Also available at http://amc.imfm.si
ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.)
ARS MATHEMATICA CONTEMPORANEA 3 (2010) 165–175
A new construction for symmetric
(4, 6)-configurations
Leah Wrenn Berman ∗
Department of Mathematics & Statistics, University of Alaska Fairbanks
Fairbanks, Alaska, USA
Nadine Alise Burtt
Department of Mathematics, University of Pittsburgh
Pittsburgh, Pennsylvania, USA
Received 16 November 2009, accepted 25 August 2010, published online 5 November 2010
Abstract
Geometric (4, 6)-configurations are collections of points and straight lines, in the Eu-
clidean plane, so that every point has four lines passing through it and every line has six
points lying on it. In this paper, we present a new construction for (4, 6)-configurations
which have high degrees of geometric symmetry, by superimposing 4-astral 4-configurations
with certain properties.
Keywords: Configurations, incidence geometry, symmetry.
Math. Subj. Class.: 51E30, 05B30
1 Introduction
A geometric (q, k)-configuration is a collection of points and straight lines in the Euclidean
plane, so that every point lies on q lines and every line passes through k points; if q = k,
we simply refer to k-configurations. If the number of points p and lines n is relevant to
the discussion, we refer to a (pq, nk) configuration. We say that a (q, k)-configuration
is symmetric if, under rotations and reflections of the plane mapping the configuration to
itself, there are fewer symmetry classes of points than the number of points in the con-
figuration, and similarly for lines: configurations that are highly symmetric have a small
number of symmetry classes of points and lines. The modern study of geometric config-
urations began about 20 years ago, with the discovery by Grünbaum and Rigby [13] of a
∗Corresponding author.
E-mail addresses: lwberman@alaska.edu (Leah Wrenn Berman), naburtt@gmail.com (Nadine Alise Burtt)
Copyright c© 2010 DMFA Slovenije
166 Ars Math. Contemp. 3 (2010) 165–175
highly symmetric drawing of a (214) configuration with three symmetry classes of points
and lines; since then, there has been considerable work investigating 4-configurations with
various properties (see, e.g., [5, 4, 6, 1, 7, 8, 11]). However, there has been relatively
little investigation of configurations where q and k are not equal and [q, k] 6= [3, 4], al-
though there are a few results in [3, 2], in a very constrained situation, where the config-
urations have as much symmetry as possible, and there are some results in Grünbaum’s
recent monograph on configurations [12, Section 4.4]. The current work presents a general
method of constructing (4, 6)-configurations with four symmetry classes of lines and six
symmetry classes of points, using as building blocks a reasonably well-understood class
of 4-configurations called 4-astral configurations. Note that two examples of the type of
(4, 6)-configurations discussed in this paper were presented without discussion in [12, Fig-
ures 4.4.8 and 4.4.10(b)].
2 Multiastral 4-configurations
To construct (4, 6)-configurations, we will use multiastral—specifically 4-astral—4-confi-
gurations as building blocks. These configurations have been studied fairly extensively (see
e.g., [4, 7, 11, 12]; in [7] they were called polycyclic and in [4, 6] they were called celestial.
The current termiology is that used in Branko Grünbaum’s recent monograph on configu-
rations [12, Sections 1.5, 3.5–3.9]; the following discussion of multiastral configurations
is adapted from that source as well, along with his survey article [11] and the first author’s
article [4].
A multiastral 4-configuration is a collection of points and straight lines in the Euclidean
plane so that every point has four lines, from each of two symmetry classes, passing through
it. Moreover, every symmetry class of points has the same number of points, say m, in it,
and the points in each symmetry class form concentric regular m-gons. The symmetry
group of the entire configuration is dm, and every line contains two points from each of
two m-gons. Multiastral 4-configurations are a generalization of astral 4-configurations,
which are 4-configurations with precisely two symmetry classes of points and two symme-
try classes of lines (see, e.g., [1, 12, 10, 9]). A 3-astral configuration is shown in Figure
1.
Multiastral configurations with h symmetry classes of points and lines are called h-
astral, and every h-astral configuration may be described by a configuration symbol of the
form
m#(s1, t1; s2, t2; . . . ; sh, th),
where there are m points in each symmetry class of points and m lines in each symmetry
class of lines. Using a configuration symbol, it is possible to construct a configuration
uniquely (although several different configuration symbols may correspond to the same
geometric configuration, depending on a choice of labelling). In order for a configuration
symbol to be valid, it must satisfy four axioms (see [11, Section 3.5] for details).
Axiom 1:
h∑
i=1
si + ti is even
Axiom 2: si 6= ti 6= si+1 for i = 1, . . . , h− 1 and, additionally, sh 6= th 6= s1
Axiom 3:
∏
cos(siπ/m) =
∏
cos(tiπ/m)
L. W. Berman and N. A. Burtt: A new construction for symmetric (4, 6)-configurations 167
Figure 1: The 3-astral 4-configuration 9#(3, 1; 2, 3; 1, 2). Lines L0 and points v0 are blue,
lines L1 and points v1 are red, and lines L2 and points v2 are green.
Axiom 4: No proper subsequence (si, ti, si+1, ti+1, ..., sj) generates a valid configuration
symbol
m#(si, ti; si+1, ti+1; ...; sj , t∗)
that satisfies Axioms 1 – 3, where 1 ≤ t∗ < m2 .
A set of lines ` = {`0, `1, . . . , `m−1} is of span s with respect to a set of vertices
{v0, v1, ..., vm−1} forming a regular m-gon if `i connects vi and vi+s, with indices taken
modulo m. Given a set of lines ` of span s, the t-th intersections of those lines, collectively
labelled (s//t) (also referred to as [[s, t]] in, e.g., [4, 6]), is found by starting at the “mid-
point” of `0 (that is, the foot of the perpendicular line to `0 that passes through the center
of the configuration) and counting leftward through t intersections of the lines ` with each
other; see Figure 2. Given an m-gon and a set of lines of span s, allowable values for t are
integers from 1 to m2 .
Given a valid configuration symbol C = m#(s1, t1; s2, t2; ...; sh, th), the construction
method to produce the configuration is as follows (adapted from the algorithm in [4]).
Step 1: Begin with m points forming the vertices of a regular m-gon. Collectively, these
vertices will be referred to as (vC)0. Typically, these vertices have coordinates(
cos
(
2πi
m
)
, sin
(
2πi
m
))
for i = 0, 1, . . . ,m− 1.
Step 2: Construct lines collectively known as (LC)0 of span s1 that connect these vertices.
Step 3: Construct the t1-st intersections of the lines (LC)0 and call them (vC)1; they have
symbol (s1//t1).
168 Ars Math. Contemp. 3 (2010) 165–175
Hs1L
Hs2L
Hs3L
Hs4L
Hs5L
Figure 2: Lines of span s with respect to points v0 and other intersections of the span s
lines, labelled (s//t). Here, m = 12, s = 3 and t = 1, 2, 3, 4, 5. The points v0 are shown
in blue.
O
A
B
B'
C
x
Figure 3: Determining the radius of a point with label C = (s//t) with respect to a regular
convex m-gon with radius r. Since the blue lines are of span s, point B′ has coordinates
(cos(2sπ/m), sin(2sπ/m)), so ∠BOA = sπm , where A is the foot of the perpendicular
from the center O to the line BB′. If OB = r, then since cos(∠AOB) = OAOB , it follows
that OA = r cos(sπ/m). Since C, which has label (s//t), is the t-th intersection of the
span s lines, ∠AOC = tπm . Therefore, cos(∠AOC) =
OA
OC , so OC = r ·
cos(sπ/m)
cos(tπ/m) . In the
diagram, m = 7, s = 3 and t = 2.
L. W. Berman and N. A. Burtt: A new construction for symmetric (4, 6)-configurations 169
Step 4: For i = 1, . . . , h−1, iteratively repeat the previous two steps: using vertices (vC)i,
construct lines Li of span si+1, and let (vC)i+1 be the ti+1-st intersections of the
lines Li, with label (si+1//ti+1).
Because the symbol is valid, the points (vC)h must coincide with the original m points
labelled (vC)0. The C subscripts may be dropped when the configuration that vi and Li
refer to is either unambiguous or irrelevant.
Adapting the discussion in [6], we say that a ray of an h-astral configuration is a diame-
tral ray of the configuration if it emanates from the center of symmetry of the configuration
(conventionally taken to be the origin) and passes through a point from the set v0. A ray
is a mid-diametral ray if it is the rotation by an angle of πm of some diameter. If diametral
rays can pass through a class of points, that class is said to be diametral or of type D, and
likewise if mid-diametral rays can pass through the points they are said to be mid-diametral
or of type MD. (In a typical configuration centered at the origin with one vertex in v0 lo-
cated at (1, 0), diametral points have angle πim with i even, while mid-diametral points have
angle πim for odd i.) If there are two classes of points and they both are diametral or both
are mid-diametral, the classes of points are the same type. In Figure 1, the points v0 and v1
(blue and red) are type D, and thus of the same type, while the points v2 are type MD.
In our construction of (4, 6)-configurations, it is useful to be able to determine the
radius of the circumcircles passing through the regular m-gons formed by the vertices vi
(the “radius of the vi”). Using elementary trigonometry, the radius of a point with label
(s//t) with respect to a regular convex m-gon of radius r is
r ·
cos
(
sπ
m
)
cos
(
tiπ
m
) ;
see Figure 3.
Let ri be the radius of vertices with label vi, and suppose r0 = 1. Because of the
iterative nature of the construction of h-astral configurations, the radius
rj =
j∏
i=1
cos
(
siπ
m
)
cos
(
tiπ
m
) .
Note that we take Axiom 3 as a necessary condition for the existence of a valid configura-
tion because of the convention that r0 = 1, the requirement that v0 = vh, and using this
value for rh.
Often, h-astral configurations are classified by considering the cohort symbolm#S;T ,
where S = {s1, . . . , sh} and T = {t1, . . . , th}. Trivial h-astral configurations are those
where S = T (as sets); Axioms 1 and 3 are satisfied without need for computation. Sys-
tematic h-astral configurations are those where S 6= T , but the values of S and T are
determined by dependence on distinct parameters. Sporadic h-astral configurations are
neither trivial nor systematic.
3 Constructing symmetric (4, 6)-configurations
Consider the trivial 4-astral configuration shown in Figure 4. This configuration consists of
four symmetry classes of points and four symmetry classes of lines, with four points, two of
each of two colors, on each line. However, there are extra four-valent intersections formed
170 Ars Math. Contemp. 3 (2010) 165–175
by the intersection of some of the lines: specifically, the blue and green lines (lines L0 and
L2) and the red and magenta lines (lines L1 and L3) intersect two at a time. Adding these
additional points would form a (4, 6)-configuration: each point would still have four lines
passing through it, but each line would now have six points, from each of three symmetry
classes, lying on it.
We can further analyze this example by realizing that the (4, 6)-configuration thus
formed may be considered as being constructed from two separate 4-astral configurations,
which have the same sets of lines (although different labels), and the same points v0 and
v2, but different points v1 and v3. Figure 5 shows such a situation: here the points v0 and
v3, colored blue and green respectively, are the same in both configurations, and as sets the
lines of the two configurations are the same as well, although the colors are switched.
Definition 3.1. Two h-astral configurations X and Y are superimposable if they have the
same sets of lines, geometrically, and the incidence structure formed from those lines and
the collection of points from both configurations is a (4, 6)-configuration.
The superimposibility of 9#(3, 1; 4, 2; 1, 3; 2, 4) and 9#(3, 2; 4, 1; 2, 3; 1, 4) is not co-
incidental: in fact, there are infinitely many such pairs of superimposable configurations.
To prove this, we will show that a particular pair of configurations X and Y is superim-
posable, by showing that they have the same set of lines and that the points (vX )0 = (vY)0
and (vX )2 = (vY)2. To do this, we will need the following lemma, slightly restated from
[6, Lemma 1]:
Lemma 3.2. For a given i, if si ≡ ti mod 2, the points labelled vi, with symbol (si//ti),
are the same type as the points labelled vi−1 (that is, (si−1//ti−1)), with indices taken
modulo h; if si 6≡ ti mod 2, then the points vi and vi−1 are of opposite type.
Theorem 3.3. Let X and Y be valid configurations with symbols X = m#(a, x1;x2, d;
b, x3;x4, c) and Y = m#(a, y1; y2, b; d, y3; y4, c). If
cos(x2π/m)
cos(x1π/m) cos(dπ/m)
=
cos(y2π/m)
cos(y1π/m) cos(bπ/m)
(3.1)
and
x1 + x2 + y1 + y2 + d+ b is even, (3.2)
then X and Y are superimposable.
Proof. Suppose that x1 + x2 + y1 + y2 + d+ b is even and
cos(x2π/m)
cos(x1π/m) cos(dπ/m)
=
cos(y2π/m)
cos(y1π/m) cos(bπ/m)
.
Let (vX )0 = (vY)0 = v0 be the set of points with coordinates(
cos
(
2πi
m
)
, sin
(
2πi
m
))
for i = 0, 1, . . . ,m− 1. By the choice of symbol, (vX )0 and (vY)0 have lines of the same
spans passing through them: that is, (LX )0 = (LY)0 and (LX )3 = (LY)3, which are lines
of spans a and c, respectively, with respect to the points v0.
L. W. Berman and N. A. Burtt: A new construction for symmetric (4, 6)-configurations 171
Figure 4: LHS: The trivial 4-astral (544, 366) configuration 9#(3, 1; 4, 2; 1, 3; 2, 4). There
are additional four-valent intersections between the lines, specifically blue-green and red-
magenta intersections, which are not points of the configuration; these are highlighted
by the gray circles. RHS: Adding in the additional intersection points leads to a (4, 6)-
configuration; the additional points are black and cyan. With different coloring, this con-
figuration is shown as Figure 4.4.10 in [12].
Figure 5: The two superimposable configurations which when combined form the (4, 6)-
configuration shown in the right-hand side of Figure 4. LHS: 9#(3, 1; 4, 2; 1, 3; 2, 4); RHS:
9#(3, 2; 4, 1; 2, 3; 1, 4).
172 Ars Math. Contemp. 3 (2010) 165–175
Figure 6: A (4, 6)-configuration formed from the nontrivial superimposable pair
12#(4, 2; 4, 3; 2, 5; 3, 1) and 12#(4, 3; 4, 2; 3, 5; 2, 1).
Figure 7: A (5, 6)-configuration formed by adding diameters to the superpo-
sition of 13#(5, 3; 4, 1; 3, 5; 1, 4) (blue, red, green, and magenta points) and
13#(5, 1; 4, 3; 1, 5; 3, 4) (blue, yellow, green, and cyan points).
L. W. Berman and N. A. Burtt: A new construction for symmetric (4, 6)-configurations 173
Note that, again by the choice of symbol, the points (vX )2 have lines (LX )2 of span
d and (LX )1 of span c passing through them; similarly, the points (vY)2 have lines (LY)2
of span d and (LY)1 of span c passing through them. Therefore, to show that X and Y
are superimposable, it suffices to show that (vX )2 is the same set of points, geometrically,
as (vY)2, which we will do by showing that they are of the same type and have the same
radius.
Lemma 3.2 states that vi−1 and vi are of the same type if si and ti have the same parity;
that is, when si + ti is even. Following the type changes through the symbol, if (vX )2 and
(vY)2 are of the same type, then a+ x1 + x2 + d and a+ y1 + y2 + b must have the same
parity, since (vX )0 = (vY)0. Since we assumed that x1 + x2 + y1 + y2 + d+ b is even, it
follows that (vX )2 and (vY)2 are of the same type.
Now, let (rX )i and (rY)i be the radii of (vX )i and (vY)i respectively. By construction,
(rX )0 = (rY)0 = 1. Then
(rX )2 = (rX )1 ·
cos(x2π/m)
cos(dπ/m)
=
cos(aπ/m) cos(x2π/m)
cos(x1π/m) cos(dπ/m)
and
(rY)2 = (rY)1 ·
cos(y2π/m)
cos(bπ/m)
=
cos(aπ/m) cos(y2π/m)
cos(y1π/m) cos(bπ/m)
.
Since we assumed that
cos(x2π/m)
cos(x1π/m) cos(dπ/m)
=
cos(y2π/m)
cos(y1π/m) cos(bπ/m)
,
it follows that (rX )2 = (rY)2, as desired.
Corollary 3.4. Superimposable configurations
X = m#(a, x1;x2, d; b, x3;x4, c)
and
Y = m#(a, y1; y2, b; d, y3; y4, c)
satisfy
cos(bπ/m) cos(x4π/m)
cos(x3π/m)
=
cos(dπ/m) cos(y4π/m)
cos(y3π/m)
. (3.3)
Proof. Since X and Y are both valid configurations, the radius r4 for each configuration
must equal 1, because v4 = v0. Since X and Y are superimposable, (vX )2 ≡ (vY)2; call
the common radius r. Then
(rX )4 = r ·
cos
(
bπ
m
)
cos
(
x3π
m
) · cos (x4πm )
cos
(
cπ
m
)
and
(rY)4 = r ·
cos
(
dπ
m
)
cos
(
y3π
m
) · cos (y4πm )
cos
(
cπ
m
) .
Since (rX )4 = (rY)4 = 1, the result follows.
174 Ars Math. Contemp. 3 (2010) 165–175
One very nice class of pairs of superimposable configurations are the trivial pairs
X = m#(a, b; c, d; b, a; d, c) and Y = m#(a, d; c, b; d, a; b, c);
these trivially satisfy Theorem 3.3. Figure 5 shows such a trivial pair. Figure 6 shows a
(4, 6)-configuration formed from the nontrivial pair 12#(4, 2; 4, 3; 2, 5; 3, 1) and 12#(4, 3;
4, 2; 3, 5; 2, 1).
4 Generalizations and open questions
Given a (4, 6)-configuration with three symmetry classes of points of one type and three of
the other type, it is possible to construct symmetric (5, 6)-configurations by adding diame-
ters; an example of such a configuration is shown in Figure 7.
In particular, consider a superimposable pair of trivial configurations X = m#(a, b;
c, d; b, a; d, c) and Y = m#(a, d; c, b; d, a; b, c). Suppose that a, b, d are of the same parity
and c is of the opposite parity to a, b, d. Applying Lemma 3.2 several times, we conclude
that (vX )0 = (vY)0 are type D, (vX )1 and (vY)1 are type D, since a ≡ b ≡ d mod 2,
(vX )2 = (vY)2 is of type MD, since c 6≡ b mod 2 and c 6≡ d mod 2, and (vX )3 and
(vY)3 are both of type MD, since a ≡ b ≡ d mod 2. Thus, in the superimposed (4, 6)-
configuration, there are three classes of points of type D, namely (vX )0, (vX )1 and (vY)1,
and three classes of points of type MD, (vX )2, (vX )3, and (vY)3, so if diameters (that is,
lines connecting the origin and points in v0) are added to the configuration, each diameter
will pass through six points.
There are several interesting ways to generalize the notion of superimposibility. For
example, consider two configurations to be superimposable if they
• share the same point sets, but different line sets, but combine into some (q, k)-
configuration
• Have different point and line sets, but still combine to form a (q, k)-configuration
Clearly, the (6, 4)-configurations formed as the polars of the (4, 6)-configurations con-
structed above may be analyzed as being formed from two superimposable 4-astral config-
urations which share the same point sets but different line sets.
Question 1. A 4-astral configuration cohortm#S;T is reducible if S = {x1, x2, i, j} and
T = {y1, y2, i, j} and m#S′;T ′ is a valid 2-astral configuration, where S′ = {x1, x2}
and T ′ = {y1, y2}. So far, the only known nontrivial superimposable pairs are reducible
(for example, the configurations which superimpose to form the (4, 6)-configuration in Fig-
ure 6 both reduce to the configuration cohort 12#{4, 4}; {5, 1}.) Are there nonreducible
pairs of superimposable 4-astral configurations?
Question 2. Are there other interesting configurations which can be formed by superposi-
tion, perhaps using h-astral configurations for h > 4?
Question 3. Is it possible to construct interesting configurations by superimposing more
than two configurations?
Question 4. Is it possible to construct 6-configurations by superimposing multiple (4, 6)-
or (6, 4)-configurations?
L. W. Berman and N. A. Burtt: A new construction for symmetric (4, 6)-configurations 175
5 Acknowledgements
The authors are grateful to the Ursinus College Summer Fellows program for providing
support for this research.
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