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ARS MATHEMATICA CONTEMPORANEA 1 (2008) 44–50
A New Class of Movable (n4) Configurations
Leah Wrenn Berman
Ursinus College, Department of Mathematics and Computer Science, P.O. Box 1000,
Collegeville, Pennsylvania, USA
Received 30 July 2007, accepted 3 July 2008, published online 8 July 2008
Abstract
A geometric (n4) configuration is a collection of n points and n lines, usually in the Eu-
clidean plane, so that every point lies on four lines and every line passes through four points.
This paper introduces a new class of movable ((5m)4) configurations—that is, configura-
tions which admit a continuous family of realizations fixing four points in general position
but moving at least one other point—including the smallest known movable (n4) configura-
tion.
Keywords: Configurations.
Math. Subj. Class.: 52C30, 52C99
A geometric (n4) configuration is a collection of n points and n lines, usually in the Eu-
clidean plane, so that every point lies on four lines and every line passes through four points.
In the last few years, there has been a fair amount of activity concerning (n4) configurations,
both in answering existence questions [4, 7] and classification questions [1, 2, 3, 6].
In [1], a class of symmetric (n4) configurations (that is, configurations with non-trivial
geometric symmetry) was introduced which were movable: they admit a continuous family
of realizations fixing four points in general position but moving at least one other point.
This paper introduces a second class of movable (n4) configurations which are also quite
symmetric and are smaller and simpler than the previous class; for example, the smallest
example of the new class is a (304) configuration, which is the smallest known movable
(n4) configuration, while the smallest example of the old class is a (444) configuration. In
addition, this new class provides examples of movable configurations which have m-fold
rotational symmetry for odd m (the previous class required m to be even). Three members
of a continuous family of (354) configurations are shown in Figure 1.
The class of configurations described here consists of ((5m)4) configurations with cyclic
symmetry; they have five symmetry classes of points and lines ifm is odd, and five symmetry
E-mail address: lberman@ursinus.edu (Leah Wrenn Berman)
Copyright c© 2008 DMFA – založništvo
L. W. Berman: A New Class of Movable (n4) Configurations 45
Figure 1: Three versions of a movable (354) configuration, with different choices of contin-
uous parameter; the discrete parameters are m = 7, a = 1, and b = 2.
46 Ars Mathematica Contemporanea 1 (2008) 44–50
classes of points and six symmetry classes of lines if m is even. In the language introduced
by Grünbaum in [5], if m is odd the configurations are (5)-astral and if m is even they are
(5, 6)-astral.
Given a regular m-gonM with vertices labelled cyclically as v0, . . . , vm−1, a diagonal
of span s is a line that connects vertices vi and vi+s. Given a set of all diagonals of span
s, we can label points formed by the intersection of a diagonal with other diagonals of the
same span; again following Grünbaum [5], we say that the t-th intersection, counted from the
center of a span s diagonal in some direction, is labelled [[s, t]].
The construction for the new class of movable configurations depends on the following
theorem from [1], listed there as Theorem 3 and given here in a restated form (See Figure 2):
Theorem 1 (Crossing Spans). Begin with a set of diagonals of span a and span b of anm-gon
M. Place an arbitrary point A on a diagonal of span a. Construct another m-gon N whose
vertices are the rotated images of A through angles of 2πim , and construct diagonals of span
b using N . Two of these span b diagonals intersect each other and a span b diagonal ofM
in the same point, and the intersection points are precisely the points labelled [[b, a]] in N .
Figure 2: Crossing spans: Construct a regular m-gon (here, m = 7) and lines of span a and b
(here, a = 2 and b = 3; the span a lines are blue and the span b lines are green). If a point is
placed arbitrarily on a line of span a and rotated to make a regular m-gon, and lines of span
b are constructed using this m-gon (the red lines), then two of these lines of span b and one
of the original lines of span b intersect in a single point (here, the innermost polygon, which
is formed as the intersections of two red lines and one green line).
The construction method is as follows.
Begin with the vertices of a regular m-gonM centered at the origin O (the black circles
in Figure 3), and choose distinct integers a and b, with 1 ≤ a, b < m2 , where if m is even, a
and b are of opposite parity. Using the points inM, construct all diagonals of span a (shown
in black in Figure 3). Place a point A0 arbitrarily on one of the diagonals, and construct the
images under rotation by 2πim about the origin to form a new m-gon A; we will label these
points cyclically as A0, A1, . . . , Am−1 (the blue squares in Figure 3). Construct the ray
−−→
OA0
and let B0 be an intersection of this ray with one of the diagonals of span a in a point other
than A0. (If m is large, there may be many choices for which line to place B0 on.) Rotate
L. W. Berman: A New Class of Movable (n4) Configurations 47
Figure 3: A diagram showing the elements of the construction of the new movable configura-
tions. Points inM are shown as black circles, points in A are shown as blue squares, points
in B are shown as green triangles, points in D are shown as yellow diamonds, and points in
E are red pentagons. The lines of span a connecting points in A are black. The lines LM are
green, LA are red, and LB are blue. The diameters are purple. In this diagram, m = 7, a = 2
and b = 3.
48 Ars Mathematica Contemporanea 1 (2008) 44–50
this point B0 around to form a second regular m-gon, B; we will label these points cyclically
as B0, B1, . . . , Bm−1 (the green triangles in Figure 3).
Now construct all diagonals of span b using the m-gons M, A, and B, and label these
sets of lines as LM, LA and LB, respectively; in Figure 3, lines in LM are green, lines in LA
are red, and lines in LB are blue.
By Theorem 1, two diagonals from LA and one diagonal from LM intersect in a single
point. Call them-gon of points formedD (the yellow diamonds in Figure 3); similarly, call E
the m-gon of points formed by the intersection of two diagonals from LB with one diagonal
of LM (the red pentagons in Figure 3).
We now have five sets of points. The points ofM have four lines passing through them,
but the points in A, B, D, and E only have three lines passing through them. To be a (n4)
configuration, every point must have four lines passing through them.
The final set of lines will be diameters—that is, lines that pass through the origin. How-
ever, which diameters need to be used and which points lie on them depends on whether m
is odd or even and on how the points of the various polygons are aligned; the construction is
slightly different depending on the parity of m.
To determine how the points in D and E are aligned with respect to the points of A and
B, consider the points of A and all the diagonals of span b using those points (the lines LA).
Given a point Ai in A and a point P in D which lies on a line of span b using the points
Ai and has label [[b, a]], the angle AiOP is πjm for some j, where j is odd if a is odd and
j is even if a is even. That is, a ray which intersects Ai also passes through a point in D,
which has label [[b, a]], if b and a have the same parity. Note that a line through the origin
that passes through a point Ai is composed of one ray through Ai and one ray with angle π
with respect to
−−→
OAi; if m is odd, this second ray does not pass through a second point in A,
while if m is even, it does. Therefore, if m is even and b and a are of the same parity, a line
through the origin passing through Ai contains two points from A, two points from B and
two points from D; this is too many points for an (n4) configuration, which is why a and b
were required to be of opposite parity when m is even.
What remains is to show how to add appropriate diameters, so that a configuration is
constructed in which every line passes through precisely four points and every point lies on
the intersection of four lines.
Case 1: m is even
Suppose in this case that m = 2k. Given a point Ai in A, the point Bi in B lies on the line
OAi. However, since m is even,M has 180◦ rotational symmetry, so the points Ai+k and
Bi+k also lie on OAi. Thus, if we add to the set of lines in the configuration the set of all
diameters of the form OAi for Ai in A, all of the points in A and B have a diameter passing
through them, and each of these diameters has four points lying on it, two from A and two
from B.
Now consider the points inD and E . Recall that both Di and Ei were chosen to be points
with label [[b, a]] based on the span b lines LA and LB; that is, pointDi is the a-th intersection
on a particular line in LA with other lines in LA, counted from the center. Moreover, sincem
is even, we chose b and a to be of opposite parity. Therefore, points Di and Ei lie on a single
line through the origin, and because of the parity constraint, this line does not pass through
Ai and Bi. However, because D and E also have 180◦ rotational symmetry, a line ODi also
passes through the points Di+k and Ei+k. Therefore, if we include these diameters as lines
L. W. Berman: A New Class of Movable (n4) Configurations 49
Figure 4: Movable configurations where m is even. Left hand side: a (304) configuration
withm = 6, a = 1, b = 2; right hand side: a (504) configuration withm = 10, a = 3, b = 4.
of the configuration, we have five classes of points (polygonsM, A , B , D, and E) and six
classes of lines (lines of span a through M, lines of span b through M, A, and B (that is,
LM, LA and LB), diameters through Ai, which contain two points from A and two points
from B, and diameters through Di, which contain two points fromD and two points from E).
Two examples of such configurations are shown in Figure 4.
Case 2: m is odd.
By construction, given a point Ai in A, the point Bi in B lies on the line OAi, since that was
how the Bi’s were chosen. BecauseM does not have 180◦ rotational symmetry (since m is
odd), no other points of A and B lie on that line.
The intersection points Di and Ei also lie on OAi. To see this, note that points in D are
of label [[b, a]] with respect to A, points in E are of label [[b, a]] with respect to B, and A and
B are aligned. If a and b are of the same parity, then points of label [[b, a]] lie on rays
−−→
OAi,
so points Di and Ei lie on the same side of the origin as Ai and Bi (see the right hand side of
Figure 5). However, if a and b are of opposite parity points Di and Ei lie on a ray that makes
an angle of π with respect to
−−→
OAi; since m is odd, this ray combines with
−−→
OAi to make a
line through the origin, with Di and Ei lying on the opposite side of the origin from Ai and
Bi (see the left hand side of Figure 5). In either case, four points of the configuration lie on
each line OAi.
Thus, we use as points of the configuration the points in the m-gonsM, A, B, D, and
E , and the lines of the configuration are the diagonals of span a usingM, the diagonals of
span b usingM,A, B (that is, LM, LA and LB), and the diametersOAi for i = 0, 1, . . . ,m.
Each diameter contains one point from A, B, D, and E .
Examples of such configurations are shown in Figure 5.
Finally, note that A0 was chosen arbitrarily; its position along the line of span a forms a
50 Ars Mathematica Contemporanea 1 (2008) 44–50
Figure 5: Movable configurations wherem is odd. Left hand side: a (354) configuration with
m = 7, a = 3, b = 2; right hand side: a (654) configuration with m = 13, a = 3, b = 5.
continuous parameter. Figure 1 shows three versions of a (354) configuration, with various
choices for the position of A0. Also, if b > a, as in the left hand side of Figure 5, then the
polygons D and E will be outside the polygons A and B.
References
[1] L. W. Berman, Movable (n4) configurations, Electron. J. Combin. 13 (2006), #R104.
[2] L. W. Berman, A characterization of astral (n4) configurations, Discrete Comput. Geom. 26
(2001), no. 4, 603–612.
[3] M. Boben and T. Pisanski, Polycyclic configurations, European J. Combin. 24 (2003), 431–457.
[4] J. Bokowski, B. Grünbaum and L. Schewe, Topological configurations (n4) exist for all n ≥ 17,
European J. Combin., to appear.
[5] B. Grünbaum, Configurations of points and lines. In The Coxeter Legacy: Reflections and Pro-
jections (C. Davis and E. W. Ellers, eds), American Mathematical Society (2006), 179–225.
[6] B. Grünbaum, Astral (n4) configurations, Geombinatorics 9 (2000), 127–134.
[7] B. Grünbaum, Which (n4) configurations exist?, Geombinatorics 9 (2000), 164–169.