicated is the case D=½, corresponding to the following figure, where (D/2+1)π = 3.92699...  .
5205.
Figure 5. D=½.
Specified in Figure 4 is the well-known case D = D6 = 2, when 6 touching circles fill up a ring (Figure 7). This case is not typical of filling, touching circles. For as is clear in Figures 2, 3 and 5, in order that the displayed numbers of unit circles close up and completely fill a ring, the diameters of the concentric circles must decrease. For example, the value D4, when 4 unit circles fill a ring, lies to the left of 1, but to the right of the abscissa to 4 on the c/2-line. Similarly, the value D9, when 9 unit circles fill a ring, lies to the left of 4, but to the right of the abscissa to 9 on the c/2-line.

Upon re-examining Figure 1, we observe that the block of 5 unit circles subtends 60o of arc of the (D+2)-circle. Hence, 30 unit circles exactly fill the ring. The correct value of D in this figure is not 17.5, but the exact value given in Equation (4) (Section D) when N=30. The labeling in Figure 1 is slightly in error, but when D=17.5, there are 30 unit circles in the ring, with space left over which, however, is insufficient for the inclusion of another unit circle.

C. The formula
(2) N = GII (D/2+1)π,
which gives the greatest possible number of (non-intersecting) unit touching circles, lying within the ring between the two concentric circles of diameters D and D+4, can be used to obtain the digits of π. For with D/2 = 10k-1, the number N=3141592... = GII 10k π is that whole number displaying the first k+1 digits of π. Thus, by just counting the greatest number of touching circles (touching one another) that can be placed in a ring between two concentric circles of radii 10k-1 and 10k+1, one obtains the first k+1 digits of π. Not a very practical scheme for obtaining the digits of π. For in using this scheme, when many digits of π are desired, one must count many touching circles along large circles. This counting task can be aided somewhat by our re-examination of Figure 1. If, for example, only a 1/6th portion of the total count is made, then the total count is six times that 1/6th count. In any event, one must choose a count of only whole numbers of touching circles. In order to obtain three digits of π, one must count 314 touching circles of diameter 2 along a circle of diameter 100. A 1/6th portion of the total count is approximately 52, and 52 touching circles span approximately a 1/6th portion of the large circle. Noting the exact portion of the large circle spanned by 52 touching circles, and increasing 52 consistent with that exact portion, will yield the correct total count of 314, when the increased value is multiplied by 6.

D. If d is the (general) diameter of touching circles, in place of 2, then Equation (2) becomes
(3) N = GII(D/d+1)π,
which gives the greatest number of touching circles of diameter d lying within two concentric circles of diameter D and D+2d. The exact relationship, involving two geometric diameters and integer N, of a completely filled ring of touching circles of diameter d, lying within two concentric circles of diameters DN and DN+d, is given by Ref. 1 page 125 (using our notation)
(4) DN/d = (1-sinπ/N)/sinπ/N.
Of necessity, N here is the greatest number of touching circles in the ring, and is a particular instance of Equation (3). Therefore, N = (DN/d+1)π, and
(5) 1 = (DN+d)π/dN.
So,


(6) 1 = CN/dN,
where CN is the circumference of the median circle, when N touching circles of diameter d exactly fill the ring. This last form illustrates the approximation claim (less, but not by much), relative to Figure 1, in Section A. We appear to have come full circle, so to speak!

E. In order to count the number N of circles, precisely filling a ring, one needs to invert the relationship in Equation (4). For this task, we introduce the useful approximation to Equation (4):
(7) DN/d ≃ N/π + π/6N - 1,
which is accurate to 1% (except for N=3) and to 1/100% for N > 30. This accuracy follows from the similarity of the two power series expansions
(8) sinπ/N = (π/N) - (1/6)(π/N)3 + ... and

(9) (π/N) / (1 + (π/N)2/6) = (π/N) - (1/6)(π/N)3 - ... .
We note that (9) is always less than (8). Solving the quadratic Equation (7) for N, and noting that N must be an integer, the equation
(10) N = NITπ/2{(DN/d + 1) + [(DN/d + 1)2 - 2/3]½}
certainly inverts (4). Here, NIT denotes the nearest integer to. For "large" values of (DN/d + 1), (10) is approximated by N = NITπ/sin(π/N), which actually holds for N > 4. The quantity
π/2{(DN/d + 1) + [(DN/d + 1)2 - 2/3]½}
is slightly less than an integer, but for N > 31, (10) is indistinguishable from
(11) N = GIIπ(D/d + 1)
where D=DN. Since GIIπ(D/d+1) is the greatest number of touching circles in a ring, by counting this number when it exceeds 30, the count is the actual number of touching circles which close up an indistinguishable (slightly smaller) neighboring ring (if need be). The neighboring ring is readily ascertained from Equation (7), with great accuracy. Equation (7) leads to a very useful trigonometric approximation
(12) sin θ ≃ θ/(1+θ3/6).


F. It might be of interest to note here, the number
(13) n = D/d + 1
(when an integer) of touching, touching circles whose centers span a subarc along the (D+d) circle. When D=d, then (13) produces three mutually tangent circles of equal diameter.
5206.
Figure 6. n=2.
When D=d, then N=6 mutually tangent, touching circles enclose the center circle of Figure 6.
5207.
Figure 7. N=6.
When D/d is a somewhat larger integer, then (13) produces interesting gears for gear-chains along the primary circle.
5208.
Figure 8. n=6.

5209.
Figure 9. n=24.
When D/d is very large, the gears become a knurling of the primary circle.
5210.
Figure 10. n large.
As n increases indefinitely, the sum, nd, of the diameters of the knurling circles tend to the arclength along the primary circle.

ACKNOWLEDGEMENTS.

I wish to thank G. William Moore, MD, PhD, for reviewing and formatting the manuscript, and G. Vincent Moore for providing the graphic art. I further wish to thank my wife of more than sixty-three years, Marilyn Struble, who rekindled in me a dormant enthusiasm in mathematics, after a sixteen-year hiatus. She had the insight to give me stimulating books on this and related topics in mathematics.

REFERENCES.



1. Coxeter HSM, Greitzer SL.
Geometry Revisited.
New Mathematical Library.
Washington, DC: Math Assn America. 1967;:.
ISBN: 0883856190, 207 pages.

2. Honsberger R.
Episodes in Nineteenth and Twentieth Century Euclidean Geometry.
New Mathematical Library. Washington DC: Math Assn America. 1996. Second printing. 2005;:.
ISBN: 0883856395, 174 pages.



Raimond A. Struble
P. O. Box 50376
Raleigh, NC 27650

Cell: 919-538-1340

email: raimondstruble@yahoo.com
email copy: George.Moore4@va.gov

June 27, 2009

Prof. Paul Yiu
Department of Mathematics
Florida Atlantic University
Boca Raton, FL 33431-0991

email: yiu@fau.edu
forumgeom@fau.edu

Dear Prof. Yiu:

Attached herewith is a copy of the manuscript, entitled "Counting Circles," which I am submitting for publication in Forum Geometricorum. This manuscript has been prepared as a Microsoft® Word .doc file. Professional-quality Figures 1-10 are available separately as .jpg files, if needed.

This manuscript has not been published, and is not under consideration in any other journal.

Thank you for your consideration.

Sincerely yours,

Raimond A. Struble, Professor Emeritus
Department of Mathematics
North Carolina State University at Raleigh


Last updated: 6/27/2009, by Raimond A. Struble, PhD.