ARS MATHEMATICA CONTEMPORANEA Also available at http://amc-journal.eu ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.) ARS MATHEMATICA CONTEMPORANEA 12 (2017) 287-299 A classification of the Veldkamp lines of the near hexagon L3 x GQ(2, 2) Richard M. Green * Department of Mathematics, University of Colorado Boulder, Campus Box 395, Boulder CO 80309-0395, U. S. A. Metod Saniga Astronomical Institute, Slovak Academy of Sciences, SK-05960 Tatranska Lomnica, Slovak Republic Received 5 October 2015, accepted 18 May 2016, published online 20 January 2017 Using a standard technique sometimes (inaccurately) known as Burnside's Lemma, it is shown that the Veldkamp space of the near hexagon L3 xGQ(2,2) features 156 different types of lines. We also give an explicit description of each type of a line by listing the types of the three geometric hyperplanes it consists of and describing the properties of its core set, that is the subset of points of L3 xGQ(2,2) shared by the three geometric hyperplanes in question. Keywords: Near hexagons, Geometric hyperplanes, Veldkamp spaces. Math. Subj. Class.: 51Exx, 81R99 1 Introduction Brouwer et al. [1] proved that there are eleven isomorphism types of slim dense near hexagons. Of these eleven, the near hexagons of sizes 27,45 and 81 are the most promising for physical applications. This paper is devoted to a study of the second of these three examples and its Veldkamp space. The first of the three examples was described in our paper [4], and we plan to study the third case in a future work. The 45 point space we study here is the product L3 xGQ(2,2), where L3 is the line containing three points and GQ(2,2) is the generalized quadrangle of order two. * Corresponding author. E-mail addresses: rmg@euclid.colorado.edu (Richard M. Green), msaniga@astro.sk (Metod Saniga) Abstract ©® This work is licensed under http://creativecommons.org/licenses/by/3.0/ 288 Ars Math. Contemp. 12 (2017) 383-413 2 Near polygons, quads, geometric hyperplanes and Veldkamp spaces In this section we gather all the basic notions and well-established theoretical results that will be needed in the sequel. A near polygon (see, e. g., [3] and references therein) is a connected partial linear space S = (P, L, I), I C P x L, with the property that given a point x and a line L, there always exists a unique point on L nearest to x. (Here distances are measured in the point graph, or collinearity graph of the geometry.) If the maximal distance between two points of S is equal to d, then the near polygon is called a near 2d-gon. A near 0-gon is a point and a near 2-gon is a line; the class of near quadrangles coincides with the class of generalized quadrangles. A nonempty set X of points in a near polygon S = (P, L, I) is called a subspace if every line meeting X in at least two points is completely contained in X. A subspace X is called geodetically closed if every point on a shortest path between two points of X is contained in X. Given a subspace X, one can define a sub-geometry SX of S by considering only those points and lines of S that are completely contained in X. If X is geodetically closed, then SX clearly is a sub-near-polygon of S. If a geodetically closed sub-near-polygon SX is a non-degenerate generalized quadrangle, then X (and often also SX) is called a quad. A near polygon is said to have order (s, t) if every line is incident with precisely s + 1 points and if every point is on precisely t +1 lines. If s = t, then the near polygon is said to have order s. A near polygon is called dense if every line is incident with at least three points and if every two points at distance two have at least two common neighbours. A near polygon is called slim if every line is incident with precisely three points. It is well known (see, e. g., [6]) that there are, up to isomorphism, three slim non-degenerate generalized quadrangles. The (3 x 3)-grid is the unique generalized quadrangle of order (2,1), GQ(2,1). The unique generalized quadrangle of order 2, GQ(2, 2), is the generalized quadrangle of the points and lines of PG(3, 2) that are totally isotropic with respect to a given symplectic form. The points and lines lying on a given nonsingular elliptic quadric of PG(5,2) define the unique generalized quadrangle of order (2,4), GQ(2,4). Any slim dense near polygon contains quads, which are necessarily isomorphic to either GQ(2,1), GQ(2,2) or GQ(2,4). Next, a geometric hyperplane of a partial linear space is a proper subspace meeting each line (necessarily in a unique point or the whole line). The set of points at non-maximal distance from a given point x of a dense near polygon S is a hyperplane of S, usually called the singular hyperplane (or perp-set) with deepest point x. Given a hyperplane H (or any subset of points C) of S, one defines the order of any of its points as the number of lines through the point that are fully contained in H (C); a point of H (C) is called deep if all the lines passing through it are fully contained in H (C). If H is a hyperplane of a dense near polygon S and if Q is a quad of S, then precisely one of the following possibilities occurs: (1) Q C H; (2) Q n H = x^ n Q for some point x of Q; (3) Q n H is a sub-quadrangle of Q; and (4) Q n H is an ovoid of Q. If case (1), case (2), case (3), or case (4) occurs, then Q is called, respectively, deep, singular, sub-quadrangular, or ovoidal with respect to H. If S is slim and Hi and H2 are its two distinct hyperplanes, then the complement of symmetric difference of Hi and H2, HiAH2, is again a hyperplane; this means that the totality of hyperplanes of a slim near polygon form a vector space over the Galois field with two elements, F2. In what follows, we shall put HiAH2 = Hi © H2 and call it the (Veldkamp) sum of the two hyperplanes. R. M. Green and M. Saniga: A classification of the Veldkamp lines 289 Finally, we shall introduce the notion of the Veldkamp space, V(T), of a point-line incidence geometry r(P, L) [2]. Here, V(T) is the space in which (i) a point is a geometric hyperplane of r and (ii) a line is the collection H'H'' of all geometric hyperplanes H of r such that H' n H'' = H' n H = H'' n H or H = H', H'', where H' and H'' are distinct points of V(r). Following [10, 8], we adopt also here the definition of Veldkamp space given by Buekenhout and Cohen [2] instead of that of Shult [11], as the latter is much too restrictive by requiring any three distinct hyperplanes H', H'' and H''' of r to satisfy the following two conditions: i) H' is not properly contained in H'' and ii) H' n H'' C H''' implies H' c H''' or H' n H'' = H' n H'''. The two definitions differ in the crucial fact that whereas the Veldkamp space in the sense of Shult is always a linear space, that of Buekenhout and Cohen needs not be so; in other words, Shult's Veldkamp lines are always of the form {H G V(r) | H D H' n H''} for certain geometric hyperplanes H' and H''. 3 The near hexagon L3 x GQ(2,2) The near hexagon L3 x GQ(2,2) has recently [9] caught an attention of theoretical physicists due to the fact that its main constituent, the generalized quadrangle GQ(2, 2), reproduces the commutation relations of the 15 elements of the two-qubit Pauli group (see, e. g., [7]), with each of its ten embedded copies of GQ(2,1) playing, remarkably, the role of the so-called Mermin magic square [5] — the smallest configuration of two-qubit observables furnishing a very important proof of contextuality of quantum mechanics. A well-known construction of GQ(2,2) identifies the points with two-element subsets of {1,2,3,4, 5,6}, with two points being collinear if and only if they are equal or disjoint. The natural action of S6 on this set of size 6 induces automorphisms of GQ(2,2). In fact, when considered in this way, S6 turns out to be the full automorphism group. It is known that every geometric hyperplane of a slim dense near polygon arises from its universal embedding. It can be shown from this that, equipped with the operation of Veldkamp sum, the Veldkamp space Vgq(2,2) is isomorphic to PG(4,2), the projective space obtained from a 5-dimensional space over F2 (see also [10]). It follows that GQ(2,2) has 25 - 1 = 31 geometric hyperplanes, which turn out to be of three types: (i) 15 perp-sets, with 7 points each; (ii) 10 grids (copies of GQ(2, 1)), with 9 points each; (iii) 6 ovoids, with 5 points each. In other words, there are three orbits of geometric hyperplanes under the action of S6. Identifying the points of GQ(2,2) with two-element subsets of the set {1, 2, 3,4,5,6} as described earlier, we find that an example of an ovoid is the set ei := {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}}. The other ovoids, e2, e3,..., e6 are obtained from e1 by acting by the transposition (1, i) for i = 2,3,..., 6 respectively. The Veldkamp sum e¿ + e¿ (for 1 < i < j < 6) is the perp-set of the point {i, j}. If we have {1, 2, 3,4, 5, 6} = {i, j,k, l,m,n} in some order, then the sum e¿ + e¿ + ek is the grid whose elements are the nine points {{a, b} : a G {i, j, k} and b G {l, m, n}}. 290 Ars Math. Contemp. 12 (2017) 383-413 It follows that the six ovoids are a spanning set for VGq(2,2). Since each point of GQ(2,2) lies in precisely two ovoids, it follows that we have the relation ei + e2 + + e4 + e5 + ee = 0, where 0 denotes the subset of GQ(2,2) consisting of all 15 points. Since we have an isomorphism VGq(2,2) — PG(4,2), it follows by a counting argument that this is the only nontrivial dependence relation between the e,, and thus that the ovoids e1,..., e5 form a basis for VGQ(2,2). The points of the near hexagon L3 x GQ(2,2) are simply the 45 ordered pairs (p, q) where p is a point of L3 and q is a point of GQ(2,2). We call a collection of 15 points (p, q) sharing the same value of p a layer of the near hexagon. A layer is an example of a quad in the sense of §2. We imagine that the points of L3 are arranged vertically, and we will sometimes use terms like "the top quad" to refer to one of the layers of the near hexagon. Two points (p1, q1) and (p2, q2) of L3 x GQ(2,2) are collinear if either (i) p1 = p2 and q1 is collinear to q2, or (ii) p1 is collinear to p2 and q1 = q2. The lines of L3 x GQ(2,2) are of two types. The type-one lines are the 15 lines of the form {(p, q) : p G L3} for a fixed point q G GQ(2,2). The type-two lines are the 45 lines of the form {(p, q) : q G L} for a fixed p G L3 and some line L of GQ(2,2). The near hexagon L3 x GQ(2,2) has a number of obvious automorphisms. One type of automorphism involves permuting the three GQ(2,2)-quads, but making no other changes. The subgroup of all such automorphisms is isomorphic to S3. Another type of automorphism involves acting diagonally on the three GQ(2,2)-quads by S6, the automorphism group of GQ(2,2). This action commutes with the action of S3 just mentioned, and produces a group of automorphisms isomorphic to S6 x S3. It turns out that this is the full automorphism group, as shown by Brouwer et al. [1]. From now on, let us denote the Veldkamp space of L3 x GQ(2,2) by V. Some features of V are close to obvious, which stems from Sec. 2. One of these is that the intersection of one of the three GQ(2, 2)-quads with a point of V (regarded as a subset of the 45 points) can take one of two forms. Either the GQ(2,2)-quad is completely filled in (i. e., it is deep), or takes the form of one of the geometric hyperplanes of GQ(2,2) (i. e., it is singular, sub-quadrangular or ovoidal). Furthermore, the Veldkamp sum of any two of the layers (regarded as subsets of GQ(2, 2) under some obvious identification) must be equal to the third layer. It follows from this that V contains 210 - 1 = 1023 points. The above discussion shows that, as an S6 x S3-module over F2, V is isomorphic to M ( N, where M is the 5-dimensional module for S6 described earlier, and N is the S3-module obtained by quotienting the 3-dimensional permutation module {f1, f2, f3} for S3 by the submodule spanned by f1 + f2 + f3. The set {f1, f2} then form a basis for N, and the set {e, ( fj : 1 < i < 5, 1 < j < 2} forms a basis for V. We will write this basis for short as {e1,... e10}, where for 1 < i < 5, e, denotes e, ( f1, and for 6 < i < 10, e, denotes ei-5 ( f2. R. M. Green and M. Saniga: A classification of the Veldkamp lines 291 4 The classification of hyperplanes The geometric hyperplanes of L3 x GQ(2,2) were classified in [9]. Up to automorphisms, there are eight types of them, denoted by Hi to H8 and described in detail in [9, Table 2]. We now explain how these eight types can be reconstructed using the results in the previous section. The description of the hyperplanes of GQ(2, 2) above can be used to identify each hyperplane with one of the 31 nontrivial set partitions of a 6-element into two pieces. If S and T are disjoint nonempty sets for which S U T = {1, 2, 3,4, 5, 6}, then we identify the pair {S, T} with the hyperplane = E ej . ies jeT If |S| > |T |, we associate the partition (|S|, |T |) of the number 6 to the set partition {S,T}. Under these identifications, the partitions of 6 given by (5,1), (4,2) and (3,3) correspond, via set partitions, to ovoids, perp sets and grids, respectively. The Veldkamp sum operation on VGq(2,2) described in the previous section may now be defined purely in terms of sets: the Veldkamp sum of the two set partitions {A|B} and {C|D} is given by {(A n C) u (B n D)|(A n D) u (B n C)}. This identification extends to a set-theoretic description of the hyperplanes of L3 x GQ(2,2). The hyperplanes of this larger space may be put into bijection with ordered quadruples of pairwise disjoint sets (A, B, C, D) such that (a) no three of the sets are empty and (b) the union of the four sets is {1,2,3,4, 5, 6}. Such a quadruple corresponds to the hyperplane given by the ordered triple of partitions ({A U B|C U D}, {A U C|B U D}, {A U D|B U C}). Here, the leftmost component of the ordered triple describes the hyperplane of GQ(2,2) appearing in the uppermost GQ(2,2)-quad of L3 x GQ(2,2), and so on. For example, if the sets C and D are empty, the top GQ(2, 2)-quad will be deep and the other two will be identical to each other, being either singular, sub-quadrangular or ovoidal. The correspondence between the ordered quadruples and the hyperplanes is four-to-one, because the quadruples (A, B, C, D), (B, A, D, C), (C, D, A, B) and (D, C, B, A) all index the same hyperplane. It follows that acting by an element of the Klein four-group V4 on an ordered quadruple leaves the corresponding hyperplane invariant. The group S6 x S4 acts on the quadruples, where S6 acts diagonally on each of the set partitions A, B, C and D, and S4 acts by place permutation. This induces an action of S6 x S4 on the hyperplanes of L3 x GQ(2,2), and since the action of V4 < S4 is trivial, this in turn induces an action of S6 x (S4/V4) = S6 x S3 on the hyperplanes, thus recovering the full automorphism group of L3 x GQ(2,2) in which S3 acts by permuting the GQ(2,2)-quads. This approach yields another way to deduce that the number of hyperplanes of L3 x GQ(2,2) is 2i0 - 1, as follows. There are 46 possible quadruples of pairwise disjoint sets (A, B, C, D) whose union is {1,2, 3,4, 5,6}, and four of these quadruples have three 292 Ars Math. Contemp. 12 (2017) 383-413 Table 1: A classification of geometric hyperplanes of L3 x GQ(2,2). Name Partition Orbit size Stabilizer Order Hi (3, 3) 30 (S3 l Z2) x S2 144 H2 (4, 2) 45 S4 x S2 x S2 96 H3 (5,1) 18 S5 x S2 240 H4 (2, 2,1,1) 270 S2 x S2 x S2 x S2 16 H5 (2, 2, 2) 90 S2 x S2 x S2 x S3 48 H6 (3,1,1,1) 120 S3 x S3 36 h7 (3, 2,1) 360 S3 x S2 12 H8 (4,1,1) 90 S4 x S2 48 empty components. Since the correspondence between quadruples and hyperplanes is four-to-one, the number of hyperplanes is (46 - 4)/4. The correspondence described above induces a natural correspondence between S6 x S4-orbits (or S6 x S3-orbits) of hyperplanes on the one hand, and partitions of 6 into two, three or four parts on the other. There are eight such partitions; they are shown in Table 1, together with their orbit sizes, stabilizers isomorphism types, stabilizer orders, and their name in the H - H8 notation of [9, Table 2]. 5 Counting and classifying different types of Veldkamp lines The orbits of lines in the Veldkamp space V may be enumerated using a standard technique sometimes (inaccurately) known as Burnside's Lemma, which proves the following. Let G be a finite group acting on a finite set X with t orbits, and for each g g G, let Xg denote the number of elements of X fixed by g. Then we have t = — ^ |Xg |. |G| 1 1 geo Furthermore, if C is a set of conjugacy class representatives of G, then we have 1 |G| E|C||X g|. geC Using this technique, we can recover known results about orbits of lines under the action of the automorphism group S6 of GQ(2,2): there are 3 orbits of hyperplanes (Veldkamp points) and 5 orbits of Veldkamp lines. We can also recover the result the Veldkamp space V has 8 orbits of hyperplanes under the automorphism group S6 x S3. The same idea can be adapted to count the orbits of Veldkamp lines of V. The counting argument is more complicated than for the case of Veldkamp points, because it is possible for a line to be fixed by a group element g without the three individual points being fixed. There are three possibilities to consider, which we denote by (1), (2) and (3) in Table 2. (1) Every point of the Veldkamp line is fixed by g. Such lines lie entirely within the fixed point space of g. Each number in the Fix(1) column is the number of lines in a projective space PG(d(g) - 1, 2), for a suitable integer d(g) depending on the conjugacy class of g. (2) One point of the Veldkamp line is fixed by g, and the other two are exchanged. To enumerate such lines, we take one point x outside the fixed point space of g. The R. M. Green and M. Saniga: A classification of the Veldkamp lines 293 other two points are the point g(x), and the point collinear with both of them (which is fixed by g). We then divide by 2 to correct for the overcount. Writing d(g) as above, it follows in each case that the entry in the Fix(2) column of g is given by (3) The element g rotates the three points of the Veldkamp line in a 3-cycle. Each entry in the Fix(3) column is a number of the form (4k — 1)/3, and the enumeration of these cases is the most complicated. An ordered Veldkamp line can be thought of as a sequence of 30 binary digits. Typically, some even number, 2k, of these bits can be chosen arbitrarily, provided that not all of them are zero, and then the rest of the structure is forced. It is then necessary to divide by 3 to correct for an overcount, by identifying an ordered Veldkamp line with each of its cyclic shifts. We identify the group S6 x S3 in the obvious way with the subgroup of S9 fixing setwise each of the subsets {1, 2,3,4, 5, 6} and {7, 8, 9}. Since there are 11 partitions of 6 and 3 partitions of 3, it follows that S6 x S3 has 33 conjugacy classes, and it is straightforward to find conjugacy class representatives. Table 2 shows the calculation for the Veldkamp lines of L3 x GQ(2,2). The grand total of proves that there are 156 orbits of Veldkamp lines of the near hexagon. All 156 types are then listed in Table 3. Here, each type is characterized by its composition (columns 9 to 16) and the properties of the core C of the line, that is the set of points that are common to all the three geometric hyperplanes of a line of the given type. In particular, for each type (column 1) we list the number of points (column 2) and lines (column 3) of the core as well as the distribution of the orders of its points. The last three columns show the intersection of C with each of the three GQ(2,2)-quads. Here, 'g-perp' stands for a perp-set in a certain GQ(2,1) located in the particular GQ(2,2), and 'unitr/tritr' abbreviates a unicentric/tricentric triad. If two or more types happen to possess the same string of parameters, the distinction between them is given by an explanatory remark/footnote. 1 2 673920 = |S6 x S3| x 156 = 720 x 6 x 156 294 Ars Math. Contemp. 12 (2017) 383-413 Table 2: Orbits of Veldkamp lines of L3 x GQ(2,2). Conjugacy class Fix(1) Fix(2) Fix(3) Size of class Product id 174251 0 0 1 174251 (12) 10795 384 0 15 167685 (12)(34) 651 480 0 45 50895 (12)(34)(56) 651 480 0 15 16965 (123) 651 0 5 40 26240 (123)(456) 1 0 85 40 3440 (1234) 35 24 0 90 5310 (1234)(56) 35 24 0 90 5310 (123)(45) 35 24 5 120 7680 (12345) 1 0 0 144 144 (123456) 1 0 5 120 720 (78) 155 496 0 3 1953 (12)(78) 155 496 0 45 29295 (12)(34)(78) 155 496 0 135 87885 (12)(34)(56)(78) 155 496 0 45 29295 (123)(78) 7 28 1 120 4320 (123)(456)(78) 0 1 5 120 720 (1234)(78) 7 28 0 270 9450 (1234)(56)(78) 7 28 0 270 9450 (123)(45)(78) 7 28 1 360 12960 (12345)(78) 0 1 0 432 432 (123456)(78) 0 1 5 360 2160 (789) 0 0 341 2 682 (12)(789) 0 0 85 30 2550 (12)(34)(789) 0 0 21 90 1890 (12)(34)(56)(789) 0 0 21 30 630 (123)(789) 1 0 85 80 6880 (123)(456)(789) 35 0 21 80 4480 (1234)(789) 0 0 5 180 900 (1234)(56)(789) 0 0 5 180 900 (123)(45)(789) 1 0 21 240 5280 (12345)(789) 0 0 1 288 288 (123456)(789) 1 6 5 240 2880 673920 R. M. Green and M. Saniga: A classification of the Veldkamp lines 295 Table 3: The types of Veldkamp lines of L3 x GQ(2,2). # of Points of Order Composition Tp Pt Ln 0 1 2 3 4 Hi H2 H3 H4 H5 He H7 Hg 1st 2nd 3rd 1 27 27 0 0 0 27 0 3 grid grid grid 2 25 24 0 0 10 10 5 2 1 full g-perp g-perp 3 23 19 0 0 12 11 0 2 - - 1 - - - - grid g-perp grid 4 21 20 0 0 6 12 3 - 3 full line line 5 21 18 0 6 0 12 3 1 1 1 - - - - - full unitr unitr 6 21 18 0 6 0 12 3 - 3 full tritr tritr 7 21 16 0 2 12 6 1 1 1 - 1 - - - - perp grid g-perp 8 21 16 0 0 18 0 3 - 3 perp perp perp 9 19 15 0 0 12 7 0 1 - - 2 - - - - grid g-perp g-perp 10 19 13 0 4 10 5 0 1 - - 2 - - - - grid g-perp g-perp 11 19 12 0 6 9 4 0 1 1 - - - - 1 - perp grid unitr 12 17 16 0 2 0 14 1 - 1 2 - - - - - full point point 13 17 12 0 2 12 2 1 - 1 - 2 - - - - perp g-perp g-perp 14 17 12 0 2 11 4 0 - 1 - 2 - - - - grid line g-perp 15 17 10 0 8 6 2 1 1 - - 1 1 - - - g-perp g-perp perp 16 17 10 1 4 10 2 0 1 - - 1 - - 1 - grid unitr g-perp 17 17 10 0 8 7 0 2 - 2 - - 1 - - - perp line perp 18 17 10 1 4 10 2 0 - 1 - 2 - - - - grid tritr g-perp 19 17 10 0 8 6 2 1 - 1 - 2 - - - - perp g-perp g-perp 20 17 9 2 6 6 3 0 1 - 1 - - - 1 - ovoid unitr grid 21 17 9 0 8 8 1 0 1 - - 1 - 1 - - perp g-perp g-perp 22 17 9 0 9 6 2 0 - 2 - - - 1 - - perp tritr perp 23 15 11 0 0 12 3 0 - - - 3 - - - - g-perp g-perp g-perp 24 15 9 0 6 6 3 0 1 - - - - - 2 - unitr grid unitr 25 15 9 0 6 6 3 0 - - - 3 - - - - g-perp1 g-perp g-perp 26 15 9 0 6 6 3 0 - - - 3 - - - - g-perp1 g-perp g-perp 27 15 8 2 4 7 2 0 - 1 - 1 - - 1 - grid tritr unitr 28 15 8 2 3 9 1 0 - 1 - 1 - - 1 - line grid unitr 29 15 8 2 4 7 2 0 - - 1 2 - - - grid unitr unitr 30 15 8 0 6 9 0 0 - - - 3 - - - g-perp g-perp g-perp 31 15 7 1 8 5 1 0 1 - - - - 1 1 - perp g-perp unitr 32 15 7 4 2 8 1 0 1 - - - - - - unitr grid unitr 33 15 7 1 8 5 1 0 - 1 - 1 - - 1 - perp unitr g-perp 34 15 7 0 9 6 0 0 - - - 3 - - - g-perp g-perp g-perp 35 15 6 2 10 1 2 0 1 - - - 1 - 1 - perp unitr g-perp 36 15 6 3 6 6 0 0 1 - - - - - - ovoid g-perp g-perp 37 15 6 2 9 3 1 0 - 1 1 - - - 1 - ovoid unitr perp 38 15 5 0 15 0 0 0 - - 3 - - - - - ovoid ovoid ovoid 39 13 8 0 4 8 0 1 - 1 - - 2 - - - perp line line 40 13 8 0 3 9 1 0 - 1 - - - - 2 - line grid point 41 13 8 0 4 7 2 0 - - - 2 1 - - - line g-perp g-perp 42 13 7 2 2 8 1 0 - - 1 1 - - 1 - grid unitr point 43 13 6 0 9 3 1 0 - 1 - - - 2 - - perp tritr tritr 44 13 6 0 9 3 1 0 - 1 - - - 2 - - perp line line 45 13 6 4 0 9 0 0 - 1 - - - - 2 - point grid tritr 46 13 6 0 10 2 1 0 - 1 - - - - 2 - perp g-perp point 47 13 6 0 9 3 1 0 - 1 - - - - 2 - perp unitr unitr 48 13 6 1 6 6 0 0 - - - 2 - 1 - - tritr g-perp g-perp 49 13 6 0 8 5 0 0 - - - 2 - 1 - - line g-perp g-perp 50 13 6 1 6 6 0 0 - - - 2 - - 1 - g-perp g-perp unitr 296 Ars Math. Contemp. 12 (2017) 383-413 Table 3: (Continued.) # of Points of Order Composition Tp Pt Ln 0 1 2 3 4 Hi H2 H3 H4 H5 He Ht Hs 1st 2nd 3rd 51 13 5 2 8 2 1 0 - 1 - - 1 1 - - perp line tritr 52 13 5 2 8 2 1 0 - - 1 1 - 1 - - perp unitr unitr 53 13 5 2 8 2 1 0 - - - 2 1 - - - tritr g-perp g-perp 54 13 5 0 11 2 0 0 - - - 2 1 - - - line g-perp g-perp 55 13 5 2 7 4 0 0 - - - 2 - 1 - - tritr g-perp g-perp 56 13 5 2 8 2 1 0 - - - 2 - - 1 - g-perp g-perp unitr 57 13 5 2 7 4 0 0 - - - 2 - - 1 - unitr g-perp g-perp 58 13 4 4 8 0 0 1 1 - - - 1 - - 1 perp unitr unitr 59 13 4 4 8 0 0 1 - 1 1 - - - - 1 perp ovoid point 60 13 4 4 8 0 0 1 - 1 - 1 - - - 1 perp unitr unitr 61 13 4 4 8 0 0 1 - 1 - - 2 - - - perp tritr tritr 62 13 4 4 7 1 1 0 - 1 - - - 2 - - tritr tritr perp 63 13 4 4 7 1 1 0 - 1 - - - - 2 - line g-perp ovoid 64 13 4 4 7 1 1 0 - 1 - - - - 2 - perp unitr unitr 65 13 4 4 6 3 0 0 - 1 - - - - 2 - tritr g-perp ovoid 66 13 4 4 8 0 0 1 - - 1 1 1 - - - perp unitr unitr 67 13 3 6 6 0 1 0 1 - - - - 1 - 1 perp unitr unitr 68 13 3 6 6 0 1 0 1 - - - - - 1 1 ovoid g-perp unitr 69 11 6 2 0 9 0 0 - - 1 - - - 2 - grid point point 70 11 5 0 7 4 0 0 - - - 1 - - 2 - g-perp g-perp point 71 11 4 2 7 1 1 0 - - 1 - 1 - 1 - perp unitr point 72 11 4 2 7 1 1 0 - - - 1 1 - 1 - line g-perp unitr 73 11 4 2 6 3 0 0 - - - 1 1 - 1 - line unitr g-perp 74 11 4 2 6 3 0 0 - - - 1 - 1 1 - unitr tritr g-perp 75 11 4 2 6 3 0 0 - - - 1 - 1 1 - line unitr g-perp 76 11 4 2 6 3 0 0 - - - 1 - - 2 - g-perp2 unitr unitr 77 11 4 2 6 3 0 0 - - - 1 - - 2 - g-perp2 unitr unitr 78 11 4 1 8 2 0 0 - - - 1 - - 2 - point g-perp g-perp 79 11 3 4 6 0 1 0 - 1 - - - - 1 1 perp point unitr 80 11 3 4 6 0 1 0 - - 1 - - 1 1 - perp unitr point 81 11 3 2 9 0 0 0 - - 1 - - - 2 - unitr unitr ovoid 82 11 3 4 6 0 1 0 - - - 2 - - - 1 unitr g-perp unitr 83 11 3 4 6 0 1 0 - - - 1 1 - 1 - tritr unitr g-perp 84 11 3 4 5 2 0 0 - - - 1 - 1 1 - tritr g-perp unitr 85 11 3 3 7 1 0 0 - - - 1 - 1 1 - line g-perp unitr 86 11 3 4 6 0 1 0 - - - 1 - - 2 - unitr3 g-perp unitr 87 11 3 4 6 0 1 0 - - - 1 - - 2 - unitr3 g-perp unitr 88 11 3 4 5 2 0 0 - - - 1 - - 2 - g-perp4 unitr unitr 89 11 3 4 5 2 0 0 - - - 1 - - 2 - g-perp4 unitr unitr 90 11 2 6 4 1 0 0 - 1 - - - - 1 1 line unitr ovoid 91 11 2 6 4 1 0 0 - - - 2 - - - 1 unitr g-perp unitr 92 11 2 6 4 1 0 0 - - - 1 1 - 1 - tritr unitr g-perp 93 11 2 6 4 1 0 0 - - - 1 - 1 1 - tritr g-perp unitr 94 11 2 6 4 1 0 0 - - - 1 - - 2 - g-perp unitr unitr 95 11 1 8 3 0 0 0 - - 2 - - - - 1 ovoid point ovoid 96 11 1 8 3 0 0 0 - - 1 - - - 2 - unitr unitr ovoid 97 11 0 11 0 0 0 0 1 2 unitr unitr ovoid 98 11 0 11 0 0 0 0 - 1 - - - - 1 1 tritr ovoid unitr 99 9 6 0 0 9 0 0 - - - - 3 - - - line line line 100 9 4 0 8 0 0 1 - 1 - - - - - 2 perp point point R. M. Green and M. Saniga: A classification of the Veldkamp lines 297 Table 3: (Continued.) # of Points of Order Composition Tp Pt Ln 0 1 2 3 4 Ht H2 H3 H4 H5 He H7 Hs 1st 2nd 3rd 101 9 3 2 6 0 1 0 - - 1 - - 1 - 1 perp point point 102 9 3 2 6 0 1 0 - - - 1 - - 1 1 point g-perp unitr 103 9 3 0 9 0 0 0 - - - - 3 - - - line line line 104 9 3 2 5 2 0 0 - - - - 2 1 - - line tritr line 105 9 3 0 9 0 0 0 - - - - 1 2 - - line line line 106 9 3 2 5 2 0 0 - - - - 1 - 2 - tritr g-perp point 107 9 3 1 7 1 0 0 - - - - 1 - 2 - point g-perp line 108 9 3 0 9 0 0 0 - - - - - 3 - - tritr tritr tritr 109 9 3 1 7 1 0 0 - - - - - 1 2 - point g-perp line 110 9 3 0 9 0 0 0 3 - unitr unitr unitr 111 9 2 4 4 1 0 0 - - - 1 - 1 - 1 line unitr unitr 112 9 2 4 4 1 0 0 - - - 1 - - 1 1 g-perp point unitr 113 9 2 4 4 1 0 0 - - - - 1 - 2 - line unitr5 unitr 114 9 2 4 4 1 0 0 - - - - 1 - 2 - line unitr5 unitr 115 9 2 4 4 1 0 0 - - - - 1 2 - - tritr tritr line 116 9 2 3 6 0 0 0 - - - - - 3 - - line line tritr 117 9 2 4 4 1 0 0 - - - - - 1 2 - tritr g-perp point 118 9 2 3 6 0 0 0 - - - - - 1 2 - tritr unitr unitr 119 9 2 4 4 1 0 0 3 - point6 g-perp unitr 120 9 2 4 4 1 0 0 3 - point6 g-perp unitr 121 9 1 6 3 0 0 0 - - - 1 1 - - 1 unitr line unitr 122 9 1 6 3 0 0 0 - - - - 3 - - - tritr tritr line 123 9 1 6 3 0 0 0 - - - - 1 2 - - line tritr tritr 124 9 1 6 3 0 0 0 - - - - 1 - 2 - line unitr unitr 125 9 1 6 3 0 0 0 - - - - - 3 - - tritr tritr tritr 126 9 1 6 3 0 0 0 - - - - - 1 2 - line unitr unitr 127 9 1 6 3 0 0 0 - - - - - 1 2 - tritr unitr unitr 128 9 1 6 3 0 0 0 3 - unitr unitr unitr 129 9 0 9 0 0 0 0 - 1 - - - - - 2 tritr point ovoid 130 9 0 9 0 0 0 0 - - 1 - - - 1 1 ovoid unitr point 131 9 0 9 0 0 0 0 - - - 1 1 - - 1 tritr unitr unitr 132 9 0 9 0 0 0 0 - - - 1 - 1 - 1 tritr unitr unitr 133 9 0 9 0 0 0 0 - - - 1 - - 1 1 unitr7 unitr unitr 134 9 0 9 0 0 0 0 - - - 1 - - 1 1 unitr7 unitr unitr 135 9 0 9 0 0 0 0 - - - - 2 1 - - tritr tritr tritr 136 9 0 9 0 0 0 0 - - - - 1 - 2 - tritr unitr unitr 137 9 0 9 0 0 0 0 - - - - - 1 2 - tritr unitr unitr 138 9 0 9 0 0 0 0 3 - unitr unitr unitr 139 7 2 2 4 1 0 0 - - - - 1 - 1 1 point unitr line 140 7 2 2 4 1 0 0 2 1 point g-perp point 141 7 1 4 3 0 0 0 - - 1 - - - - ovoid point point 142 7 1 4 3 0 0 0 - - - - - 1 1 1 line unitr point 143 7 1 4 3 0 0 0 2 1 unitr8 unitr point 144 7 1 4 3 0 0 0 2 1 point unitr8 unitr 145 7 0 7 0 0 0 0 - - - 1 - - - unitr unitr point 146 7 0 7 0 0 0 0 - - - - 1 - 1 1 tritr point unitr 147 7 0 7 0 0 0 0 - - - - - 1 1 1 tritr point9 unitr 148 7 0 7 0 0 0 0 - - - - - 1 1 1 tritr point9 unitr 149 7 0 7 0 0 0 0 2 1 point10 unitr unitr 150 7 0 7 0 0 0 0 2 1 point10 unitr11 unitr 298 Ars Math. Contemp. 12 (2017) 383-413 Table 3: (Continued.) # of Points of Order Composition Tp Pt Ln 0 1 2 3 4 Hi H2 H3 H4 H5 He H 7 Hg 1st 2nd 3rd 151 7 0 7 0 0 0 0 2 1 point10 unitr11 unitr 152 5 1 2 3 0 0 0 - - - - 1 - - 2 line point point 153 5 0 5 0 0 0 0 1 - 2 tritr point point 154 5 0 5 0 0 0 0 1 2 unitr point point 155 3 1 0 3 0 0 0 3 point point point 156 3 0 3 0 0 0 0 3 point point point Explanatory remarks: xTwo (25) or no two (26) of the g-perps are such that their centers are joined by a type-one line. 2 The center of the g-perp does (77) or does not (76) lie on the type-one line passing through the center of one of the two unicentric triads. 3The centers of the two unicentric triads are (86) or are not (87) joined by a type-one line. 4One line (88) or no line (89) of the g-perp is incident with the type-one line passing through the center of one of the two unicentric triads. 5The five type-one lines through the points of the two triads do (114) or do not (113) cut a doily-quad in an ovoid. 6One line (120) or no line (119) of type-two through the point is incident with the type-one line through the center of the g-perp. 7One (133) or none (134) of the unicentric triads is such that the type-one lines through two of its points pass through the centers of the other two triads. 8The centers of the two unicentric triads are (143) or are not (144) joined by a type-one line. 9The point does (147) or does not (148) lie on the type-one line passing through a center of the tricentric triad. 10The point does (149) or does not (150 and 151) lie on the type-one line passing through the center of one of the two unicentric triads. 11 The centers of the two unicentric triads do (150) or do not (151) belong to the same grid-quad. Acknowledgment This work already started in 2009, when the second author was a fellow of the Cooperation Group "Finite Projective Ring Geometries: An Intriguing Emerging Link Between Quantum Information Theory, Black-Hole Physics and Chemistry of Coupling" at the Center for Interdisciplinary Research (ZiF) of the University of Bielefeld, Bielefeld, Germany. It was also supported in part by the VEGA Grant Agency, grant No. 2/0003/16. R. M. Green and M. Saniga: A classification of the Veldkamp lines 299 References [1] A. E. Brouwer, A. M. Cohen, J. I. Hall and H. A. Wilbrink, Near polygons and Fischer spaces, Geom. Dedicata 49 (1994), 349-368, doi:10.1007/BF01264034. [2] F. Buekenhout and A. M. Cohen, Diagram geometry, volume 57 of Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. 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