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OPTIMAL PACKINGS OF TWO TO FOUR EQUAL CIRCLES ON
ANY FLAT TORUS
MADELINE BRANDT, WILLIAM DICKINSON, ANNAVICTORIA ELLSWORTH,
JENNIFER KENKEL, AND HANSON SMITH
Abstract. We find explicit formulas for the radii and locations of the circles
in all the optimally dense packings of two, three or four equal circles on any flat
torus, defined to be the quotient of the Euclidean plane by the lattice generated
by two independent vectors. We prove the optimality of the arrangements
using techniques from rigidity theory and topological graph theory.
1. Introduction and Main Result
We consider a problem from discrete geometry in the area of equal circle pack-
ing in a domain. The domain that we consider is the quotient of the Euclidean
plane by a full-rank lattice, called a flat torus. We determine all optimally dense
arrangements of two, three, and four equal circles on any flat torus. Theorem 1.1
states explicit formulas for the optimal radius on any torus. The complexity of the
expressions for the optimal radii is a consequence of the explicit formulas for the
coordinates of the circle centers achieving the optimal densities given in Section 7.
The geometry of a flat torus influences the nature of the optimal packings it
admits. Therefore, we must first carefully describe how the optimal arrangements
break the moduli space of flat tori apart. The moduli space of un-oriented flat
tori can be represented as a strip in the x-yplane where x2+y2≥1, y > 0,
and 0 ≤x≤1
2. Within this moduli space the optimal arrangements determine
regions, bounded by line and circle segments, where the optimal radii are governed
by different formulas. These regions are defined in Table 1 and pictured in Figure 1.
Using these regions we can state the main result of this article.
Date: May 15, 2019.
2010 Mathematics Subject Classification. 52C15.
Key words and phrases. Equal circle packing, flat torus, packing graph, rigidity theory.
The authors were partially supported by National Science Foundation grants DMS-0451254
and DMS-1262342. The second author would like to thank David Austin for reading drafts of this
article and writing a Python interface for using the program PiScript (written by Bill Casselman)
that we used to produce the illustrations in this article. We all express our gratitude to Bob
Connelly for inspiring this work. We also thank the referees for their comments that improved
this work.
1
arXiv:1708.05395v2 [math.MG] 14 May 2019
2 BRANDT, DICKINSON, ELLSWORTH, KENKEL, AND SMITH
2
3
√3
1
2
(1
2,√3
2)
(1
2,1 + √3
2)
R22
R12
2
3
1 + √3
2
√3
1
2
(1
2,√3
2)
(1
2,3√3
2)
(−1
2,√3
2)
R23
R13
R33
2
3
2√3
2
√3
1
2
(1
2,√3
2)
(1
2,5√3
6)
(1
2,1 + 3√3
2)
R14
R24
R34
R44
Figure 1. The different expressions for the optimal radii from
Theorem 1.1 break the moduli space for un-oriented tori into var-
ious regions for 2 circles (left), 3 circles (middle) and 4 circles
(right). The unfilled dots on the perimeter of each moduli space
indicate tori on which the optimal packing is one in which each cir-
cle is tangent to six circles (i.e. the packing lifts to the triangular
close packing in the plane).
Two Circles
Region R12√1−x2≤y < q1−(x−1
2)2+√3
2
Region R22q1−(x−1
2)2+√3
2≤y
Three Circles
Region R13√1−x2≤y < q1
3−x2+√3
3
Region R23q1
3−x2+√3
3≤y < √1−x2+√3
Region R33√1−x2+√3≤y
Four Circles
Region R14√1−x2≤y < 2−x
√3
Region R242−x
√3≤y < q1
3−x−1
22+√3
2
Region R34q1
3−x−1
22+√3
2≤y < q1−x−1
22+3√3
2
Region R44q1−x−1
22+3√3
2≤y
Table 1. These are the equations describing the boundaries of
each region in Figure 1. These regions break the moduli space
for un-oriented flat tori (x2+y2≥1, y > 0, and 0 ≤x≤1
2) into
various regions bounded by line and circle segments. The subscript
in the region name refers to the number of circles in the packing.
OPTIMAL PACKINGS ON FLAT TORI 3
Theorem 1.1. Let Pk(x, y)be a packing of kequal circles on a flat torus that is the
quotient of the plane by the vectors v1=h1,0iand v2=hx, yiwhere x2+y2≥1,
y > 0, and 0≤x≤1
2. If rk(x, y)is the least upper bound of the radius then we have
the following expressions for the optimal radii in different regions given explicitly
in Table 1 and shown in Figure 1.
r2(x, y) = (√x2+y2
4yp(x−1)2+y2in Region R12;
1
2in Region R22.
r3(x, y) =
√x2+y2
2(y+√3x)rx+1
22+y−√3
22in Region R13;
1
6r9x2+y−p3+4y2−12x22in Region R23;
1
2in Region R33.
r4(x, y) =
√x2+y2
2(y−√3x)rx−1
22+y−√3
22in Region R14;
qA2−√A2
2−16B2
4√2
where A2= 2 y√3−x2−(x−1)√3 + y2+ 3
and B2= (x2+y2)(x−3
22+y−√3
22)
in Region R24;
qA3−√A2
3−16B3
8√2
where A3= 9 + 5y2−(2x−1)2
and B3=(x−2)2+y2(x+ 1)2+y2
in Region R34;
1
2in Region R44.
The expressions from Theorem 1.1 for the optimal radii agree with the work
of Heppes ([Hep99]) who presents the optimal radii1and arrangements for two,
three and four equal circle packings on any rectangular torus (the quotient of the
plane by perpendicular lattice vectors). Heppes used the same techniques as Melis-
sen ([Mel97]) who determined the optimally dense arrangements of 1 to 4 equal
circles in a square flat torus (the quotient of the plane by unit and perpendicular
lattice vectors). Both Heppes and Melissen utilized a technique similar to the one
commonly used in proving the optimality of an arrangement of equal circles packed
into a unit square. This technique centers on knowing a candidate for the densest
arrangement of circles in the square or rectangle which establishes a lower bound
on the diameter of circles in the densest arrangement. One next cleverly uses this
diameter to partition the square or rectangle into regions with an appropriate di-
ameter to prove the global optimality of the candidate arrangement. The approach
employed in this article is fundamentally different. Here we prove the optimality of
the arrangements of 2, 3 and 4 equal circles on a flat torus using techniques from
rigidity theory and topological graph theory.
Besides the work of Melissen and Heppes, packings on flat tori have been stud-
ied by other authors. Przeworski ([Prz06, Theorem 2.3]) determines the optimal
1There is a typo in Heppes’ formulas for the optimal radius for the rectangular tori on the
edges of region R13and R14(in the notation of this article) for three and four circles. Each of
his formulae for these radii need to be multiplied by 1
4.
4 BRANDT, DICKINSON, ELLSWORTH, KENKEL, AND SMITH
arrangements of two equal circles on any flat torus. Dickinson et al. ([DGKX11a,
DGKX11b]) determine the optimal packings of 1–5 equal circles on the square flat
torus and 1–6 equal circles on a triangular flat torus (the quotient of the plane
where the lattice vectors are unit and form an angle of π
3). The arrangements and
radii from these articles agree with the results presented here. In addition, Musin
and Nikitenko ([MN16]) use similar techniques coupled with a computer algorithm
to numerically determine the optimal arrangements of 6, 7 and 8 equal circles on a
flat square torus. Lubachevsky et al. ([LGS97]) explore packings with large num-
bers (50-10,000) of equal circles packed on a square torus (among other domains).
They used a billiards algorithm to discover their arrangements and they discuss
large scale patterns as there is little hope of proving optimality. For similar explo-
rations from a physics point of view, see the article by Donev et al. ([DTSC04]).
Articles [RS97, GR08] explore optimal packings of squares on the square flat torus.
The present work use similar techniques as in [DGKX11a], [DGKX11b], and
[MN16] to discover and prove the optimality of equal circle packings on any flat
torus. In Section 2, we review basic terms and definitions and outline the structure
of the proof of Theorem 1.1. Section 3 recalls results from rigidity theory and delin-
eates the combinatorial properties of packing graphs associated to optimally dense
packings whose packing graphs do not contain a loop. Section 4 characterizes all
equal circle packings that contain at least one self-tangent circle or whose packing
graphs contain at least one loop. We use topological graph theory in Section 5
to enumerate of all 2-cell embeddings of combinatorial graphs onto a topological
torus and conclude that section with tools that eliminate many of the embedded
graphs from being associated with any equal circle packing on any torus. Section 6
contains a discussion on the remaining embedded graphs that are not associated
with an optimal equal circle packing (including those that are locally but not glob-
ally maximally dense). Finally, in Section 7 we demonstrate realizations of the
remaining embedded graphs as optimal equal circle packings and thus prove the
existence of packings achieving the optimal radii given in Theorem 1.1 and give the
coordinates of the circle centers.
2. Basic Notions And An Overview Of The Proof Of Theorem 1.1
In this section we review some terminology and recall some basic facts about
circle packings. We then outline the proof of Theorem 1.1.
2.1. Basic Notions. The quotient of the Euclidean plane by a lattice generated
by two independent vectors v1and v2is called a flat torus. When the vectors
are perpendicular the quotient is called a rectangular flat torus; when they are
perpendicular and unit, the quotient is called a square flat torus; when they are
unit and form an angle of π
3the quotient is called a triangular flat torus. A
fundamental domain of a flat torus is the set of points in the Euclidean plane,
{t1v1+t2v2|t1, t2∈R,0≤t1, t2<1}. To specify the location of a circle on torus,
we will actually give a location in the universal cover (the Euclidean plane) which
will serve as a representative of the equivalence class of the lifts of the location from
the torus. In a slight abuse of nomenclature, we shall also say that two points in the
Euclidean plane that are in the same equivalence class are also lifts of each other
(this implies that they differ by a vector in the lattice). Notice that we can lift any
packing from a flat torus to a periodic packing in the Euclidean plane by lifting all
the circles in the packing in all possible ways. This means that the density of a
OPTIMAL PACKINGS ON FLAT TORI 5
packing on a flat torus must be less than the maximum density of a packing of the
Euclidean plane. The L. Fejes T´oth-Thue Theorem ([FT64, Thu10]) states that
the density of all packings of equal circles in the Euclidean plane is less than the
density of the triangular close packing, where each circle is tangent to six others.
The triangular close packing has density π
√12 and therefore the packing density on
the torus cannot exceed this bound.
The standard basis for a lattice is one where v1=h1,0iand v2=hx, yiwhere
x2+y2≥1, y > 0 and −1
2< x ≤1
2. Every oriented lattice can be transformed by
scaling and the action of SL(2,Z) to have a standard basis (see, for example, [Prz06,
Theorem 2.3] or [Jos06, Theorem 2.7.1]). As we are working with unoriented tori,
we restrict to the region where 0 ≤x≤1
2. The optimal arrangements naturally
break this restricted moduli space of flat tori into regions with different expressions
for the optimal radius. See Figure 1.
For a given flat torus, an arrangement of equal circles forms a packing on the
torus if the interiors of the circles are disjoint. The density of a packing is
the ratio of the area of the circles to the area of the flat torus. Notice that the
transformations used to put a lattice (with a periodic circle packing) into a standard
form preserve the density of that packing. We define two packings with the same
number of circles to be -close if there is a one-to-one correspondence between
the circles, so that corresponding circles have centers that are all within a distance
of of each other. We define a packing Pto be optimal or locally maximally
dense if there exists an > 0 so that all -close packings of equal circles have a
packing density no larger than that of P. A packing Qis globally maximally
dense if it is the densest possible packing. Rather than searching directly for the
globally maximally dense packings, our techniques allow us to determine all the
locally maximally dense arrangements of a fixed number of circles on a given flat
torus, from which we determine the globally maximally dense packing(s).
The main structure that allows us to form a list of all the locally maximally dense
packings for a fixed number of circles on a flat torus is the graph of a packing. Given
a packing Pon a flat torus, the packing graph associated to P, denoted GP,
has geometric vertices and edges defined as follows. (This is also sometimes called
a kissing or contact graph as in [MN16].) The center of each circle in the packing is
associated to a vertex (with a location in the torus) of GPand two (not necessarily
distinct) vertices of GPare connected with an edge (with a length as a line/geodesic
segment on the torus) if and only if the corresponding circles are tangent to each
other. Further, all circle-circle tangencies lead to an edge and this allows pairs of
vertices to have more than one edge connecting them. Thus each packing of equal
circles on a flat torus is naturally associated to an embedding of a graph on a flat
torus where all the edges are equal in length. Throughout this article we allow the
graphs to be multigraphs that can possibly contain multiedges and loops.
It is important to note that it is possible to have a free circle or rattler in an
optimal arrangement; that is, a circle that is not tangent to any other circle. For
packing on a torus, a circle that is only self-tangent is still considered to be free.
The work of Schaer ([Sch65]) on a unit square, Musin ([MN16]) on a square torus,
and Melissen ([Mel93]) on an equilateral triangle, prove that at least one of the
globally optimally dense packings of seven equal circles in each of these domains
contains a free circle. In the present work we discover a family of locally maximally
dense equal circle packings with four circles that admits a fifth circle that is free.
6 BRANDT, DICKINSON, ELLSWORTH, KENKEL, AND SMITH
See the comments after Theorem 3.2 in Section 3 for more discussion and the top
row of Figure 10 for an image.
2.2. Overview of the proof of Theorem 1.1. By viewing the associated packing
graph of an optimal packing as type of tensegrity framework (an embedded graph
with additional structure) on a torus, Connelly ([Con88]) has proven a lower bound
on the number of circle-circle tangencies. His result implies that in an optimal
packing of nequal circles (without any free circles) on a flat torus there must be
2n−1 or more edges in the associated packing graph. Using the fact that each circle
in a torus can be tangent to at most six circles, we can deduce that in an optimal
packing of nequal circles (without any free circles) on a flat torus there must be
3nor fewer edges. This means that there are a finite number of graphs that can
be associated to an optimal packing of nequal circles on a flat torus. By studying
all of these possible graphs, and all the possible ways they can be embedded on a
torus, we are led to all the locally (and globally) optimally dense packings.
More specifically, this leads us to the following outline of the proof of Theo-
rem 1.1. (Note that the case of n= 2 is handled differently in Subsection 7.1 and
those packings with self-tangent circles are handled separately in Section 4.)
(1) Step one is divided into two parts and the details are provided in Section 3.
(a) Make a list of all the possible graphs that could be associated to an
optimal packing of n= 3 or n= 4 circles on a flat torus. That is, list
all the combinatorially distinct graphs with n= 3 or n= 4 vertices
and between 2n−1 and 3nedges. There are 862 of these combinatorial
graphs.
(b) Not all of the combinatorial graphs from step 1a can be associated to
an optimal packing. We use Proposition 3.5 to eliminate all but 23 of
the combinatorial graphs. This is shown in Table 2 and all 23 of them
are shown in Figure 3.
(2) Step two is divided into two parts and the details are provided in Section 5.
(a) For each of the combinatorial graphs remaining from step 1b, we use
techniques from topological graph theory (specifically Edmond’s per-
mutation technique [Edm60]) to enumerate all the possible 2-cell em-
beddings of it onto a topological torus. Note that a combinatorial
graph can embed in multiple ways. There are 103 of these embedded
combinatorial graphs (or simply embedded graphs).
(b) Not all of these embedded combinatorial graphs can associated to an
equal circle packing (optimal or not). We use Propositions 5.1 and 5.2
to eliminate all but 27 of the embedded combinatorial graphs. This is
shown in Table 3 and all 27 of them are shown in Figures 6 and 7.
(3) For each of the embedded combinatorial graphs remaining from step 2b
we must answer some questions: Does the embedded graph correspond to
an equal circle packing? If so, is the packing optimal or not? If so, is
the packing locally or globally optimal? As discussed in Section 6, 15 of
the embedded combinatorial graphs either do not correspond to an equal
circle packing or, if they do, the corresponding packing is locally, but not
globally optimal. Section 7 handles the remaining 12 embedded combina-
torial graphs that are associated to the globally optimal packings. In this
section, explicit coordinates for the locations of centers of the circles are
OPTIMAL PACKINGS ON FLAT TORI 7
given and, thus, the optimal radii at all locations in the moduli space can
be determined.
Therefore, because we started with all possible combinatorial graphs that could be
associated with an optimal packing and we embedded those combinatorial graphs
in all possible ways, we have determined all the globally (and locally) packings on
any torus. Note: As a torus is compact and the packing radius determined by a
collection of npoints on a torus is continuous, we know that there is a globally
optimal packing of nequal circles on that torus.
3. Results From Rigidity Theory
We now discuss some tools from rigidity theory, including some useful results
from [DGKX11b, Section 3] and [MN16, Section 3]. See these references for more
details. In particular, we state propositions that establish an upper and lower
bound on the number of edges a packing graph associated to an optimal packing
must contain. This allows us to create a short finite list (Figure 3) of combinatorial
multigraphs without loops (loops are handled in Section 4) each of which is a
candidate to be the packing graph of an optimal packing. Afterwards we complete
Step 1 of the proof (following the overview of the proof found in Subsection 2.2).
3.1. Tools From Rigidity Theory. Rigidity theory involves the study of tenseg-
rity frameworks. For our purposes, we specialize to strut tensegrity frameworks,
which are essentially graphs embedded in a Riemannian manifold with some addi-
tional structure. To be a strut tensegrity framework, each edge in the graph is not
allowed to decrease in length as the location of its endpoint vertices change. We can
view the (embedded) packing graph of a circle packing as a strut framework on a
torus. This is appropriate because as we move the vertices of a circle packing graph
to try and improve the density, we want the length of the edges to either increase
(or remain unchanged), in order to possibly increase (or maintain) the density. The
motions (if any) of the vertices that respect the distance constraints between the
vertices are called flexes. A flex that is not induced from a family of rigid motions
of the torus is called a non-trivial flex. If there are no non-trivial flexes, then the
strut framework is called rigid. Combining the rigidity theory ideas with the ideas
of circle packing, we have the following (for a complete proof see [DGKX11a, Prop.
3.1]).
Proposition 3.1. If the strut tensegrity framework associated to a circle packing
Pis rigid, then the circle packing Pis locally maximally dense.
The flexes of a framework can be linearized and will lead to another notion of
rigidity that will turn out to be equivalent in our case. If you consider the time
zero derivative of a flex at each vertex, then you obtain a collection of vectors. This
collection of vectors must satisfy a system of linear homogeneous (strut) inequalities
which result from the flex respecting the distance constraints. A collection of vectors
that satisfy the strut inequalities and are not the time zero derivative of a family
of rigid motion of the torus at each vertex is called a non-trivial infinitesimal flex.
If there are no non-trivial infinitesimal flexes, then the strut framework is called
infinitesimally rigid. The connection between infinitesimal rigidity and rigidity
of a framework is well studied and Connelly ([Con88]) has proven that a strut
tensegrity framework is rigid if and only if it is infinitesimally rigid. Determining
the infinitesimal rigidity of an arrangement is the same as the feasibility part of
8 BRANDT, DICKINSON, ELLSWORTH, KENKEL, AND SMITH
linear programming, is straightforward to check, and, with Proposition 3.1, enables
us to easily check when a given circle packing is locally maximally dense.
We are now in a position to state the main result from Connelly ([Con88]),
specialized to the context of flat tori, which is almost the converse of Proposition 3.1.
Theorem 3.2 (Connelly).If Pis a packing that is locally maximally dense on a flat
torus, then there is a sub-packing Qof Psuch that the associated strut tensegrity
framework is infinitesimally rigid and the circles not in Q(possibly an empty set)
are prevented from increasing their radius (i.e. are free circles).
Motivated by Theorem 3.2 we now discuss those locally maximally dense packing
with disconnected packing graphs.
3.1.1. Optimal Packings With A Disconnected Packing Graph Or That Admit An-
other Circle. If we remove any free circles from a locally maximally dense arrange-
ment, then we obtain a locally maximally dense packing for fewer circles in the flat
torus. Conversely, if there is room for another circle (free or not) in a globally max-
imally dense packing of ncircles, then adding this circle gives us an arrangement
realizing the globally maximal density for n+1 circles. Heppes ([Hep99])and Melis-
sen ([Mel97]) exploited this when they noticed that their globally optimal packings
for 3 circles in the boundary of region R13that is along the yaxis admit another
circle. That is, the globally maximally dense packings along this edge have room
for a fourth circle (creating an additional 4 tangencies so the circle is not free). This
leads to the globally maximally dense arrangements in the corresponding locations
in region R14on the yaxis for 4 circles.
The present work gives more examples of this phenomenon. The family of pack-
ings for 3 circles that is globally optimally dense for region R13also extends to a
locally optimally dense packing on the left of the yaxis (this is the light gray region
in the middle of Figure 1). It turns out that this locally optimal two parameter
family of 3 circles packing admits a fourth circle (with 4 additional tangencies)
that becomes globally optimally dense packings for 4 circles. See the top row of
Figure 11. With the addition of this circle, these packings occupy region R14on
the right of Figure 1.
The only time that a free circle (or an additional circle) might be able to be
added to an arrangement is when there is a face in the packing graph consisting of
seven or more edges ([MN16, Prop. 3.6]). There are two cases in the present work
where this happens in an optimal packing.
•The optimal packing occupying region R13(where it doesn’t admit the ad-
ditional circle) and the adjacent gray region (seen in the middle of Figure 1
which is discussed in the paragraph above) for 3 circles.
•In one locally (but not globally) maximally dense family of packings of 4
circles. This does admit a free circle. See the top row of Figure 10. This
packing is a locally optimally dense packing of 5 circles which is beyond
the scope of the present work.
It should be noted that there are two arrangements of 3 circles that contain a regular
hexagonal face with room for an additional circle (with 6 additional tangencies).
Adding the fourth circle to these arrangements yields the triangular close packing
with 4 circles on tori corresponding to the unfilled points on the vertical axes on
the right side of Figure 1. There are a handful of locally optimal 4 circle packings
OPTIMAL PACKINGS ON FLAT TORI 9
with a regular hexagon and adding a circle leads to the triangular close packing on
5 circles.
In this article, we will determine all the locally maximally dense arrangements
for 3 and 4 equal circles without free circles. This, coupled with the discussion
here, means that we will have created an exhaustive list of locally maximally dense
packings. Therefore, for the remainder of this article, we assume that all of our
graphs are connected. As noted earlier, Przeworski [Prz06, Theorem 2.3] has solved
this problem for 2 equal circles and, for completeness we analyze this case using
our methods (see Subsection 7.1) and we reach the same conclusion as Przeworski.
3.1.2. Characterizing The Packing Graphs Associated To Optimal Packings. Now
we observe that we can find a lower bound on the number of edges (and their
arrangement) incident to a vertex in the packing graph associated to a locally
maximally dense packing with no free circles.
Proposition 3.3. If Pis a locally maximally dense packing of circles on a flat
torus with no free circles, then no circle in Phas its points of tangency contained
in a closed semi-circle. In particular, every circle is tangent to at least three circles.
Proof. If there were such a circle in a locally maximally dense packing, then the
packing graph would not be infinitesimally rigid violating Theorem 3.2. For more
details see [DGKX11b, Section 3].
Connelly ([Con90]) proves a lower bound on the number of edges that a packing
graph must contain in order for the associated packing to be locally maximally
dense.
Proposition 3.4 (Connelly).If Pis a locally maximally dense packing of ncircles
on a flat torus with no free circles, then the packing graph associated to Pcontains
at least 2n−1edges.
In the case of 3 and 4 equal circles (unlike the case of 5 or more circles on a
square torus, see [DGKX11a, Prop. 4.4]) it is possible that a circle can be tangent
to another circle in two different ways. However, we can eliminate the possibility of
two circles being tangent in three (or more) ways in the case of 4 (or more) circles
and restrict the arrangements in the case of 3 circles to the triangular close packing.
B1
B2
B3
A
u1
u2
P
Figure 2. A packing in which circle Ais tangent to circle Bin
at least three ways.
10 BRANDT, DICKINSON, ELLSWORTH, KENKEL, AND SMITH
Proposition 3.5. Given an equal circle packing on a flat torus with ncircles.
•If n≥4, then no two circles can share three or more tangencies.
•If n= 3, then no two circles can share four or more tangencies. Further if
there are two circles that share three tangencies, then every circle is tangent
to six circles (i.e. the packing lifts to the triangular close packing).
Proof. Suppose circle Ais tangent to circle Bin 3 (or more ways) on a torus that
is the quotient of the Euclidean plane by lattice Λ. Lift circle Ainto the Euclidean
plane and consider the triangle formed by the centers of the lifts of circle Bthat
are tangent to the lift of A. Call these centers B1,B2, and B3. Let u1(u2) be
the vector of Λ that connects B2and B1(B2and B3) respectively. Let Pbe the
Λ-lattice parallelogram formed by B1,B2,B3and B2+u1+u2. See Figure 2.
By Pick’s theorem the area of P,AP, is equal to AF D(i+b
2−1) where AF D is
the area of a fundamental domain of lattice Λ, i(b) is the number of lattice points
interior (on the boundary) of P. Using the minimum number of lattice points of
Λ inside and on the boundary of P, we have that AP≥AFD . Further if dis the
common diameter of the circles, then the geometry of the packing implies that the
maximum area of APis 3√3
2d2and we have the following bounds on the density, ρ,
of the packing.
π
√12 ≥ρ=Area covered by ncircles
AF D ≥nπ d
22
AP≥nπ d
22
3√3
2d2≥nπ
3√12
For n≥4 this is a contradiction and for n= 3 all of the inequalities become
equalities and the density of the packing implies that the arrangement must lift to
the triangular close packing. This also implies that it is impossible for a packing
of three circles on a flat torus to have a pair of circles tangent in four or more
ways.
These three propositions and the observation that in the triangular close packing
(the most dense packing of equal circles in the plane) each circle is tangent to six
others, helps us list important properties of the packing graphs of locally maximally
dense arrangements ncircles on the torus. This is summarized in the following
proposition.
Proposition 3.6. Given a locally maximally dense packing, P, of n≥3equal
circles without any free or self-tangent circles on a flat torus, the packing graph GP
satisfies the following three conditions.
(1) GPis connected, contains no loops, and contains at least 2n−1and at
most 3nedges.
(2) Every vertex of GPis connected to at least three and at most six others.
(3) No pair of vertices of GPis connected by 3 or more edges, except if n=
3and the packing is the triangular close packing implying that all three
vertices of GPhave degree 6.
3.2. Step 1 of the proof of 1.1. The number of multigraphs (we allow multi-
ple edges between vertices because these correspond to pairs of circles tangent in
multiple ways) with a fixed number of vertices and edges is well studied. Using
Dr. Gordon Royle’s data posted on the web ([Roy]) or the program Nauty ([MP14]
with the gtools geng and multig) we can make a list of the combinatorial graphs
that satisfy the first condition of Proposition 3.6. (This is step 1a of the overview of
OPTIMAL PACKINGS ON FLAT TORI 11
Number of graphs
satisfying condition
1 of Prop. 3.6
Number of graphs
satisfying conditions
1 & 2 of Prop. 3.6
Number of graphs
satisfying all condi-
tions of Prop. 3.6
3 Vertices 37 10 3
4 Vertices 825 102 20
Table 2. The number of combinatorial graphs remaining after
parts of Proposition 3.6. See Figure 3 for visualizations.
the proof found in Subsection 2.2.) Both the posted data and the results of Nauty
yielded the same number of graphs shown in the first column in Table 2. Using
Wolfram’s Mathematica ([Inc]) we wrote a routine to remove the combinatorial
graphs that didn’t satisfy conditions two and three of Proposition 3.6. See Figure 3
for visualizations of the remaining combinatorial graphs. (This is step 1b of the
overview of the proof found in Subsection 2.2.)
4. Packings with self-tangent circles
In this section, we completely characterize all optimal packings of 3 or 4 equal
circles whose packing graph contains a loop. This leads us to justify the assumption
used outside of this section that all packing graphs do not contain a loop. On
the torus, locally maximally dense packings with free circles are closely related to
packings that have self-tangent circles. This is because if you have a torus with
basis v1=h1,0iand v2=hx, yiwhere x2+y2≥1, y > 0, and 0 ≤x≤1
2and
the length of v2is long enough, all the circles can achieve the maximum radius
of 1
2and become self-tangent and free. In this section we determine the boundary
between where all the circles are self-tangent (see Figure 4) and where none of the
circles are self-tangent. We show that the region allowing self-tangent circles is the
only one where packing graphs with a loop are realizable; this means that we have
characterized all packings with a loop in them and need not consider combinatorial
graphs with loops outside of this section. Throughout this section we assume that
the torus is the quotient of the Euclidean plane by the vectors v1and v2that
satisfy the restrictions given in this paragraph. We begin with some observations
about packings that contain self-tangent circles.
Proposition 4.1. Let Pbe a packing of equal circles on a flat torus. The following
four statements are equivalent.
(1) There exists a circle in Pthat is self-tangent.
(2) The combinatorial multigraph associated to Pcontains a loop.
(3) The common radius of the circles in Pis 1
2.
(4) All the circles of Pare self-tangent.
Proof. All statements follow from the fact that this is an equal circle packing and
the observation that v1is a shortest vector of length one so lifts of circles that differ
by this lattice element must have radius 1
2. In this case all circles in the packing
must be self-tangent.
12 BRANDT, DICKINSON, ELLSWORTH, KENKEL, AND SMITH
CG12,2
CG23,3
CG31,1
CG48,5
CG54,1
CG62,2
CG72,2
CG812,2
CG94,4
CG105,2
CG117,0
CG124,2
CG132,2
CG147,0
CG157,0
CG167,4
CG176,0
CG182,1
CG197,0
CG203,2
CG213,0
CG223,0
CG232,2
Figure 3. The 23 combinatorial graphs (CG) with three or four
vertices satisfying Proposition 3.6. The first number in the sub-
script is the number of distinct unlabeled and unoriented toroidal
embeddings. The second number in the subscript is the number
of distinct unlabeled and unoriented toroidal embeddings remain-
ing after applying Proposition 5.1. (Applying Prop. 5.2 eliminates
more.) The remaining toroidal embeddings are shown in Figures 6
and 7. See Section 5 for more details about these subscript num-
bers.
Observe that Proposition 4.1 implies that if there is a loop in the combinatorial
graph associated to a packing P(or the packing radius is 1
2), then Pis locally
maximally dense. This is because 1
2is the absolute upper bound of the radius in
any packing on the tori we are considering so any nearby packing has density less
or equal to that of P.
Now we examine the structure of packings with a self-tangent circle and consider
the cases where such a packing contains or does not contain a free circle. The
intuition behind the first part of Proposition 4.2 is to first note that all the circles
in the packing have radius 1
2and therefore each circle is self-tangent, so it and
its lifts create a “layer” in the flat torus – see Figure 4. Next, in order to fit the
packing on the “smallest” flat torus, the layers must be stacked so that most circles
are tangent to the circle above and below in two different ways forming a section
of triangular close packing at the “bottom” of the torus.
OPTIMAL PACKINGS ON FLAT TORI 13
Layer 1
Layer n
Layer 1
Layer n
α
Cn
C1
αCn
C1
Figure 4. The lattice associated to an optimal packing with an
odd number (left) or an even number (right) of equal circles with
packing radius 1
2with no free circles. Notice how all the circles are
self-tangent and each forms a layer so that towards the bottom of
the torus the layers are arranged to form part of a triangular close
packing.
Proposition 4.2. Let Pbe a packing of n≥2equal circles on a flat torus where
the combinatorial multigraph associated to Pcontains a loop.
(1) If there are no free circles in P, then there exists αwith π
3≤α≤π
2
such that v2=hx, n−1
2√3 + sin (α)iwhere x=1
2−cos (α)for neven or
x= cos (α)for nodd.
(2) If there is a free circle in P, then all circles in Pare free and there exists
π
3≤α≤π
2such that v2=hx, yiwhere x=1
2−cos (α)(for neven) or
x= cos (α)(for nodd) and y > n−1
2√3 + sin (α).
Proof. First we begin by showing that there exists a cyclic ordering of the circles
C1, C2, C3, . . . , Cnwhere circle Ciis tangent to itself and can only be tangent to
Ci−1or Ci+1 for 1 ≤i≤n(indices of the circles counted modulo n). To see
this notice that as the combinatorial multigraph associated to Pcontains a loop,
by Proposition 4.1, all circles are self-tangent. For each circle in the packing, this
means that the toroidally embedded packing graph contains an essential cycle (a
closed path not homotopic to a point) of length one consisting only of the edge
corresponding to the self-tangency. Cutting the torus along these essential cycles
results in ncylinders. This establishes a cyclic ordering of the circles. As all
tangencies in this packing must occur in some cylinder, we know that, beside being
tangent to itself, Cican only be tangent to the circles forming the edges of the
cylinders with the essential cycle associated to Ci, namely Ci−1and Ci+1.
To establish item (1) of the proposition, assume that there are no free circles in P
and by the remark immediately preceding this proposition, Pis locally maximally
dense. Theorem 3.2 with the assumption that there are no free circles implies that
the entire strut tensegrity framework associated Pmust be infinitesimally rigid (not
just some subgraph). This implies that each circle in the packing must have at least
three tangencies not including the self-tangency, because the strut inequality from
14 BRANDT, DICKINSON, ELLSWORTH, KENKEL, AND SMITH
the self-tangency is trivial. The only way for this to be true is if each circle Ciis
tangent at least once (and at most twice) to each of Ci−1and Ci+1.
Now we observe that there can be at most one isuch that Ciis tangent once
to Ci+1. Suppose not, then there exists two natural numbers, iand j(i < j ≤n),
such that Ciis tangent only once to Ci+1,Cjis tangent only once to Cj+1 and for
each natural number m,i+ 1 ≤m≤j−1, Cmis tangent twice to Cm+1. In this
case there is a non-trivial infinitesimal flex (i.e. a non-trivial assignment of vectors
to the circle centers that satisfies the strut inequalities – See Section 3 for more
details) that is a non-zero constant vector at the centers of Cm(i+ 1 ≤m≤j−1)
and the zero vector at the other circle centers. For the non-zero constant, choose a
vector that makes an obtuse angle with both the single edge between Ciand Ci+1
and the edge between Cjand Cj+1. Such a choice always exists except in the case
when these two edges are parallel (in which case you would choose the non-zero
vector to be perpendicular to both). Hence there is at most one circle Cithat is
tangent once to Ci+1.
For the purpose of determining the lattice that defines the torus assume, by
renumbering if necessary, that the possibly single tangency between circles occurs
between Cnand C1. By lifting the packing to the plane and by reflecting if necessary
we can assume that the angle, α, between the horizontal and the edge between Cn
and C1(measured clockwise for neven and measured counterclockwise for nodd) is
between π
3and π
2. In this case v2has the form given in item (1) of this proposition.
See Figure 4.
For item (2), suppose that Pcontains a free circle, Ci. As this circle is free we
can translate it to break all tangencies (if any) with this circle (except the self-
tangency). Next, we can translate (perpendicular to v1) all other circles a little bit
using the space created when all tangencies with Ciwere broken. In this way we
see that all circles are free in this packing. To determine the lattice of the torus for
this packing notice that as the circles are free we can move them into the positions
so that there are two tangencies between Ciand Ci+1 for 1 ≤i<n. This results
in a situation similar to that which is pictured in Figure 4 except that there are
no tangencies between circle Cnand C1. Notice that the angle αis the angle such
that the xcoordinate of v2is either cos(α) (nodd) or 1
2−cos (α) (neven). The
bound on ycoordinate follows.
Finally we use the constructions, results, and observations that arise in the rest
of this article and Proposition 4.2 to observe that all locally and globally optimal
packings on tori in the region described in item (2) must have a loop in the packing
graph. Therefore we have completely characterized the tori on which there exists a
locally maximally dense packing that contains a self-tangent circle for 3 or 4 equal
circles. Notice that the descriptions of the regions for odd and even numbers of
circles given in Propostion 4.3 agree with the descriptions of regions R22,R33, and
R44given in Table 1.
Proposition 4.3. Let Pbe a locally maximally dense packing of n= 3 or n= 4
equal circles on a flat torus. If the flat torus is the quotient of the plane with
respect to a lattice with basis vectors v1=h1,0iand v2=hx, yiwhere there exists
π
3≤α≤π
2such that x=1
2−cos (α)(for neven) or x= cos (α)(for nodd) and
y > n−1
2√3 + sin (α), then there is loop in the combinatorial multigraph associated
to Pand Pcontains only free circles.
OPTIMAL PACKINGS ON FLAT TORI 15
B
A
C
D
B
A
C
D
Figure 5. Toroidal embeddings of CG5 (left) and CG4 (right)
that do not correspond to any equal circle packing graph on any
torus. The edges AB and CD are parallel because of the rhombus
in the embedded graph, but there is no edge AD, so neither of
these can be packing graphs associated to an equal circle packing.
Proof. Suppose not, then there is no loop in the associated combinatorial multi-
graph. For n= 3,4 all possible multigraphs without loops were considered as
packing graphs in the remainder of this article and none of them embed locally
maximally densely on a torus with basis v1and v2as described in this proposition.
Therefore there must be a loop in the combinatorial multigraph. If Pcontains no
free circles, then Proposition 4.2 item (1) gives the form of the basis vectors for the
torus, but these are incompatible with the basis vectors for the torus described in
this proposition, therefore there must be at least one free circle. Proposition 4.2
item (2) implies that all the circles in Pare free.
Propositions 4.3 and 4.2 implies that we do not have to consider loops in the
packing graph explorations in the remaining sections of this article.
5. Results From Topological Graph Theory
This section is broken into two parts mirroring two parts of step 2 of the overview
of the proof from Subsection 2.2. First, we apply previously known techniques
for determining all the 2-cell embeddings of a combinatorial multigraph onto a
topological torus. Second, we apply several propositions that prohibit a toroidally
embedded graph from being associated with an optimal equal circle packing. This
leaves us with a short list of toroidally embedded graphs that potentially could be
associated with an optimal equal circle packing. See Figures 6 and 7.
5.1. Step 2a of the proof of Theorem 1.1. A graph embedded in a torus is
a 2-cell embedding if each connected region (face) determined by removing the
embedded graph from the torus is homeomorphic to an open disk. As noted
in [DGKX11b, MN16] if a packing on the torus is locally maximally dense, then
the associated packing graph is a 2-cell embedding. One tool for enumerating all
the 2-cell embeddings of a graph on any surfaces is Edmonds’ permutation tech-
nique ([Edm60]) which is outlined in [DGKX11b, Section 5]. (We use the same
C++ implementation, adapted for multigraphs, in this article.) Essentially this is
a brute force technique where, at each vertex, all possible orderings of the adjacent
vertices are considered (call rotation schemes). Once a face walking algorithm is
executed and the Euler characteristic is computed, all the possible toroidal embed-
dings can be selected. As the graphs we are dealing with are small, this brute force
16 BRANDT, DICKINSON, ELLSWORTH, KENKEL, AND SMITH
method converges. There are more efficient algorithms, see [KK05, Sect. 13.4].
The software written by Kocay ([Koc07]) was instrumental in visualizing and com-
puting the these embedded graphs. The paper, Embeddings of small graphs on the
torus by Gagarin et al. ([GKN03]), present an independently created table listing
the number of toroidal embeddings of all vertex-transitive graphs (with fewer than
12 vertices) onto the torus. The results of our program were checked against this
table and, when our brute force program converged, it agreed with the results in
this paper.
5.2. Step 2b of the proof of Theorem 1.1. Once we have a list of all possible
toroidal embeddings we ask if the embedded graph could be associated to an opti-
mal equal circle packing. The following two propositions eliminate many of these
potential packing graphs.
Proposition 5.1. If a graph embedded on a torus contains a vertex surrounded by
any one of the following face patterns, then the embedded graph cannot be the graph
associated to a locally maximally dense equal circle packing. The forbidden face
patterns are (1) two triangles and a polygon, (2) three triangles and a polygon, (3)
five triangles, (4) four triangles and a quadrilateral, (5) six polygons with at least
one non-triangle, (6) a triangle, a quadrilateral and a polygon, (7) two triangles
and two quadrilaterals, (8) three quadrilaterals, or (9) seven (or more) polygons.
For a proof of Proposition 5.1 see [DGKX11b, Prop. 6.1]. Proposition 5.2 is
another useful tool for eliminating embedded graphs from being associated with
any equal circle packing. It is illustrated in Figure 5. Table 3 shows how these two
propositions cut down on the number of embedded graphs that might correspond
to an optimal equal circle packing.
Proposition 5.2. Suppose that an embedded graph corresponds to an equal circle
packing and has a pair of edges, AB and CD, that when lifted to the plane determine
lines that are parallel. If Aand Dare on the same side of line ←→
BC (in the plane)
and BC is another edge of the graph, then AD is also an edge in the graph.
Proof. Suppose that an embedded graph corresponds to the packing graph of an
equal circle packing on a torus and that A,B,C, and Dsatisfy the conditions stated
in the proposition. See either part of Figure 5. This implies that AB,BC, and CD
is a chain of edges each of which has length to equal the common diameter, d, of
the circles. The geometry of this situation forces vertices Aand Dto be distance
dapart. This implies that the equal circles centered at these vertices are tangent
and that there is an edge between them.
6. Non-optimal and Non-globally Maximally Dense Packings
In this section we discuss the details associated with first part of Step 3 of
the overview of the proof found in Subsection 2.2. We discover those embedded
combinatorial graphs on three and four vertices that do not always embed in a
globally maximally dense way. In summary, of the 27 embedded combinatorial
graphs shown in Figures 6 and 7:
•Seven of the embedded graphs cannot be associated to an equal circle
packing on any torus (ECG4–3, ECG4–4, ECG9–2, ECG9–3, ECG10–1,
ECG10–2, ECG12–1);
OPTIMAL PACKINGS ON FLAT TORI 17
Number of distinct
toroidal embeddings
Number of embed-
dings remaining af-
ter Prop. 5.1
Number of embeddings
remaining after using
Props. 5.2 and 5.1
3 Vertices 6 6 6
4 Vertices 97 31 21
Table 3. The number of embedded graphs that might correspond
to an optimal packing after applying Propositions 5.2 and 5.1. See
Figures 6 and 7 for visualizations of the remaining embedded
graphs.
ECG1–1
int(R13)∪Ebl
ECG1–2
int(R23)∪El
ECG2–1
R23∩R13
ECG2–2
Not LMD
ECG2–3
R23∩x=1
2
ECG3–1
(1
2,√3
2)
Figure 6. The 6 embedded combinatorial graphs (ECG) on three
vertices that remain after applying Propositions 5.1 and 5.2. On
the first line, the number before the dash refers to the combi-
natorial graph (CG) that led to the embedding and the number
after the dash is the embedding number. The second line indi-
cates more about the embedding that is useful in understanding
the regions Ri3of Theorem 1.1. If there is a globally optimal
packing whose associated packing graph is the one under consid-
eration, then the tori containing that packing occupies a region in
the moduli space and it is loosely indicated. For example, ECG1–1
indicates “int(R13)∪Ebl” which means the interior of region R13
and the bottom (b) and left (l) edges of that region. If the embed-
ding corresponds to an equal circle packing, but it is not locally
maximally dense (LMD), then it reads “Not LMD”.
•Five of the embedded graphs lead to a family of equal circle packing(s) on
some tori, but the packings are never locally maximally dense (ECG2–2,
ECG4–1, ECG6–1, ECG9–4, ECG13–2);
•One of the embedded graphs leads to a family of equal circle packings that
are only locally maximally dense and not ever globally maximally dense
(ECG4–2); and
•Two of the embedded graphs lead to families of equal circle packings that
are locally (and not globally) maximally dense on some tori and are globally
maximally dense in other tori (ECG1–1, ECG9–1).
Several of the embedded combinatorial graphs are not associated to an equal
circle packing on any torus. The methods used to prove this are ad hoc but usually
18 BRANDT, DICKINSON, ELLSWORTH, KENKEL, AND SMITH
ECG4–1
Not LMD
ECG4–2
LMD Not GMD
ECG4–3
ECG4–4
ECG6–1
Not LMD
ECG7–1
int(R34)∪Er
ECG9–1
int(R24)∪Er
ECG9–2
ECG9–3
ECG9–4
Not LMD
ECG10–1
ECG10–2
ECG12–1
ECG13–1
R34∩x= 0
ECG13–2
Not LMD
ECG16–1
R24∩R34
ECG18–1
int(R14)∪Eb
ECG20–1
R14∩x= 0
ECG20–2
R14∩R24
ECG23–1
(1
2,√3
2)
ECG23–2
(0,2
√3)
Figure 7. The 21 embedded combinatorial graphs (ECG) on four
vertices that remain after applying Propositions 5.1 and 5.2. On
the first line, the number before the dash refers to the combi-
natorial graph (CG) that led to the embedding and the number
after the dash is the embedding number. The second line indi-
cates more about the embedding that is useful in understanding
the regions Ri4of Theorem 1.1. If there is a globally optimal
packing whose associated packing graph is the one under consid-
eration, then the tori containing that packing occupies a region in
the moduli space and it is loosely indicated. For example, ECG23–
1 indicates 1
2,√3
2which means that the packing is only possible
at this location in the moduli space. If the embedding corresponds
to an equal circle packing, but it is not locally/globally maximally
dense (LMD/GMD), then it reads “Not LMD/GMD”. If the sec-
ond line is blank, then the embedding doesn’t correspond to an
equal circle packing on any torus.
involve showing that if all the edges in the embedding have equal length, then there
are a pair of unconnected vertices that are forced to be too close together. For
example, we can eliminate the embedding ECG10–1 in Figure 8 from being associ-
ated to any equal circle packing on any torus by making the following observations.
(Note: in the discussion below, when we say that two edges in the torus are parallel,
we mean that when lifted to the Euclidean plane, they determine two lines that are
OPTIMAL PACKINGS ON FLAT TORI 19
A′A
B
C
DD′
Figure 8. The embedding ECG10–1 doesn’t correspond to any
equal circle packing on any torus.
ECG2–2 ECG4–1 ECG6–1 ECG9–4 ECG13–2
Figure 9. These are the five embedded combinatorial graphs that
correspond to equal circle packings that are not locally maximally
dense. Note that the two rightmost embeddings are different be-
cause the underlying combinatorial graphs are different.
parallel.) If this was associated to some equal circle packing, then all edge lengths
would be equal and dashed segments AA0and DD0would be equal in length and
parallel (when lifted to the plane) because their endpoints differ by the same lattice
vector. This makes triangles 4AA0Cand 4DD0Bcongruent which implies that
←−→
D0Bis parallel to ←→
AC or ←−→
A0Cdepending on which side of ←−→
DD0that the point B
is located. If ←−→
D0Bis parallel to ←→
AC, then Proposition 5.2 applies to the chain of
edges AC,C B, and BD0. If ←−→
D0Bis parallel to ←−→
A0C, then Proposition 5.2 applies to
the chain of edges CA0,A0B, and BD0. This eliminates this embedding. Embed-
dings ECG9–2, ECG9–3, ECG10–2, and ECG12–1 can be eliminated in a similar
way. The arguments to eliminate the embedded combinatorial graphs ECG4–3 and
ECG4–4 are more involved.
The embedded combinatorial graphs ECG2–2, ECG4–1, ECG6–1, ECG9–4, and
ECG13–2 correspond to circle packings that are not locally maximally dense. To
see that they correspond to equal circle packings see the circle packings in Figure 9
which shows the circle packings associated to these embedded graphs. None of these
packings are locally maximally dense because there is a non-trivial infinitesimal flex
for each. See Section 3 for a few details and [DGKX11a, Sec. 3] for complete details
and references. Roughly stated the non-trivial flex for the packing on the far left
of Figure 9 involves rotating the circles about the center of the equilateral triangle
in the packing graph. For the remaining ones, the non-trivial flex involves fixing
a circle or ‘row’ of circles and sliding another row. For example in the packing on
the second to the left, if you fix the circle at the lower left corner of a fundamental
20 BRANDT, DICKINSON, ELLSWORTH, KENKEL, AND SMITH
1
3
1 1
2
3
1
2
3
11
1
1
2
3
3
4
1
1
2
3 3
4
3
11 1
2
3
4
1 1
2
3 3
4
1
3
1
1
1
1
2
2
3 3
3
4 4
1
3
1 1
2
3
1
2
1
1
2
3
3
4
1
1
2
3
4
1
2
3
4
1 1
2
3 3
4
1
3
1
1
1
1
2
2
3 3
3
4 4
Figure 10. A locally and non-globally maximally dense equal cir-
cle packing associated to ECG4–2. On the right side, in black, we
see the region of the moduli space occupied by these packings. In
the top left we can see that it is possible to add a fifth circle (in
gray with dashed border) that is not constrained by its neighbors
(it is a rattler or free circle). It is not possible to add a sixth cir-
cle in the remaining empty space and have the packing be locally
maximally dense. In other regions of the moduli space adding a
fifth circle is not possible (bottom left).
domain, then the remaining circles can all be ‘slid’ upwards. In the language of the
article [Con90], these arrangements lack a proper stress.
The remainder of the embedded combinatorial graphs embed in a locally maxi-
mally dense way on the tori for which they are equal circle packings. First we had
to determine which region of the moduli space they occupy. To do this we con-
strained the embedding so that all lengths were equal and all the angles between
adjacent edges were between π
3(included) and π(excluded). If an angle is πor
greater then the packing is not locally maximally dense by Prop. 3.3. Under these
constraints we figured out which region of the moduli space the tori containing the
packing occupied. This is how the regions Rik(and boundaries) and the regions
in shown in Figures 10 and 11 were determined. To figure out if the equal circle
packing was locally maximally dense, we applied Theorem 6.2 (due to Roth and
Whiteley) and Lemma 6.2 from [Con88]; we found a rigid ordering and a proper
stress in the regions of the moduli space indicated. See [Con88] for examples and
complete details.
The embedded graph for ECG4–2 corresponds to a locally maximally dense pack-
ing that is not globally maximally dense. In Figure 10 you can see the associated
OPTIMAL PACKINGS ON FLAT TORI 21
Figure 11. The equal circle packings corresponding to ECG1–
1 (top row) and ECG9–1 (bottom row) are globally maximally
dense in some regions of the moduli space but in others are only
locally maximally dense. Where the gray regions overlap the strip
0≤x≤1
2they are globally maximally dense, but in the other
regions they are only locally maximally dense. For the packing in
the upper left, in the non-globally maximally dense region, there
is room for another circle (shown in gray with a dashed border).
Adding this circle results in a globally maximally dense packing
that occupies region R14(and four additional tangencies so that
the circle is not free).
equal circle packing and the region of the moduli space of flat tori that it can oc-
cupy. We could transform the region outside of the strip 0 ≤x≤1
2back into this
strip in the moduli space, but it would overlap itself and it is more convenient to
view it this way. We note that there is room for a fifth circle that is free in some
of the packings. This leads to a locally maximally dense packing of 5 circles that
contains a free circle.
Finally the combinatorial graphs ECG9–1 and ECG1–1 correspond to packings
that are locally maximally dense and in some regions of the moduli space are
globally maximally dense but not in others. See Figure 11.
7. Existence and Descriptions of the Globally Optimal Packings
In this section we discuss the details associated with last part of Step 3 of the
overview of the proof found in Subsection 2.2. We prove the existence of packings
that achieve the radii from Theorem 1.1 in the regions indicated. To eliminate
the translational symmetry we assume that there is always a circle centered at
22 BRANDT, DICKINSON, ELLSWORTH, KENKEL, AND SMITH
2
√3
1
2
(1
2,√3
2)
(1
2,1 + √3
2)
R22
R12
22 22 2 22 22
1
2
2
1 1
2
2
2
2
1 1 1
2
2
2
2
2
2
1 1
2
2
2
2
1
2
2
1
1
2
2
2
1
1
1
1
2
2
2
2
2
2
1
1
1
1
2
2
2
2
2
2
1
1
2
2
2
1
1
2
2
1
1
1
1
2
2
2
2
1
1
1
1
1
1
2
2
2
2
2
2
1
1
1
1
2
2
2
2
1
1
2
2
1
2
1 1
2 2
1 1 1
2 2 2
1 1
2 2
1
2
1
1
1
1
1
1
2
2
2
2
2
2
2
2
2
Figure 12. On the left is a typical optimal packing of two equal
circles in region R12. On the right is the corresponding location in
the moduli space of the torus.
the origin and to simplify the presentation of the coordinates, define the quantity
RRi
k=q16(rRi
k)2−1 where rRi
kis the expression for the optimal radius for kequal
circles in region Rik. Note that the formulas given in Theorem 1.1 for rk(x, y) are
all lower bounded by 1
4so RRi
kis always real and positive. Let CRi
kbe the list of
the centers for kcircles in region Rikup to equivalence. That is, the coordinates
given are in the plane and determine an equivalence class of points in the plane
that correspond to the location of the circle centers in the torus. Note that in the
regions R22,R33and R44(and all of lower edges of these regions – See Figure 1)
all the circles in the optimal arrangements are self-tangent with radius 1
2. The
arrangements in these regions (except on the lower edge) are far from unique –
every circle is free. See Section 4 for complete details.
7.1. Two Equal Circles. The case of 2 equal circles is included for the sake of
completeness. The optimal arrangements and radii are proved in [Prz06]. However,
the tools outlined in this article imply that after fixing the location of the center of
the first circle at the origin, the location of the second circle center in the optimal
arrangement must be at the circumcenter of the triangle with vertices the origin,
v1and v2. If this were not the case, then there would be fewer than 3 tangencies
and the arrangement could not be locally maximally dense. Note that the bound
in Proposition 3.6 part (1) is true even for n= 2. This implies that
CR1
2=(0,0),1
2,RR1
2
2.
In this region (including the lower edge without the left endpoint and the right edge
without the upper endpoint) there are three tangencies. Along the left edge (except
for the upper endpoint), when the torus is rectangular, there are four tangencies
and along the top edge (excluding the left endpoint) all the circles are self-tangent
with radius 1
2and there are five tangencies. The only exception to this when the
lattice has v2=h0,√3iand the packing forms the triangular close packing (each
circle is tangent to six others) with six tangencies. See Figure 12 for a typical
optimal packing in this region.
OPTIMAL PACKINGS ON FLAT TORI 23
2
2
√3
1
2
(1
2,√3
2)
R23
12
3
1 1
2 2
3 3
1 1 1
222
3 3 3
1 1 1
2 2
3 3
1 1
2
3
1
1
2
2
3
3
1
1
1
1
2
2
2
2
3
3
3
3
1
1
1
1
1
2
2
2
2
3
3
3
3
1
1
1
2
2
3
3
1
1
2
2
3
3
1
1
1
1
2
2
2
2
3
3
3
3
1
1
1
1
1
1
2
2
2
2
2
2
3
3
3
3
3
3
1
1
1
1
1
2
2
2
2
3
3
3
3
1
1
1
2
2
3
3
12
3
1 1
2 2
3 3
1 1 1
2 2 2
333
1 1
2 2
3 3
12
3
1
1
1
1
1
1
2
2
2
2
2
2
3
3
3
3
3
3
2
2
√3
1
2
(1
2,√3
2)
R23
31
3 3
1 1
333
1 1 1
3 3
1 1
3
2
3
3
1
1
2 2
3
3
3
3
1
1
1
1
222
3
3
3
3
3
3
1
1
1
1
1
1
2 2
3
3
3
3
1
1
1
1
2
3
3
1
2
2
3
3
3
1
1
1
1
2
2
2
2
3
3
3
3
3
3
1
1
1
1
1
1
1
1
2
2
2
2
3
3
3
3
3
3
1
1
1
1
1
2
2
3
3
3
1
2
2
2
3
3
1
1
1
2
2
2
2
2
2
3
3
3
3
1
1
1
1
1
2
2
2
2
2
2
2
2
2
3
3
3
3
3
3
1
1
1
1
1
2
2
2
2
2
2
3
3
3
3
1
1
1
2
2
2
3
3
1
2
2
3
1 1
2
2
2
2
3 3
1 1 1
2
2
2
2
2
2
3 3 3
1 1
2
2
2
2
3 3
1
2
2
3
1
1
1
1
1
1
1
2
2
2
2
2
2
3
3
3
3
3
3
3
3
3
Figure 13. The top row shows a typical optimal packing (asso-
ciated to ECG1–1) of three equal circles in region R13and corre-
sponding location in the moduli space. The bottom row shows the
same images only for optimal packings in region R23(associated
to ECG1–2).
7.2. Three Equal Circles. For three equal circles, notice that the moduli space is
broken into three regions and one can check that the radius is a continuous function
in the moduli space. It is interesting to note that on the boundary between regions
R13and R23, the radius is actually constant at 1
√12 . To establish the existence of
the packing consider the following locations for the circles:
CR1
3=(0,0),1
2,−RR1
3
2, √3RR1
3+ 1
4,√3−RR1
3
4!
CR2
3=(0,0),1
2,−RR2
3
2,1
2,RR2
3
2.
7.2.1. Region R13: Three Equal Circles. The typical packing in the interior of this
region (and including the left and lower edges without the right endpoint) has five
tangencies and is associated to ECG1–1. Along the upper edge of this region a sixth
tangency is formed so that circle 3 is tangent to circle 1 in two different ways so that,
with the edges from circle 2, two equilateral triangles are formed (associated to ECG
2–1). The only exception to this is when the lattice is triangular (i.e. v2=h1
2,√3
2i)
and the packing forms the triangular close packing with nine tangencies (associated
to ECG3–1). A typical optimal packing from region R13is shown the bottom row
of Figure 13.
24 BRANDT, DICKINSON, ELLSWORTH, KENKEL, AND SMITH
1
2
2 2
3
3
4 4
1 1
2
2 2
3 3
3
4 4
1
2
2
34
1
1
1
2
2 2
3 3
3
4
4 4
1
1
2
2
3
4
4
1 1
2
3 3
4
4 4
1 1
2
3 3
4
4
1
2
3
4
1
1
1
1
2
2 2
3 3
3
4
4 4
1
2 2
3
3
4
1 1
2 2
3 3
3
4 4
1
2
3 3
4
1
1
1
2
2
2
2
3
3
4 4
1
1
2
3
4
1 1
2 2
3 3
1 1
2
3 3
1
3
1
1
1
1
2
2
2
3 3
3
4 4
1
2 2
3
1 1
2 2
3
1
2
1
1
1
2
2
2
2
3
4 4
1
1
2
2
3
4
1 1
2 2
1 1
2 2
1
2
1
1
1
1
2
2
2
2
3 3
4 4
Figure 14. On the left (middle/right) is a typical optimal packing
of four equal circles in region R14(R24/R34). These packings are
associated to ECG18–1 (ECG9–1/ECG7–1).
7.2.2. Region R23: Three Equal Circles. The typical packing in the interior of this
region (and including the left edge without the endpoints) has five tangencies and
is associated to ECG1–2. See the top row of Figure 13. The right edge (excluding
the endpoints) of this region adds a sixth tangency so that the packing graph is a
union of rhombi when circle 2 is tangent twice to circle 3 (associated to ECG2–3).
Along the upper edge of this region all circles are self tangent with radius 1
2and
there are eight tangencies except at the righthand endpoint (where v2=h1
2,3√3
2i)
where the triangular close packing is formed with nine tangencies. See the left side
of Figure 4 for a typical packing along this circular edge. Note that this packing
graph is not on our list in Figures 6 or 7 because all the circles are self-tangent and
the packing graph contains a loop.
7.3. Four Equal Circles. For 4 circles, remarkably, the locations of the centers
of circles of the optimal arrangements in regions 1 and 2 is the same relative to the
corresponding radius, so the following is true for i= 1 and i= 2:
CRi
4=((0,0),1
2,RRi
4
2, 1−√3RRi
4
4,RRi
4+√3
4!, 3−√3RRi
4
4,3RRi
4+√3
4!).
For region 3, the centers are
CR3
4=(0,0),1
2,RR3
4
2,0, RR3
4,1
2,3RR3
4
2.
7.3.1. Region R14: Four Equal Circles. The optimal packings in the interior of this
region (and including the lower edge without the right endpoint) have nine tan-
gencies and the packing graph is the union of two triangles and three rhombi and
is associated to ECG 18–1. Along the left edge (excluding the upper endpoint),
circle 4 becomes tangent to the lift of circle 1 at v1+v2and the packing graph
creates an Archimedean-like tiling of four equilateral triangles and 2 rhombi (as-
sociated to ECG20–1). Along the top edge (excluding both endpoints), circle 4
becomes tangent to the lift of circle 1 at v2and the packing graph creates a differ-
ent Archimedean-like tiling of four equilateral triangles and 2 rhombi (associated
to ECG20–2). The exception to this is when the lattice becomes triangular (with
v2=h1
2,√3
2ior v2=h0,2
√3i) and the packing forms the triangular close packing
OPTIMAL PACKINGS ON FLAT TORI 25
with twelve tangencies (associated to ECG23–1 or ECG23–2). A typical optimal
packing from region R14is shown the left side of Figure 14.
7.3.2. Region R24: Four Equal Circles. On the interior of this region (and including
the right edge without the lower endpoint) the optimal packings have 8 tangencies
and is associated to ECG9–1. Along the top edge (excluding the left endpoint),
circle 2 becomes tangent to circle 3 in two different ways and the packing graph
is a tiling of 4 triangles and a hexagon (associated to ECG16–1). It is interesting
to note that on the boundary between regions R24and R34, the radius is actually
constant at 1
√12 . A typical optimal packing from region R24is shown the middle
portion of Figure 14.
7.3.3. Region R34: Four Equal Circles. The typical optimal packing in the interior
of this region (and including the right edge without either endpoint) has seven
tangencies and associated to ECG7–1. See the right side of Figure 14. Along left
edge (without either endpoint), circle 1 becomes tangent to circle 4 in two different
ways and the packing graph is the union of four rhombi (associated to ECG 13–1).
Along the upper edge of this region all circles are self tangent with radius 1
2and
there are eleven tangencies except at the lefthand endpoint (where v2=h0,2√3i)
where the triangular close packing is formed with twelve tangencies. See the right
side of Figure 4 for a typical packing along this circular edge. Note that this packing
graph is not on our list in Figures 6 or 7 because all the circles are self-tangent and
the packing graph contains a loop.
In conclusion, we have used rigidity theory to delineate the properties of the
combinatorial graphs associated to a locally maximally dense packing with three or
four equal circles. There was a list of 23 combinatorial graphs that could possibly
be associated to such an optimal equal circle packing. Using Edmond’s Permuta-
tion Technique we were able to make a list of 103 distinct topologically embedded
toroidal graphs. We were able to eliminate many of these using various techniques
and this left us with 15 toroidally embedded graphs that could be associated to an
optimal equal circle packing. One of these was locally (and not globally) maximally
dense, 2 embedded locally (and not globally) maximally densely on some tori and
globally maximally densely on others, and the remaining 12 embedded in a globally
maximally dense way. This exhaustive search had to include all globally maximally
dense packings (which have to exist because our packing domain is compact) and
therefore proves that the radii in Theorem 1.1 are globally optimal.
References
[Con88] Robert Connelly. Juxtapositions rigides de cercles et de sph`eres. I. Juxtapositions
finies. Structural Topology, (14):43–60, 1988. Dual French-English text.
[Con90] Robert Connelly. Juxtapositions rigides de cercles et de sph`eres. II. Juxtapositions in-
finies de mouvement fini. Structural Topology, (16):57–76, 1990. Dual French-English
text.
[DGKX11a] William Dickinson, Daniel Guillot, Anna Keaton, and Sandi Xhumari. Optimal pack-
ings of up to five equal circles on a square flat torus. Beitr. Algebra Geom., 52(2):315–
333, 2011.
[DGKX11b] William Dickinson, Daniel Guillot, Anna Keaton, and Sandi Xhumari. Optimal pack-
ings of up to six equal circles on a triangular flat torus. J. Geom., 102(1-2):27–51,
2011.
26 BRANDT, DICKINSON, ELLSWORTH, KENKEL, AND SMITH
[DTSC04] Aleksandar Donev, Salvatore Torquato, Frank H. Stillinger, and Robert Connelly.
A linear programming algorithm to test for jamming in hard-sphere packings. J.
Comput. Phys., 197(1):139–166, 2004.
[Edm60] J. Edmonds. A combinatorial representation for polyhedral surfaces. Notices Amer.
Math. Soc., 7:646, 1960.
[FT64] L. Fejes T`oth. Regular Figures, volume 48 of International Series of Monographs on
Pure and Applied Mathematics. Macmillan, New York, 1964.
[GKN03] Andrei Gagarin, William Kocay, and Daniel Neilson. Embeddings of small graphs on
the torus. Cubo, 5(2):351–371, 2003.
[GR08] Thierry Gensane and Philippe Ryckelynck. Square packings in the flat torus. Geom-
binatorics, 17(4):141–156, 2008.
[Hep99] Alad´ar Heppes. Densest circle packing on the flat torus. Period. Math. Hungar.,
39(1-3):129–134, 1999. Discrete geometry and rigidity (Budapest, 1999).
[Inc] Wolfram Research, Inc. Mathematica, Version 12.0. Champaign, IL, 2019.
[Jos06] J¨urgen Jost. Compact Riemann surfaces. Universitext. Springer-Verlag, Berlin, third
edition, 2006. An introduction to contemporary mathematics.
[KK05] William Kocay and Donald L. Kreher. Graphs, algorithms, and optimization. Dis-
crete Mathematics and its Applications (Boca Raton). Chapman & Hall/CRC, Boca
Raton, FL, 2005.
[Koc07] William Kocay. Groups & graphs – software for graphs, digraphs, and
their automorphism groups. Match, 58(2):431–443, 2007. Presented at the
conference “Computers in Scientific Discovery” in 2006, available from
http://www.combinatorialmath.ca/G&G/.
[LGS97] Boris D. Lubachevsky, Ron L. Graham, and Frank H. Stillinger. Patterns and struc-
tures in disk packings. Period. Math. Hungar., 34(1-2):123–142, 1997. 3rd Geometry
Festival: an International Conference on Packings, Coverings and Tilings (Budapest,
1996).
[Mel93] Hans Melissen. Densest packings of congruent circles in an equilateral triangle. Amer.
Math. Monthly, 100(10):916–925, 1993.
[Mel97] J. B. M. Melissen. Packing and covering with circles = Pakken en overdekken met
cirkels : (met een samenvatting in het Nederlands) / door Johannes Bernardus
Marinus Melissen. PhD thesis, Universiteit Utrecht, Utrecht, 1997.
[MN16] Oleg R. Musin and Anton V. Nikitenko. Optimal packings of congruent circles on a
square flat torus. Discrete Comput. Geom., 55(1):1–20, 2016.
[MP14] Brendan D. McKay and Adolfo Piperno. Practical graph isomorphism, II. J. Symbolic
Comput., 60:94–112, 2014.
[Prz06] Andrew Przeworski. Packing disks on a torus. Discrete Comput. Geom., 35(1):159–
174, 2006.
[Roy] Gordon Royle. Gordon royle’s small multigraphs. http://school.maths.uwa.edu.
au/~gordon/remote/graphs/multigraphs/index.html. Accessed: 2019-05-03.
[RS97] Andrew V. Reztsov and Ian H. Sloan. On 2D packings of cubes in the torus. Proc.
Amer. Math. Soc., 125(1):17–26, 1997.
[Sch65] J. Schaer. The densest packing of 9 circles in a square. Canad. Math. Bull., 8:273–277,
1965.
[Thu10] Axel Thue. Ein Fundamentaltheorem zur Bestimmung von Ann¨aherungswerten aller
Wurzeln gewisser ganzer Funktionen. J. Reine Angew. Math., 138:96–108, 1910.
OPTIMAL PACKINGS ON FLAT TORI 27
Department of Mathematics, University of California, Berkeley, Berkeley, CA 94720-
3840
E-mail address, Madeline Brandt: brandtm@berkeley.edu
Department of Mathematics, Grand Valley State University, Allendale, MI 49401
E-mail address, William Dickinson: dickinsw@gvsu.edu
254 Windsor St Unit 3L, Cambridge, MA 02139
E-mail address, AnnaVictoria Ellsworth: AnnaVictoria.ellsworth@gmail.com
Department of Mathematics, University of Utah, Salt Lake City, Utah 84112-0090
E-mail address, Jennifer Kenkel: kenkel@math.utah.edu
Department of Mathematics, University of Colorado, Boulder, Colorado 80309-0395
E-mail address, Hanson Smith: hanson.smith@colorado.edu
