103-configurations and projective realizability of multiplied configurations

Des. Codes Cryptogr. (2009) 51:45–54DOI 10.1007/s10623-008-9242-2

103-configurations and projective realizabilityof multiplied configurations

Krzysztof Petelczyc · Małgorzata Prazmowska

Received: 22 May 2008 / Revised: 3 September 2008 / Accepted: 3 September 2008 /Published online: 7 October 2008© Springer Science+Business Media, LLC 2008

Abstract Some remarks on 103-configurations which contain the complete graph K4 aregiven, on their representations, and on projective realizability. Results are applied to show aclass of configurations that cannot be realized in any Pappian projective space.

Keywords 103-configuration · Projective plane · Veblen configuration ·Correlatively multiplied configuration

Mathematics Subject Classifications (2000) 51A45 · 51E10 · 05B30

1 Introduction

It is known that there are exactly ten 103 configurations and 9 of them can be realized in areal projective plane, while exactly one cannot be projectively realized (cf. [1,4,5,10]). Thisexceptional one, which we denote by V, will also play an important role in our further inves-tigations. Though V cannot be realized in a real plane, it can be realized in Desarguesianplanes. More precisely, V cannot be realized on a Pappian plane and it can be realized onevery non Pappian Desarguesian plane (cf. 2.9).

The best known and the most important in projective geometry is the Desargues config-uration. This one can be characterized as a suitable “closure” of the complete K4-graph andthis observation (cf. [2,14]) was a starting point for further generalizations and consideringso called combinatorial Grassmann spaces (cf. [11]).

Communicated by D. Ghinelli.

K. Petelczyc (B) · M. PrazmowskaInstitute of Mathematics, University of Bialystok, Akademicka 2, 15-246 Bialystok, Polande-mail: [email protected]

M. Prazmowskae-mail: [email protected]

123

46 K. Petelczyc, M. Prazmowska

It was noted in [1] that V also can be represented as a closure of K4. A representation ofthe Kantor 103G-configuration as a closure of K4 was presented in [12]. After a careful anal-ysis of all the 103-configurations we have realized that exactly 6 of them can be representedas a closure of K4; namely those which contain a 4-clique.We begin our note with statingprecisely what exactly such a “closure” means (Construction 2.3) and what are the obtainedconfigurations.

Another representation of the Desargues configuration as a perspective of two trianglesalso can be generalized (Construction 2.6). It is clear that such a “generalized (or “twisted”)perspective” must contain a 4-clique: a perspective centre and one of the perspective triangles.There are exactly three 103-configurations that can be represented as a twisted perspectiveof two triangles. In our note we briefly characterize their geometry.

Finally, we note that V can be found in many so called “correlatively multiplied” con-figurations (see [9]). From this we derive immediately that these multiplied configurationscannot be realized in a Pappian projective space. Perhaps the most intriguing is the fact thatno series of correlatively multiplied cyclic projective planes can be embedded into a Pappianprojective space.

2 103-configurations that contain a 4-clique

Let us begin with simple observations.

Lemma 2.1 Let a 103-configuration N contain a 4-clique (i.e. a 4-set such that any two itselements are collinear, and no three are on any line). Then after removing the points and linesof this clique a Veblen configuration (see Fig 2, sometimes called also a Pasch configuration)arises.

Proof Evident: after removing given clique the remaining points and lines yield a (62 43)-configuration, that is exactly the Veblen configuration. ��Lemma 2.2 A 103-configuration contains a 4-clique iff it contains a Veblen configuration.

Construction 2.3 Let us write X = {1, 2, 3, 4} for the set of vertices of the graph K4. Theedges of K4 are the 2-subsets of X . On every edge e of K4 we put a new point e∞. Then thefamily L∗ := {{a ∪ a∞}: a ∈ ℘2(X)

}of lines of size 3 is obtained. It is natural to identify

e∞ = {i, j} ∈ ℘2(X) where e joins i, j ∈ X (comp. [6]). Finally, let us introduce on theset ℘2(X) of 2-subsets of X the structure V = 〈℘2(X),L〉 of the Veblen configuration (inother words: we group the elements of ℘2(X) into triples so that the Veblen configuration isobtained).

The configuration with the point set X ∪℘2(X) and the line set L ∪ L∗ is a closure of thegraph K4.

Note that the closure constructed above depends entirely on the labelling of the pointsof the Veblen configuration by the elements of ℘2(X) i.e. on the family L. This closurewill be denoted by D(L) (or D(V), where V = 〈℘2(X),L〉 is the corresponding Veblenconfiguration). Clearly, D(L) is a 103-configuration.

Two auxiliary definitions are also useful. For a set X and positive integer k we write

Gk(X) = 〈℘k(X), {℘k(A) : A ∈ ℘k+1(X)}〉and

G∗k (X) = 〈℘k(X), {{a ∈ ℘k(X) : A ⊂ a} : A ∈ ℘k−1(X)}〉.

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103-configurations and projective realizability of multiplied configurations 47

a

b

c

d

e

f

a’

b’

c’

d’

AB

C

D

A’

B’

C’

D’

E’

F’

Fig. 1 The configuration V

Note, that if |X | = 4 then G2(X) and G∗2(X) are Veblen configurations.

Let |X | = 4. It is known that D(G2(X)) is the Desargues configuration. The unique103-configuration which is not realizable in a real projective space is the configurationD(G∗

2(X)) (cf. [1]).We denote the obtained configuration by V (see Fig. 1).Note that V is entirely determined by its (unique!) 4-clique. From this we directly derive

Fact 2.4 Aut(V) ∼= S4, namely, any automorphism of V is uniquely determined by anarbitrary permutation of its 4-clique.

In the next step we note the following

Fact 2.5 Let V = 〈℘2(X),L〉 be a Veblen configuration. Then after a permutation of the setX the configuration V is one of the six presented in Fig. 2.

Proof The proof consists in elementary combinatorics. ��Considering the list of all the 103-configurations, determining which of them contain a

4-clique, and observing possible labeling V of the Veblen configuration presented in Fig. 2we see that in the corresponding cases we arrive to the following configuration N = D(V)(the names “headdress”, “kepi”, “basinet”, “fez”, and “nightcap” are used after [7]):

(i; V = G2(X)) N is the Desargues configuration (cf. Fig. 3). N is self dual andcontains five 4-cliques.

(ii; V = G∗2(X)) N is the V-configuration, called also the nightcap configuration

(see Fig. 4). It is self dual.(iii; V = PB({1, 2})) N is the 103G Kantor configuration, or the kepi configuration (rep-

resented also as the Veronese configuration V3(3); cf. [12,13], seeFig. 5). It is self dual. It contains six 4-cliques.

(iv; V = µ(PB({1, 2}))) N is the headdress configuration (see Fig. 6). It contains exactlyone 4-clique. Its dual is presented in (vi).

123


Fig. 2 Various labeling of the points of the Veblen Figure by the elements of ℘2(X), where X = {1, 2, 3, 4}.The symbols of the form PB({i, j}) used here merely indicate some way of labeling, despite that they havetheir strict meaning in the more general context of so called multiveblen configurations (cf. [12]). The mapµ : ℘2(X

′) � a −→ X ′ \ a ∈ ℘2(X′) is a bijection, and µ(V) stands for the image of V under this bijection.

V6 is equivalent under a bijection of X to µ(V5)

{3,4}

{1,4}

{1,3}

{2,3}

{2,4}

{1,2}{1}

{3}

{4}

{2}

Fig. 3 The Desargues configuration

(v; V = V5) N is the self dual fez configuration (Fig. 7). It contains two4-cliques.

(vi; V = µ(V5)) N is the basinet configuration (see Fig. 8). It contains exactly oneK4-clique. Its dual is the one considered in (iv).

Another well known representation of the Desargues configuration refers to a perspectiveof two triangles.

Construction 2.6 Let two triangles e1, e2, e3 and d1, d2, d3 have a perspective centre pi.e. let them be inscribed into three lines K1, K2, K3 with the common point p such that

123


{3,4}

{1,4}

{1,3}

{2,3}

{2,4}

{1,2}

{1}{3}

{4}{2}

Fig. 4 The nightcap configuration

{3,4}

{1,4}

{1,3}{2,3}

{2,4}

{1,2}

{1}

{3}

{4}{2}

Fig. 5 The Kantor 103G-configuration

ei , di ∈ Ki for i = 1, 2, 3; set e4 = e1, d4 = d1. Denote the corresponding sides of thetriangles by Ei , Di : Ei = ei , ei+1, Di = di , di+1.

Next, let ϕ be a bijection of the set {1, 2, 3} and let ci be a point on Ei , Dϕ(i) for i = 1, 2, 3.Finally, let the points ci yield a new line L of size 3. The obtained 103-configuration will bedenoted by �(ϕ).

Representation 2.7 Let ϕ be as in 2.6. Then exactly one of the following possibilities arises:ϕ = id, ϕ is a transposition, ϕ is a cycle of length 3. In the corresponding cases, without lossof generality we can take ϕ as below and we obtain the following configuration N = �(ϕ)

representing (possibly) “twisted” perspective.

(i; ϕ = id) Then N is the classical Desargues configuration.

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Fig. 6 The headdressconfiguration

{3,4}

{1,4}

{1,3}

{2,3}{2,4}

{1,2}{1}

{3} {4}

{2}

{3,4}

{1,4}

{1,3}

{2,3}

{2,4}

{1,2}{1}

{3}

{4}

{2}

Fig. 7 The fez configuration

{3,4}

{1,4}

{1,3}

{2,3}

{2,4}

{1,2}

{1}{3}

{4}{2}

Fig. 8 The basinet configuration

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Fig. 9 The 103-systems of triangle perspectives

(ii; ϕ = (1)(2, 3)) Then N is the 103G-configuration of Kantor. The axis L joins the com-mon point of exactly one pair of corresponding sides E1, D1 and two“twisted” intersections E3 ∩ D2 and E2 ∩ D3.

(iii; ϕ = (1, 3, 2)) Then N is the fez configuration. No two corresponding sides of the tri-angles intersect in N.

Drawings illustrating these types of perspective are given in Fig. 9.

Note that all the configurations of the above list were already mentioned when we havedetermined closures of K4. It is evident, as a triangle together with respective perspectivecenter yields in a configuration a 4-clique.

Remark 2.8 Let ϕ be as in 2.6. Let us relabel the points and lines of N = �(ϕ) as follows.Ai = Ei , ai = ei , Bi = Dϕ(i), and Bi ,Bi+1 intersect in bi+1 for i = 1, 2, 3. The two (line)triangles A1, A2, A3 and B1, B2, B3 have L as the perspective axis: the three common pointsof the pairs of corresponding sides lie on L . Moreover, there is a permutation ψ of the set{1, 2, 3} such that the three lines ai , bψ(i) have a point in common. We see that ψ = ϕ−1.From this we easily derive that the configuration �(ϕ) is self dual.

Let N = 〈S,G, 〉 be a configuration. A projective embedding of N into a projective spaceP is an injective map that associates with the elements of S points of P and with the elementsof G lines of P and which preserves in both directions the incidence. A configuration N canbe projectively realized if there is a projective embedding of N into a Desarguesian projectivespace.

Proposition 2.9 The configuration V cannot be embedded into a Pappian projective space.It can be embedded into a projective plane over a field F iff F is not commutative.

Proof Let V be represented as a closure of the K4 graph with the vertices X = {0, 1, 2, 3},i.e. let V = D(G∗

2(X)); adopt notation of 2.3. Let us write S(i) = {a ∈ ℘2(X) : i ∈ a} forthe corresponding lines of G∗

2(X). Suppose that φ is an embedding of V into a Desarguesianprojective space P.

In the first step we note that φ(V) lies on a plane of P; namely, on the plane of P spannedby the triangle (φ(0), φ({0, 1}), φ({0, 2})). Consequently, we can assume that P is a plane.

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Since a Desarguesian plane is homogeneous on lines we can take the lineφ(S(0)) improperand continue our reasoning in a Desarguesian affine plane, which should contain all the pointsof φ(V) except φ({0, 1}), φ({0, 2}), and φ({0, 3}). Let us introduce a coordinate system onthis plane so as φ(0) = [0, 0], φ(1) = [1, 0], and φ(2) = [0, 1]. In this system φ({0, 1})is the direction of the x-axis and φ({0, 2}) is the direction of the y-axis. Moreover, the lineφ(1, 2) = φ(1), φ(2) has equation y = −x + 1 and then φ({1, 2}) = [d,−d + 1] forsome scalar d �= 0, 1. Since φ(S(1)) = φ({0, 1}), φ({1, 2}) we get the equation of φ(S(1)):y = −d + 1; analogously we get the equation of φ(S(2)): x = d . Let the line φ(0, 3) haveequation y = αx for a scalar α �= 0; then φ(3) = [b, αb] for a scalar b �= 0. Then the lineφ(1, 3) has equation y = αb(b − 1)−1x − αb(b − 1)−1 and the line φ(2, 3) has equationy = (αb − 1)b−1x + 1. From this we compute:

φ({1, 3}) = [(b − 1)b−1α−1(1 − d)+ 1,−d + 1],φ({2, 3}) = [d, (αb − 1)b−1d + 1].

Let M = φ({1, 3}), φ({0, 3}); clearly, it is a line through φ({1, 3}) parallel to φ(0, 3) andthus

M has equation y = αx + 1 − d − α((b − 1)b−1α−1(1 − d)+ 1).

On the other hand, M should be the line φ(S(3)) i.e. φ({2, 3}) should lie on it. This leadsto the equation

(αb − 1)b−1d = αd − d − α((b − 1)b−1α−1(1 − d)+ 1),

which is equivalent to

(b−1α−1 − α−1b−1)d = b−1α−1 − α−1 − 1. (1)

The plane P is Pappian iff the coordinate field of P is commutative. Under the assumptionof commutativity Eq. (1) reduces to 1 − b = αb, which gives α = 1−b

b . However, one cancompute that this solution leads to an additional incidence: φ(3) lies on φ(1, 2), which isimpossible.

Assume that the coordinate field F of P is not commutative, let α, b be a pair of scalarssuch that αb �= bα; then t := b−1α−1 −α−1b−1 �= 0. Set u := b−1α−1 −α−1 − 1. Supposethat the point φ(3) lies on φ(1, 2); this gives α = (1−b)b−1, but then t = 0, which is impos-sible and therefore φ(3) is not on φ(1, 2); moreover, u �= 0 as well. Next, set d = t−1u; it isclear that d �= 0, d �= 1, and (1) is satisfied. From elementary geometry we can deduce thatno additional incidence appears and thus with given parameters α, b we obtain an embeddingof V into the projective plane over F. ��

As an example to the proof of 2.9 we can take the affine plane P defined over the field ofquaternions Q. (the standard notation i, j, k for the solutions of x2 = −1 in Q is used) andadopt α = i , b = j , d = 1

2 + 12 j − 1

2 k. This gives

φ(3) = [ j, k], φ({1, 2}) =[

1

2+ 1

2j − 1

2k,

1

2− 1

2j + 1

2k

],

φ({2, 3}) =[

1

2+ 1

2j − 1

2k,

1

2+ j

1

2k

], φ({1, 3}) =

[1

2+ 1

2j + k,

1

2− 1

2j + 1

2k

].

A straightforward computer aided computation shows that (1) is satisfied and therefore anembedding of V into P is obtained.

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3 Correlatively multiplied configurations and their projective realizability

Let M = 〈S,G〉 be a self-dual partial linear space, let � be its correlation, and k be an integer,k > 2. The structure M∗ = k �� M is defined as follows (comp. [9]). Let M∗ = Ck × M ,G∗ = Ck × G.

We apply the following convention:

– (i, a) is a point, where i ∈ Ck and a ∈ M ;– [i, l] is a line, where i ∈ Ck and l ∈ G.

According to this terminology the relation∗

of incidence of M∗ is characterized by thecondition

(i, a)∗ [ j, l] iff, either i = j and a l, or i = j + 1 and a = �(l) (2)

Then we set

M∗ = 〈M∗,G∗, ∗〉.The structure M∗ = k �� M is referred to as a correlative multiplying of M.

Proposition 3.1 Let M be a self dual partial linear space with a correlation �. Assume thatM contains a 4-clique (equivalently: assume that M contains a Veblen configuration). Thenk �� M contains the configuration V.

Consequently, k �� M cannot be realized in a Pappian projective space.

Proof Let M contain a 4-clique X = {a, b, c, d} such that a, b A, b, c B, c, d C ,

d, a D, a, c E , b, d F , and set M∗ = k �� M. Then for some i = 0, . . . , k − 1we obtain the 4-clique {(i, x) : x ∈ X} in M∗. Moreover, the point (i + 1, �(y)) lies on theline [i, y] for all y ∈ {A, B,C, D, E, F} = Y . The points (i + 1, �(y)) form the Veblenfigure since the lines [i, y] are the edges of K4-graph. Next set S = {(i, x) : x ∈ X} ∪ {(i +1, �(y)) : y ∈ Y }, L = {[i, y] : y ∈ Y } ∪ {[i + 1, �(x)] : x ∈ X}. Then V ∼= 〈S,L, ∗〉 isembedded into M∗. This justifies the claim. Analogous reasoning applies when M containsa Veblen configuration. ��

Considering the examples of correlatively multiplied structures investigated in [9] we notethat most of them cannot be projectively realized in Pappian spaces.

Corollary 3.2 Let M∗ = k �� M, where M is any one of the below list

(a) a Pappian projective plane;(b) the generalized Desargues configuration Gk(X), where |X | = 2k + 1 (cf. [11]);(c) the configuration H T (q) with prime power q > 3 (the affine plane over G F(q) with

one line direction deleted. comp. [3]).

Then M can, and M∗ cannot be projectively realized in a Pappian space.

Remark 3.3 The statement of 3.2 does not mean that there is no projective embedding ofany correlatively multiplied structure. The simplest example is given by series of cyclicallyinscribed polygons (cf. [8,9]): the 93 configuration that is presented in the form 3 �� C3 canbe projectively realized.

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103-configurations and projective realizability of multiplied configurations