ISRN Geometry

VolumeΒ 2012Β (2012), Article IDΒ 560760, 10 pages

doi:10.5402/2012/560760

## Geometric Realization of Some Triangle-Free Combinatorial Configurations

Department of Mathematics, University of Washington, Seattle, WA 98195-4350, USA

Received 17 April 2012; Accepted 15 May 2012

Academic Editors: E.Β Gutkin, B.Β Mohar, and J. S.Β Snoeyink

Copyright Β© 2012 Branko GrΓΌnbaum. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The main purpose of this paper is to illustrate the mutual benefit to combinatorics
and geometry by considering a topic from both sides. Al-Azemi and Betten enumerate the distinct *combinatorial* (22_{3}) configurations that are triangle free.
They find a very large number of such configurations, but when taking into
account the automorphism group of each, they find two cases in which there is
only a single configuration. On the heuristic assumption that an object that is
unique in some sense may well have other interesting properties, the *geometric*
counterparts of these configurations were studied. Several unexpected results and
problems were encountered. One is that the combinatorially unique (22_{3}) configuration
with automorphisms group of order 22 has three distinct geometric realizations
by astral configurations.

In [1], Al-Azemi and Betten consider *combinatorial configurations* (22_{3}). Here by () we mean an incidence structure of distinct *points* and distinct *blocks*, each block contains points, each point belongs to blocks, and each pair of distinct points belongs to at most one block. A *triangle* in a configuration is a set of three points , , , and three blocks , , , such that for ,ββ is incident with if and only if . A configuration with no triangles is said to be *triangle-free*. There exists considerable literature on triangle-free combinatorial configurations; in particular, data on () configurations for were given by Betten et al. in [2], and this has been extended in [1] to the case . The latter paper also determines (among other results) the classification of the 157,211 triangle-free configurations (22_{3})
by the size of the group of automorphisms of the configurations. Two
particularly interesting results are that there is a single
configuration with group of order 11, and a single configuration with
group of order 22. The present note arose from the question whether
these two special configurations have interesting geometric
representations, in particular, with isometric symmetries group
isomorphic with the automorphisms group or one of its nontrivial
subgroups. Our main result is that the answers are affirmative. Some
explanations of the geometric setting are required.

A *geometric configuration* () is a collection of distinct points and distinct *straight* lines in the Euclidean plane, each point lying on exactly lines and each line going through exactly points. In an obvious way, each geometric configuration is
isomorphic to (can be interpreted as) a combinatorial configuration. The
converse is well-known not to be valid. In fact, the determination of
general conditions for a combinatorial configuration to be realizable by
a geometric one is one of the deepest problems regarding
configurations. More details on the concepts involved, and on the
results known in this connection, may be found in [3].

*astral*if and its points and lines form two transitivity classes under rotational symmetries, each consisting of points or lines, respectively. As explained in detail in [3, Chapter 2], the points of such a configuration are situated at the vertices of two concentric regular -gons. The configuration is denoted if the following conditions are all satisfied; Figure 1 provides an example illustrating 11#(4,2;3) while Figure 2 shows the simplified notation we will use throughout the paper:(i)the points in one of the transitivity classes are labeled consecutively (clockwise, or else counterclockwise) for ; (ii)a line denoted (for ) passes through the points labeled and (subscripts mod ); (iii) the line passes through one point of the other class; (iv) points and determine a line ; (v)the line passes through a point .

Depending on the transitivity class selected, and on clockwise or counterclockwise orientation, up to four different symbols correspond to each astral configuration. Usually one selects the lexicographically highest among them.

Conversely, given a symbol , one may start constructing the corresponding geometric configuration. As becomes obvious at once, it is necessary to determine where on the line does the point lie. This leads to a quadratic equation for the ratio (which is the same for all ) in which divides the segment ; This is to ensure that condition (v) is satisfied. Then the resulting construction depends on the roots of this equation. If there are two distinct real roots there are two geometrically distinct but combinatorially isomorphic geometric configurations ; if there is a single root (of multiplicity 2) there is only one geometric configuration; if the roots are conjugate complex numbers there is no geometric configuration in the real Euclidean plane. All the astral configurations are also easily seen to be selfdual. Additional details and properties of astral and other () configurations are given in Chapter 2 of [3].

By construction, all astral configurations have as geometric symmetries all rotations of the cyclic group and no mirror symmetries. In the case we are interested, this is the cyclic group . It is easy to determine what are the configurations that are triangle-free. A consideration of all the relevant cases leads to exactly six symbols that yield triangle-free configurations. These are:

As visible from Figure 3, each symbol in the first row above corresponds to a pair of geometrically distinct (but combinatorially isomorphic) configurations. Figures 2 and 4 show geometric configurations that correspond to the symbols in the second row, which lead to one configuration for each symbol. As is easily verified (see Figures 2, 3, and 4), all the configurations in the both rows are triangle-free and have geometric symmetry group , hence with automorphism group containing .

The isomorphism of the astral configuration 11#(4,2;3) with the combinatorial configuration described in Section A.4 of [1] is shown in Figure 2.

Guided by the generators (in (A.13) and (A.14) of [1]) of the automorphism group of the configuration in A.4 of [1], it was possible to find the geometric basis for the larger, 22-element group, of automorphisms of 11#(4,2;3). An additional generator corresponds to a βsplit-mirrorβ reflection: one transitivity class of points is reflected in one mirror, the other in an appropriate different mirror. This is illustrated in Figure 5. This automorphism was not anticipated by the previous geometric knowledge. While not an isometry, it obviously has a geometric context.

What about the other two configurations in the second row, shown in Figure 4? It is easy to find "split-mirror" reflections for each, hence they also have automorphism groups of order 22. This seems to be in contradiction with the uniqueness result of [1]; however, that quandary is resolved by showing that each is, in fact, combinatorially isomorphic with 11#(4,2;3), as expected from the combinatorial uniqueness result. The isomorphism is documented and illustrated in Figure 6. Thus we have the following proposition.

Proposition 1. *The unique triangle-free combinatorial configuration (22 _{3}) with automorphism group of order 22 found in [1] has precisely three geometrically distinct realizations by astral configurations.*

Here, and in Proposition 2, the assertion that the precise number of geometrically distinct astral configurations is 3 (resp., 6) follows from well-known facts about that kind of configurations.

It should be noted that although there has been no claim in the literature that geometric astral configurations with different symbols are combinatorially distinct, this was the unstated belief held by at least some workers in the field (including the present writer). Hence this isomorphism is rather unexpected. On the other hand, it should be noted that for () astral configurations such behavior has been noted earlier. Figure of [3] shows three geometrically distinct configurations that are combinatorially isomorphic; the same holds for their polars.

Turning to the astral configurations in Figure 3, we begin by observing that the first one, 11#(4,2;1)β², is isomorphic with the combinatorial configuration with automorphism group described in Section A.2 of [1], and given by the blocks listed there in (A.5). This is illustrated in Figure 7. The configuration 11#(4,2;1)β²β² is isomorphic to 11#(4, 2; 1)β², hence to the unique configuration with automorphism group just mentioned.

Concerning the other two pairs of configurations in Figure 3, it turns out that the situation is analogous to the one discussed previously.

Proposition 2. *The unique triangle-free combinatorial configuration (22 _{3}) with automorphism group of order 11 found in [1]
has precisely six geometrically distinct realizations by astral
configurations; they form three pairs of configurations with the same
symbol.*

This is illustrated in Figure 8. The authorβs attempts to establish isomorphism between configurations from different pairs were successful for the pairs 11#(4,2;1) and 11#(4,3;2). However, the third pair 11#(5,3;1) resisted the search for isomorphism. The author is obliged to Dr. Marko Boben for the isomorphism on which Figure 8(b) is based.

Beyond this, the geometric and combinatorial results concerning cases of uniqueness deserve to be extended to other values of n, especially in cases with prime , as does the existence of βsplit-mirrorβ reflections in other astral () configurations. Also, the behavior of () configurations (for ) deserves to be studied as to the possibilities of geometrically distinct realizations of given combinatorial configurations. Similar investigations regarding topological configurations (formed by points and pseudolines) should also be rewarding.

*Conjecture β1*. All () astral configurations with admit βsplit-mirrorβ reflections and have automorphism groups with elements.

*Multiastral configurations *are defined in section 2.9 of [3] as generalizations of astral configuration; essentially, they have as points sets of vertices of concentric regular n-gons, with suitable restrictions. We may conjecture the following.

*Conjecture β2*. The automorphism group of any multiastral configuration has at most elements.

A different challenge is posed by the question
whether there are geometric realizations of the three triangle-free
combinatorial configurations (22_{3}) with automorphism group of
order 24. If any is found to exist, one would wish to see how
geometrically symmetric each may be. But if no realization existsβthis
would resolve the old conjecture that every 3-connected combinatorial () configuration with is geometrically realizable with points and straight lines in the Euclidean plane.

(*Note added April 8, 2012*:
Dr. Marko Boben has found geometric realizations of all three of these
configurations, albeit with trivial group of isometric symmetries.)

Boben et al. [4] investigated the existence or nonexistence of geometric triangle-free configurations () for all ; many details of the history of the topic may also be found in this paper. Related more general results are in [3] and in Boben et al.βs paper [5]. Although it was easy to determine directly that there are no triangles in any of the (22_{3})
configurations listed above, I am grateful to Dr. Marko Boben for a
list of general criteria for the presence of a triangle in an astral
configuration. These are analogous to the criteria in [5], and they confirmed my checking by hand that the configurations in question are triangle-free.

#### References

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