In the geometry of circle packings in the

Euclidean plane In mathematics, the Euclidean plane is a Euclidean space of dimension two. That is, a geometric setting in which two real quantities are required to determine the position of each point ( element of the plane), which includes affine notions of ...

, 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 greater than or equal to every element of . Dually, a lower bound or minorant of is defined to be an element ...

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 rational number written as a fraction where the numerator is one and the denominator is a positive integer. A unit fraction is therefore the reciprocal of a positive integer, 1/''n''. Examples are 1/1, 1/2, 1/3, 1/4, 1/5, et ...

\frac

where

F_i

is the

i

Fibonacci number In mathematics, the Fibonacci numbers, commonly denoted , form a sequence, the Fibonacci sequence, in which each number is the sum of the two preceding ones. The sequence commonly starts from 0 and 1, although some authors start the sequence from ...

. 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 Degeneracy, degenerate, or degeneration may refer to: Arts and entertainment * Degenerate (album), ''Degenerate'' (album), a 2010 album by the British band Trigger the Bloodshed * Degenerate art, a term adopted in the 1920s by the Nazi Party i ...

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. Adj ...

. 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, mu ...

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 City University of New York Graduate Ce ...

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. Thurston ...

's conjecture that circle packings can be used to approximate

conformal map In mathematics, a conformal map is a 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-preserving) at a point u_0\in ...

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, a closed-form expression is a mathematical expression that uses a finite number of standard operations. It may contain constants, variables, certain well-known operations (e.g., + − × ÷), and functions (e.g., ''n''th roo ...

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 that can be embedded in the plane, i.e., it can be drawn on the plane in such a way that its edges intersect only at their endpoints. In other words, it can be drawn in such a way that no edges cross ...

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 {{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 {{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