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ARS MATHEMATICA CONTEMPORANEA 6 (2013) 253–259
On geometric trilateral-free (n3) configurations
Michael W. Raney
Georgetown University, Washington, DC, USA
Received 14 November 2011, accepted 20 March 2012, published online 28 October 2012
Abstract
This note presents the first known examples of a geometric trilateral-free (233) config-
uration and a geometric trilateral-free (273) configuration. The (273) configuration is also
pentalateral-free.
Keywords: Configurations, trilaterals.
Math. Subj. Class.: 05B30, 51E30, 05C38
1 Introduction
A (combinatorial) (n3) configuration is an incidence structure consisting of n distinct
points and n distinct lines for which each point lies on exactly three lines, each line is
incident with exactly three points, and any two points are incident with at most one com-
mon line. If an (n3) configuration may be depicted in the Euclidean plane using points
and (straight) lines, it is said to be geometric. As observed in [5] (pg. 17–18), it is evident
that every geometric (n3) configuration is combinatorial, but the converse of this statement
does not hold.
Adopting the terminology from [2], we say that a g-lateral in a configuration is a cycli-
cally ordered set {p0, l0, p1, l1, . . . , lg−2, pg−1, lg−1} of pairwise distinct points pi and
pairwise distinct lines such that pi is incident with li−1 and li for each i ∈ Zg . Hence a 3-
lateral is a trilateral, or triangle, a 4-lateral is a quadrilateral, and a 5-lateral is a pentalateral,
according to the previously established nomenclature. A configuration is g-lateral-free, for
a particular g ∈ {3, 4, 5}, if no g-lateral exists within the configuration.
Several recent papers (see [1], [3]) have examined triangle-free (n3) configurations.
The smallest example of a triangle-free configuration is the Cremona-Richmond (153) con-
figuration. A theorem mentioned in [5] (Theorem 5.4.3, pg. 333) states the following:
Theorem 1.1. For every n ≥ 15 except n = 16 and possibly n = 23 and n = 27, there
are geometric trilateral-free (n3) configurations.
E-mail address: mwr23@georgetown.edu (Michael W. Raney)
Copyright c© 2013 DMFA Slovenije
254 Ars Math. Contemp. 6 (2013) 253–259
In this note we provide new examples of a geometric, triangle-free (233) configura-
tion and a geometric, triangle-free (273) configuration, so that this theorem may now be
modifed:
Theorem 1.2. For every n ≥ 15 except n = 16, there are geometric trilateral-free (n3)
configurations.
Additionally, the (273) configuration is also pentalateral-free. It serves as the smallest
known example of a geometric (n3) configuration that is both 3-lateral-free and 5-lateral-
free; the formerly smallest known example of such a configuration is a (513) configuration
[2].
2 The examples
Configuration tables and diagrams of both of these new configurations C1 and C2 are pro-
vided below, and rational coordinates for their geometric realizations are given. Verification
that the former configuration is trilateral-free, and that the latter configuration is trilateral-
free and pentalateral-free, has been conducted using Mathematica.
2.1 C1, a geometric triangle-free (233) configuration1 1 1 2 2 3 3 4 4 5 5 6 6 7 8 9 9 11 12 12 15 18 212 8 16 13 17 4 6 7 20 10 14 7 9 10 11 10 19 14 13 15 16 19 22
3 21 20 19 23 5 22 15 23 18 16 8 12 13 17 11 21 22 14 18 17 20 23

PD
Fi
ll P
DF
 E
dit
or
 w
ith
 Fr
ee
 W
rit
er 
an
d T
oo
ls
M. W. Raney: On geometric trilateral-free (n3) configurations 255
Point Coordinates Point Coordinates Point Coordinates
1 (33/4, 29/4) 2 (7, 7) 3 (2, 6)
4 (4, 6) 5 (5, 6) 6 (2, 5)
7 (3, 5) 8 (6, 5) 9 (1, 4)
10 (3, 4) 11 (542/97, 4) 12 (0, 3)
13 (3, 3) 14 (455/97, 3) 15 (0, 2)
16 (445/97, 2) 17 (462/97, 2) 18 (0, 1)
19 (1, 1) 20 (1132/291, 1) 21 (1, 0)
22 (2, 0) 23 (1876/485, 0)
2.2 C2, a geometric triangle-free, pentalateral-free (273) configuration1 1 1 2 2 3 3 4 4 4 5 6 7 7 7 10 10 11 12 13 13 15 16 18 19 22 252 10 20 5 9 6 8 5 14 21 8 9 8 11 23 11 17 14 20 14 16 24 17 21 20 23 26
3 13 27 19 12 15 25 6 17 24 18 22 9 16 26 12 19 22 23 15 25 27 18 26 21 24 27

Point Coordinates Point Coordinates Point Coordinates
1 (0, 8) 2 (3, 8) 3 (4, 8)
4 (2, 7) 5 (3, 7) 6 (5, 7)
7 (1, 6) 8 (4, 6) 9 (5, 6)
10 (0, 5) 11 (1, 5) 12 (6, 5)
13 (0, 4) 14 (2, 4) 15 (8, 4)
16 (1, 3) 17 (2, 3) 18 (7, 3)
19 (3, 2) 20 (6, 2) 21 (7, 2)
22 (5, 1) 23 (6, 1) 24 (8, 1)
25 (4, 0) 26 (7, 0) 27 (8, 0)
256 Ars Math. Contemp. 6 (2013) 253–259
3 Motivation for the results
Both C1 and C2 have arisen serendipitously in conjunction with the author’s study of magic
(n3) configurations. An (n3) configuration is said to be magic if it is possible to assign the
integers {1, 2, . . . , n} as labels for its n points, where each integer is used exactly once,
in such a manner that the sum of the point labels along each line of the configuration is
always the same magic constant, M . Since each point of the configuration is involved in
three such sums, we see that
nM = 3
n∑
i=1
i = 3
n(n+ 1)
2
M =
3
2
(n+ 1)
Hence n must be odd (and at least 7) for a magic configuration to be possible. The smallest
example of a magic configuration turns out to one of the 31 (113) configurations. Its
combinatorial table is 1 1 1 2 2 2 3 3 3 4 46 7 8 5 6 7 4 5 7 5 6
11 10 9 11 10 9 11 10 8 9 8

This configuration is (113)17, according to the (113) configuration labeling scheme initi-
ated in [4] and referenced in [6],[7].
Magic (n3) configurations have not, to the author’s knowledge, been previously considered
in the literature on configurations, although other magic configurations such as magic stars
have been studied [8].
C1 is dual to the magic (233) configuration1 1 1 2 2 2 3 3 3 4 4 4 5 5 5 6 6 6 7 7 7 8 913 14 15 11 12 15 10 15 16 12 13 14 8 9 14 9 10 13 8 10 11 12 11
22 21 20 23 22 19 23 18 17 20 19 18 23 22 17 21 20 17 21 19 18 16 16

with magic constant 32 (23 + 1) = 36. Also, C2 is dual to the magic (273) configuration1 1 1 2 2 2 3 3 3 4 4 4 5 5 5 6 6 6 7 7 7 8 8 8 9 9 914 17 18 15 16 18 13 16 17 11 12 17 10 12 18 10 11 16 11 14 15 12 13 15 10 13 14
27 24 23 25 24 22 26 23 22 27 26 21 27 25 19 26 25 20 24 21 20 22 21 19 23 20 19

M. W. Raney: On geometric trilateral-free (n3) configurations 257
with magic constant 32 (27 + 1) = 42. This means that in each case there exists an iso-
morphism between Ci and its dual that interchanges the roles of points and lines while
preserving incidence structure. We say that the dual configuration of a magic configuration
is comagic; hence C1 and C2 are comagic. So for both C1 and C2 it is possible to label its
lines in such a manner that the sum of the labels of the three lines incident to any point of
the configuration is always the same magic constant, again 32 (n+ 1).
It turns out that a diagram associated with a comagic configuration may be conveniently
constructed. Suppose that (x1 x2 x3)T is a line in the configuration table of the original
magic configuration, where x1 < x2 < x3. It follows that the point (x1, x2, x3) ∈ R3 lies
in the plane {(x, y, z) ∈ R3 : x + y + z = 32 (n + 1)}. After plotting each corresponding
point in this plane, for k = 1, . . . , n we connect three points with an arc (labeled k) if
the three points share k as a coordinate. We thereby produce a diagram within the plane
x+ y + z = 32 (n+ 1).
Next, we project the diagram onto the xz-plane by simply eliminating the y-coordinate.
No information about the configuration is lost when doing this, since for any point we may
recapture x2 = 32 (n + 1) − x1 − x3. Below is a diagram for C1 achieved in this fashion
with each (x1, x3) point indicated.
Observe that this diagram has three nonlinear arcs. After some algebraic manipulation in-
volving shifting seven of the 23 points, we find that it is possible to recast the diagram so
that all of the arcs indeed are straight lines. After rescaling the points (via the transforma-
tion (x, z) 7→ (x − 1, 23 − z)) we arrive at the geometric realization for C1 provided in
Section 2.1.
We again depict the diagram for C1, this time with its associated magic line labeling.
258 Ars Math. Contemp. 6 (2013) 253–259
When undergoing this process for the (273) configuration, we discover pleasantly that
no shifting of arcs is required. This is a consequence of each line (x1 x2 x3)T sat-
isfying the conditions 1 ≤ x1 ≤ 9, 10 ≤ x2 ≤ 18, and 19 ≤ x3 ≤ 27. After
lopping off the x2-coordinates and rescaling the resulting points (via the transformation
(x, z) 7→ (x− 1, z − 19)) we arrive at the geometric realization for C2 provided in Section
2.2.
We display the diagram of C2 again with its associated magic line labeling.
M. W. Raney: On geometric trilateral-free (n3) configurations 259
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