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In the geometry of circle packings in the
Euclidean plane In mathematics, a Euclidean plane is a Euclidean space of Two-dimensional space, dimension two, denoted \textbf^2 or \mathbb^2. It is a geometric space in which two real numbers are required to determine the position (geometry), position of eac ...
, the ring lemma gives a
lower bound In mathematics, particularly in order theory, an upper bound or majorant of a subset of some preordered set is an element of that is every element of . Dually, a lower bound or minorant of is defined to be an element of that is less th ...
on the sizes of adjacent circles in a circle packing.


Statement

The lemma states: Let n be any integer greater than or equal to three. Suppose that the unit circle is surrounded by a ring of n interior-disjoint circles, all tangent to it, with consecutive circles in the ring tangent to each other. Then the minimum radius of any circle in the ring is at least the
unit fraction A unit fraction is a positive fraction with one as its numerator, 1/. It is the multiplicative inverse (reciprocal) of the denominator of the fraction, which must be a positive natural number. Examples are 1/1, 1/2, 1/3, 1/4, 1/5, etc. When a ...
\frac where F_i is the ith
Fibonacci number In mathematics, the Fibonacci sequence is a Integer sequence, sequence in which each element is the sum of the two elements that precede it. Numbers that are part of the Fibonacci sequence are known as Fibonacci numbers, commonly denoted . Many w ...
. The sequence of minimum radii, from n=3, begins Generalizations to three-dimensional space are also known.


Construction

An infinite sequence of circles can be constructed, containing rings for each n that exactly meet the bound of the ring lemma, showing that it is tight. The construction allows halfplanes to be considered as degenerate circles with infinite radius, and includes additional tangencies between the circles beyond those required in the statement of the lemma. It begins by sandwiching the unit circle between two parallel halfplanes; in the geometry of circles, these are considered to be tangent to each other at the
point at infinity In geometry, a point at infinity or ideal point is an idealized limiting point at the "end" of each line. In the case of an affine plane (including the Euclidean plane), there is one ideal point for each pencil of parallel lines of the plane. Ad ...
. Each successive circle after these first two is tangent to the central unit circle and to the two most recently added circles; see the illustration for the first six circles (including the two halfplanes) constructed in this way. The first n circles of this construction form a ring, whose minimum radius can be calculated by
Descartes' theorem In geometry, Descartes' theorem states that for every four kissing, or mutually tangent circles, the radii of the circles satisfy a certain quadratic equation. By solving this equation, one can construct a fourth circle tangent to three given ...
to be the same as the radius specified in the ring lemma. This construction can be perturbed to a ring of n finite circles, without additional tangencies, whose minimum radius is arbitrarily close to this bound.


History

A version of the ring lemma with a weaker bound was first proven by
Burton Rodin Burton Rodin is an American mathematician known for his research in conformal mappings and Riemann surfaces. He is a professor emeritus at the University of California, San Diego. Education Rodin received a Ph.D. at the University of California, ...
and
Dennis Sullivan Dennis Parnell Sullivan (born February 12, 1941) is an American mathematician known for his work in algebraic topology, geometric topology, and dynamical systems. He holds the Albert Einstein Chair at the Graduate Center of the City University ...
as part of their proof of
William Thurston William Paul Thurston (October 30, 1946August 21, 2012) was an American mathematician. He was a pioneer in the field of low-dimensional topology and was awarded the Fields Medal in 1982 for his contributions to the study of 3-manifolds. Thurst ...
's conjecture that circle packings can be used to approximate
conformal map In mathematics, a conformal map is a function (mathematics), function that locally preserves angles, but not necessarily lengths. More formally, let U and V be open subsets of \mathbb^n. A function f:U\to V is called conformal (or angle-prese ...
s. Lowell Hansen gave a
recurrence relation In mathematics, a recurrence relation is an equation according to which the nth term of a sequence of numbers is equal to some combination of the previous terms. Often, only k previous terms of the sequence appear in the equation, for a parameter ...
for the tightest possible lower bound, and Dov Aharonov found a
closed-form expression In mathematics, an expression or equation is in closed form if it is formed with constants, variables, and a set of functions considered as ''basic'' and connected by arithmetic operations (, and integer powers) and function composition. ...
for the same bound.


Applications

Beyond its original application to conformal mapping, the circle packing theorem and the ring lemma play key roles in a proof by Keszegh, Pach, and Pálvölgyi that
planar graph In graph theory, a planar graph is a graph (discrete mathematics), graph that can be graph embedding, embedded in the plane (geometry), plane, i.e., it can be drawn on the plane in such a way that its edges intersect only at their endpoints. ...
s of bounded degree can be drawn with bounded
slope number In graph drawing and geometric graph theory, the slope number of a graph is the minimum possible number of distinct slopes of edges in a drawing of the graph in which vertices are represented as points in the Euclidean plane and edges are represe ...
.


References

{{reflist, refs= {{citation , last = Aharonov , first = Dov , doi = 10.1080/17476939708815009 , issue = 1–4 , journal = Complex Variables , mr = 1624890 , pages = 27–31 , title = The sharp constant in the ring lemma , volume = 33 , year = 1997 {{citation , last1 = Aharonov , first1 = D. , last2 = Stephenson , first2 = K. , issue = 3 , journal = Algebra i Analiz , mr = 1466797 , pages = 104–140 , title = Geometric sequences of discs in the Apollonian packing , url = https://mi.mathnet.ru/aa782 , volume = 9 , year = 1997 {{citation , last = Hansen , first = Lowell J. , doi = 10.1080/17476938808814284 , issue = 1 , journal = Complex Variables , mr = 946096 , pages = 23–30 , title = On the Rodin and Sullivan ring lemma , volume = 10 , year = 1988 {{citation , last1 = Keszegh , first1 = Balázs , last2 = Pach , first2 = János , author2-link = János Pach , last3 = Pálvölgyi , first3 = Dömötör , editor1-last = Brandes , editor1-first = Ulrik , editor1-link = Ulrik Brandes , editor2-last = Cornelsen , editor2-first = Sabine , contribution = Drawing planar graphs of bounded degree with few slopes , doi = 10.1007/978-3-642-18469-7_27 , location = Heidelberg , mr = 2781274 , pages = 293–304 , publisher = Springer , series = Lecture Notes in Computer Science , title = Graph Drawing: 18th International Symposium, GD 2010, Konstanz, Germany, September 21-24, 2010, Revised Selected Papers , volume = 6502 , year = 2011, arxiv = 1009.1315, isbn = 978-3-642-18468-0 {{citation , last1 = Rodin , first1 = Burt , author1-link = Burton Rodin , last2 = Sullivan , first2 = Dennis , author2-link = Dennis Sullivan , issue = 2 , journal = Journal of Differential Geometry , mr = 906396 , pages = 349–360 , title = The convergence of circle packings to the Riemann mapping , url = https://projecteuclid.org/euclid.jdg/1214441375 , volume = 26 , year = 1987, doi = 10.4310/jdg/1214441375 , doi-access = free {{citation , last = Stephenson , first = Kenneth , isbn = 978-0-521-82356-2 , mr = 2131318 , publisher = Cambridge University Press , title = Introduction to Circle Packing: The Theory of Discrete Analytic Functions , title-link = Introduction to Circle Packing , year = 2005; see especially Lemma 8.2 (Ring Lemma)
pp. 73–74
and Appendix B, The Ring Lemma
pp. 318–321
{{citation , last = Vasilis , first = Jonatan , doi = 10.1007/s10711-010-9545-0 , journal = Geometriae Dedicata , mr = 2795235 , pages = 51–62 , title = The ring lemma in three dimensions , volume = 152 , year = 2011, s2cid = 120113578 Circle packing Lemmas Fibonacci numbers Geometric inequalities