Also 
available 
at 
http://amc.imfm.si 


ARS 
MATHEMATICA 
CONTEMPORANEA 
1 
(2008) 
44–50 


ANew Class of Movable(n4) 
Configurations 

Leah Wrenn Berman 

Ursinus College, Departmentof Mathematics and Computer Science,P.O. Box 1000, 
Collegeville,Pennsylvania, USA 

Received30July2007, accepted3July2008, published online8July2008, 
adjusted for printing 25 August 2008* 


Abstract 

Ageometric(n4) 
configuration is a collection of n 
points and n 
lines, usually in the Euclidean
 plane, so thatevery point lies on four lines andevery line passesthroughfour points. 
This paper introduces a new class of movable ((5m)4) 
configurations—that is, configurations
 which admita continuousfamilyof realizations fixing four pointsin general position 
but moving at least one other point—including the smallest known movable(n4) 
configuration.
 

Keywords: Configurations. 
Math. Subj. Class.: 52C30, 52C99 

Ageometric(n4) 
configuration is a collection of n 
points and n 
lines, usually in the Euclidean
 plane, so that every point lies on four lines and every line passes through four points. 
In the lastfew years, there has beenafair amountof 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: theyadmita continuousfamily 
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 
ofa continuousfamilyof (354) 
configurations are shown in Figure 1. 

The class of configurations described here consists of ((5m)4) 
configurations with cyclic 
symmetry; theyhave five symmetry classes of points and lines if m 
is odd, and five symmetry 
classes of points and six symmetry classes of lines if m 
is even. In the language introduced 

For technical reasons, all figures in the printed version appear on pages 45, 47 and 49. The original online 
version version can be found at http://amc.imfm.si/index.php/amc/article/viewFile/17/26 
. 
E-mail address: lberman@ursinus.edu (Leah Wrenn Berman) 

Copyright czni.

 
2008 DMFA – zalo.stvo 


L.W. Berman:ANew ClassofMovable (n4) 
Configurations 
Figure 1: Three versions of a movable (354) 
configuration, with different choices of continuous 
parameter; the discrete parameters are m 
=7, a 
=1, and b 
=2. 


Figure 2: Crossing spans: Construct a regular m-gon (here, m 
=7)and lines of spana 
and b 
(here, a 
=2 
and b 
=3;the spana 
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). 


Ars Mathematica Contemporanea1(2008) 44–50 

by Gr¨unbaum 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-gon M 
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 
samespan;again followingGr¨unbaum[5],wesaythatthe 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 Theorem3andgiven hereina restated form (See Figure2): 

Theorem1 (Crossing Spans). Begin witha setofdiagonalsof span a 
and span b 
of an m-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 2i 
, and construct diagonals of span 
m 


b 
using N 
. Two of these span b 
diagonals intersect eachother and a span b 
diagonal of M 
in the same point, and the intersection points are precisely the points labelled [[b, 
a]] 
in N 
. 

The construction method is as follows. 
Begin with the vertices of a regular m-gon M 
centered at the origin O 
(the black circles 

m 


in Figure 3), and choose distinct integers a 
and b, with 1 
= 
a;b 
< 
, where if m 
is even, a 


2 


and b 
are of opposite parity. Using the points in M, 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 

2i 


images under rotation by about the origin to form a new m-gon A;we will label these 

m 


..... 


pointscyclically 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 manychoices for which line to place B0 
on.) Rotate 
this point B0 
around to forma 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 the m-gon of points formed D 
(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 inA, 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. However, 
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 inD 
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 onalineof span b 
using the points 
Ai 
and has label [[b, 
a]], the angle AiOP 
is j 
for some j, where j 
is odd if a 
is odd and 

m 


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 p 



L.W. Berman:ANew ClassofMovable (n4) 
Configurations 
Figure3:Adiagramshowingthe elementsofthe constructionofthenewmovable configurations. 
Points in M 
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. 


Ars Mathematica Contemporanea1(2008) 44–50 

..... 


with respect to OAi;ifm 
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 manypoints 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 in D 
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 
ona particular linein LA 
with other lines in LA, counted from the center. Moreover, since m 
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 
of the configuration, we have five classes of points (polygons M, 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 from D 
and two points from E). 
Two examples of such configurations are shown in Figure 4. 

Case 2: m 
is odd. 

By construction,givena point Ai 
in A, the point Bi 
in B 
lies on the line OAi, since that was 
how the Bi’s were chosen. Because M 
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 p 
with respect to OAi;sincem 
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-gons M, A, B, D, and 
E, and the lines of the configuration are the diagonals of span a 
using M, the diagonals of 
span b 
using M, A, B 
(that is, LM;LA 
and LB), and the diameters OAi 
for i 
=0, 
1;:::;m. 
Each diameter contains one point from A, B, D, and E. 


L.W. Berman:ANew ClassofMovable (n4) 
Configurations 
Figure 4: Movable configurations where m 
is even. Left hand side: a (304) 
configuration 
with m 
=6, a 
=1, b 
=2;right hand side: a(504) 
configuration with m 
= 
10, a 
=3, b 
=4. 


Figure 5: Movable configurations where m 
is odd. Lefthand side:a (354) 
configuration with 
m 
=7, a 
=3, b 
=2;right hand side: a(654) 
configuration with m 
= 
13, a 
=3, b 
=5. 


Ars Mathematica Contemporanea1(2008) 44–50 

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 
continuous parameter. Figure1shows threeversionsofa (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 andT. Pisanski, Polycyclic configurations, EuropeanJ. Combin. 24 (2003), 431–457. 
[4] J. Bokowski,B.Gr¨n 
= 
17, 
unbaum andL. Schewe,Topological configurations (n4) 
exist for all 
EuropeanJ. Combin., to appear. 

[5] B. Gr¨unbaum, Configurations of points and lines. In The Coxeter Legacy: Reflections and Projections 
(C.Davis andE.W. Ellers, eds), American Mathematical Society (2006), 179–225. 
[6]B.Gr¨unbaum, Astral (n4) 
configurations, Geombinatorics 9(2000), 127–134. 
[7] B.Gr¨
unbaum, Which (n4) 
configurations exist?, Geombinatorics 9(2000), 164–169.