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ARS MATHEMATICA CONTEMPORANEA 2 (2009) 1–25
A catalogue of simplicial arrangements in the
real projective plane
Branko Grünbaum
University of Washington, Department of Mathematics, Seattle, WA 98195, USA
Received 29 August 2008, accepted 15 January 2009, published online 27 February 2009
Abstract
An arrangement is the complex generated in the real projective plane by a finite family
of straight lines that do not form a pencil. The faces of an arrangement are the connected
components of the complement of the union of lines. An arrangement is simplicial if all its
faces are simplices (triangles). The present paper updates the only previous catalogue of
simplicial arrangements, published almost 40 years ago, and presented in a very condensed
form. The simplicial arrangements often provide optimal solutions for various problems.
A problem that seems very hard is to decide whether the collection presented here is a
complete listing of simplicial arrangements.
Keywords: Simplicial arrangement.
Math. Subj. Class.: 51M16
1 Introduction
An arrangement is the complex generated in the real projective plane by a family of straight
lines that do not form a pencil. The vertices of an arrangement are the intersection points
of two or more lines, the edges are the segments into which the lines are partitioned by
the vertices, and the faces are the connected components of the complement of the set of
lines generating the arrangement. Figure 1 shows an arrangement generated by five lines;
it has five vertices, twelve edges, and eight faces. A face vector of an arrangement is
a triplet f = (f0, f1, f2), where f0 is the number of vertices, f1 the number of edges,
and f2 the number of faces of the arrangement. For the arrangement in Figure 1 we have
f = (5, 12, 8). As is well known, Euler’s theorem implies that f0 − f1 + f2 = 1 for every
arrangement.
An arrangement is simplicial if all faces are triangles. Clearly, for every simplicial
arrangement 2f1 = 3f2. Simplicial arrangements were first introduced by Melchior [8]; an
extensive account appeared in [5].
E-mail address: grunbaum@math.washington.edu (Branko Grünbaum)
Copyright c© 2009 DMFA
2 Ars Math. Contemp. 2 (2009) 1–25
Throughout the paper we find it convenient to represent the real projective plane in
the old-fashioned way as the extended Euclidean plane. This consists of all the points of
the real Euclidean plane, together with the “ideal points-at-infinity” (each of which cor-
responds to a pencil of all mutually parallel lines) and with the “line-at-infinity” (that is
formed by the totality of the points-at-infinity). This way of representing those projective
arrangements that we are interested in avoids the need for an algebraic presentation and
makes easily possible the visual verification of the simplicial character of the arrangements
we are considering.
Three infinite families of (isomorphism classes of) simplicial arrangements are known;
we denote them by R(0), R(1) and R(2). Besides these families of simplicial arrange-
ments, only 90 other simplicial arrangements are known; we call them sporadic arrange-
ments.
Family R(0) consists of all near-pencils. A near-pencil denoted A(n, 0) consists of
n − 1 lines that have a point in common; the last (“exceptional”) line is not incident with
that point. Two examples, with n = 5, are shown in Figure 1. Near-pencil simplicial
arrangementsA(n, 0) exist for all n ≥ 3. For each n, allA(n, 0) are isomorphic; for n ≥ 5
there are uncountably many projectively inequivalent arrangements A(n, 0).
∞
Figure 1: Two illustrations of near-pencils A(5, 0) from the family R(0). These (as well
as all the other diagrams) are set in the extended Euclidean plane, which is a convenient
model of the real projective plane. In the diagram at left, some of the triangles are the usual
Euclidean triangles, while each of the other triangles consists of two components, disjoint
in the Euclidean plane but contiguous “at-infinity”; each of these triangles is bounded by a
finite segment and by two rays each of two lines. In the diagram at right, the exceptional line
is the “line-at-infinity”. Each of the triangles has as sides two Euclidean rays and a segment
of the line-at-infinity; it has one vertex in the finite plane and two vertices at-infinity. In
this illustration, and throughout the catalogue, the inclusion of the line-at-infinity as one of
the lines of the arrangement is indicated by the infinity symbol∞.
The family R(1) consists of simplicial arrangements denoted A(n, 1); these are most
easily described as follows. The arrangement A(n, 1) exists for all even n = 2m, with
m ≥ 3. Starting with a regular convex m-gon in the Euclidean plane, A(n, 1) is obtained
by taking the m lines determined by the sides of the m-gon together with the m lines of
mirror symmetry of that m-gon. The arrangements A(8, 1) and A(10, 1) are illustrated in
Figure 2.
The familyR(2) consists of simplicial arrangements denotedA(n, 1), for n = 4m + 1
with m ≥ 2. The arrangementA(4m+1, 1) is obtained fromA(4m, 1) in theR(1) family
by adjoining the “line-at-infinity” in the extended Euclidean plane model of the projective
B. Grünbaum: A catalogue of simplicial arrangements in the real projective plane 3
Figure 2: A view of the arrangements A(8, 1) and A(10, 1) from the familyR(1) .
plane. An example with m = 3 is shown in Figure 3.
∞
Figure 3: A view of the arrangement A(13, 1) from the familyR(2).
Sporadic arrangements are rather mysterious. There is no known explanation why the
ones that exist do exist, or why others do not. In particular, there is no known explanation
for the observation that no sporadic simplicial arrangement found so far has more than 37
lines. The restriction cannot be related to the topology of the projective plane, since there
are several additional infinite families of simplicial arrangements of pseudolines, besides
many additional sporadic ones.
The present compilation of data about simplicial arrangements arose from several aims.
First, there has been no detailed published account of the known simplicial arrangements
beyond the paper [5], published more that a third of a century ago. The collection in which
the paper appeared is not widely available, and I have to admit that the presentation there
is not “user-friendly”. Moreover, although the number of sporadic arrangements is still the
same as quoted there, two changes have occurred. One pair of arrangements — denoted
A2(17) andA7(17) in the paper — have been found to be isomorphic, as reported in [7] and
[1, p. 64]. On the other hand, an additional arrangement A(16, 7) was found, as indicated
in [6, pages 7 and 9]. The complete list of the presently known (isomorphism classes of)
simplicial arrangements is given in the table below, and the 90 sporadic members (as well
4 Ars Math. Contemp. 2 (2009) 1–25
as selected other arrangements) are illustrated by one or more diagrams at the end of this
catalog. For each arrangement we also list in the table several data, as well any comments
that may seem appropriate.
I conjecture that the present list is a complete enumeration of isomorphism classes of
sporadic arrangements; the language throughout the paper is simplified by assuming that
this is a fact, even though it is the most attractive open problem concerning simplicial
arrangements.
The second reason for this collection is the possibility that a better presentation than
the one in [5] may generate more interest in the challenging topic. One way of improving
the presentation is by noticing that the simplicial arrangements form a partially ordered
set, with respect to the relation in which lines of one are a subset of lines of another. The
interesting aspect is that there is only a small number of arrangements that are maximal with
respect to the number of lines. This is visible from the Hasse diagram shown in Figure 4.
Indeed, from their construction it is clear that none of the arrangements in the families
R(0), R(1) and R(2) is maximal, while the diagram shows there are only ten sporadic
maximal ones. In the diagram, the maximal arrangements are indicated by bold-framed
numerals. The numerals with shaded backgrounds indicate “pseudo-minimal” sporadic
simplicial arrangements, that is, arrangements that do not contain as subarrangements any
sporadic arrangements. A few arrangements from the families R(1) and R(2) are also
shown in cases where this seemed appropriate.
An additional reason is that the simplicial arrangements, or arrangements that are nearly
simplicial, appear as examples or counterexamples in many contexts of combinatorial ge-
ometry and its applications. This is illustrated, among others, in [6], [7], [1], [2, p. 825],
[3, p. 86], [4, Section 5.4].
The list of simplicial arrangements that follows includes all the arrangements with at
most 37 lines. The sporadic arrangement denoted A(n, k) coincides with the arrangement
denoted Ak(n) in [5], except that A(17, 7) is A9(17) because A7(17) has been shown
to be isomorphic to A2(17) and was deleted. The first column, n, denotes the number
of lines, the second column the notation for the isomorphism class of the arrangement.
The third column shows the f -vector f = (f0, f1, f2), the number of vertices, edges, and
faces of the arrangement. The fourth column, t = (t2, t3, t4, . . . ), list the values of tj ,
that is the number of vertices incident with precisely j of the lines. The fifth column,
r = (r2, r3, r4, . . . ), similarly gives the value of rj , the number of lines each of which
is incident with precisely j of the vertices. Some of the t- and r-vectors contain many
consecutive zeros; in order to save space, the presence of p consecutive zeros has been
indicated by 0p. The last column serves to indicate which of the arrangements are non-
sporadic, as well as which are maximal (signaled by M ) or pseudominimal (indicated by
m).
As is visible from the list, most of the arrangements are characterized by their f - and t-
vectors. In a majority of the other cases, the r-vectors distinguish between non-isomorphic
arrangements. In only four cases are these vectors not sufficient. In such cases a criterion
that describes the difference is explained following the diagrams of the arrangements in
question.
B. Grünbaum: A catalogue of simplicial arrangements in the real projective plane 5
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
34
37
4
1
4
5
4
2
2
2
2
1
3
3
23
1
2
3
2
5
1
1
3 2
1
1 2 3
1 2 3
2
2 3
6 2 3
9 6 2 4 3
6 7 8 3
1 7 3 6 4 5
4 3 5
6 35 4
2 4
1
3
7 64 2
4 3
4 2 3
4 5 6
3 4 5
3
3 2
2
3
5
7
1
1
1
4
4
5
1
7
2
8
2
5
2
1
1
16
Figure 4: A Hasse diagram of the partially ordered set of sporadic simplicial arrangements,
together with a few arrangements from the families R(1) and R(2). The arrangement
A(n, k) is indicated by the entry k in row n.
6 Ars Math. Contemp. 2 (2009) 1–25
2 The list of simplicial arrangements with n ≤ 37
n A(n, 0) f = (f0, f1, f2) t = (t2, t3, t4, . . . ) r = (r2, r3, r4, . . . ) Notes
3 A(3, 0) (3, 6, 4) (3) (3) R(0)
4 A(4, 0) (4, 9, 6) (3, 1) (3, 1) R(0)
5 A(5, 0) (5, 12, 8) (4, 0, 1) (4, 0, 1) R(0)
6 A(6, 0) (6, 15, 10) (5, 0, 0, 1) (5, 0, 0, 1) R(0)
A(6, 1) (7, 18, 12) (3, 4) (0, 6) R(1)
7 A(7, 0) (7, 18, 12) (6, 0, 0, 0, 1) (6, 0, 0, 0, 1) R(0)
A(7, 1) (9, 24, 16) (3, 6) (0, 4, 3) m
8 A(8, 0) (8, 21, 14) (7, 04, 1) (7, 04, 1) R(0)
A(8, 1) (11, 30, 20) (4, 6, 1) (0, 2, 6) R(1)
9 A(9, 0) (9, 24, 16) (8, 05, 1) (8, 05, 1) R(0)
A(9, 1) (13, 36, 24) (6, 4, 3) (0, 0, 9) R(2)
10 A(10, 0) (10, 27, 18) (9, 06, 1) (9, 06, 1) R(0)
A(10, 1) (16, 45, 30) (5, 10, 0, 1) (0, 0, 5, 5) R(1)
A(10, 2) (16, 45, 30) (6, 7, 3) (0, 0, 6, 3, 1)
A(10, 3) (16, 45, 30) (6, 7, 3) (0, 1, 3, 6)
11 A(11, 0) (11, 30, 20) (10, 07, 1) (10, 07, 1) R(0)
A(11, 1) (19, 54, 36) (7, 8, 4) (0, 0, 4, 4, 3)
12 A(12, 0) (11, 33, 22) (11, 08, 1) (11, 08, 1) R(0)
A(12, 1) (22, 63, 42) (6, 15, 0, 0, 1) (0, 0, 3, 3, 6) R(1)
A(12, 2) (22, 63, 42) (8, 10, 3, 1) (0, 0, 3, 3, 6)
A(12, 3) (22, 63, 42) (9, 7, 6) (0, 0, 3, 3, 6)
13 A(13, 0) (12, 36, 24) (12, 09, 1) (12, 09, 1) R(0)
A(13, 1) (25, 72, 48) (9, 12, 3, 0, 1) (0, 0, 3, 0, 10) R(2)
A(13, 2) (25, 72, 48) (12, 4, 9) (0, 0, 3, 0, 10)
A(13, 3) (25, 72, 48) (10, 10, 3, 2) (0, 0, 1, 4, 8)
A(13, 4) (27, 78, 52) (6, 18, 3) (04, 13) m
14 A(14, 0) (14, 39, 26) (13, 010, 1) (13, 010, 1) R(0)
A(14, 1) (29, 84, 56) (7, 21, 0, 0, 0, 1) (0, 0, 0, 7, 0, 7) R(1)
A(14, 2) (29, 84, 56) (11, 12, 4, 2) (0, 0, 1, 4, 4, 4, 1)
A(14, 3) (30, 87, 58) (9, 16, 4, 1) (04, 11, 3)
A(14, 4) (29, 84, 56) (10, 14, 4, 0, 1) (0, 0, 0, 4, 6, 4) m
15 A(15, 0) (15, 42, 28) (14, 011, 1) (14, 011, 1) R(0)
A(15, 1) (31, 90, 60) (15, 10, 0, 6) (04, 15) m
A(15, 2) (33, 96, 64) (13, 12, 6, 2) (0, 0, 1, 4, 2, 4, 4)
A(15, 3) (34, 99, 66) (12, 13, 9) (04, 9, 3, 3)
A(15, 4) (33, 96, 64) (12, 14, 6, 0, 1) (04, 10, 4, 1)
A(15, 5) (34, 99, 66) (9, 22, 0, 3) (04, 9, 3, 3) m
16 A(16, 0) (16, 45, 30) (15, 012, 1) (15, 012, 1) R(0)
A(16, 1) (37, 108, 72) (8, 28, 04, 1) (0, 0, 0, 4, 4, 0, 8) R(1)
A(16, 2) (37, 108, 72) (14, 15, 6, 1, 1) (0, 0, 1, 2, 4, 2, 7)
A(16, 3) (37, 108, 72) (15, 13, 6, 3) (04, 10, 0, 6)
A(16, 4) (36, 105, 70) (15, 15, 0, 6) (04, 10, 5, 0, 0, 1)
A(16, 5) (37, 108, 72) (14, 16, 3, 4) (0, 0, 0, 2, 4, 8, 0, 2) m
Continued on next page
B. Grünbaum: A catalogue of simplicial arrangements in the real projective plane 7
n A(n, 0) f = (f0, f1, f2) t = (t2, t3, t4, . . . ) r = (r2, r3, r4, . . . ) Notes
A(16, 6) (37, 108, 72) (15, 12, 9, 0, 1) (04, 7, 6, 3)
A(16, 7) (38, 111, 74) (12, 19, 6, 0, 1) (0, 0, 0, 3, 3, 2, 8) m
17 A(17, 0) (17, 48, 32) (16, 013, 1) (16, 013, 1) R(0)
A(17, 1) (41, 120, 80) (12, 24, 4, 0, 0, 0, 1) (04, 8, 0, 9) R(2)
A(17, 2) (41, 120, 80) (16, 16, 7, 0, 2) (0, 0, 1, 0, 6, 0, 10)
A(17, 3) (41, 120, 80) (18, 12, 7, 4) (04, 8, 0, 9)
A(17, 4) (41, 120, 80) (16, 16, 7, 0, 2) (0, 0, 1, 0, 6, 0, 10)
A(17, 5) (41, 120, 80) (16, 18, 1, 6) (04, 6, 8, 1, 0, 2)
A(17, 6) (42, 123, 82) (16, 15, 10, 0, 1) (04, 6, 3, 7, 0, 1)
A(17, 7) (43, 126, 84) (13, 22, 7, 0, 1) (04, 6, 0, 10, 0, 1) was A9(17)
A(17, 8) (43, 126, 84) (14, 20, 7, 2) (0, 0, 0, 0, 1, 8, 8) m
18 A(18, 0) (18, 51, 34) (17, 014, 1) (17, 014, 1) R(0)
A(18, 1) (46, 135, 90) (9, 36, 05, 1) (04, 9, 0, 0, 9) R(1)
A(18, 2) (46, 135, 90) (18, 18, 6, 3, 1) (04, 3, 3, 12) m
A(18, 3) (46, 135, 90) (19, 16, 6, 5) (04, 6, 2, 6, 3, 1)
A(18, 4) (46, 135, 90) (18, 19, 3, 6) (04, 3, 9, 3, 0, 3)
A(18, 5) (46, 135, 90) (18, 19, 3, 6) (04, 3, 9, 3, 0, 3)
A(18, 6) (47, 138, 92) (18, 16, 12, 0, 1) (04, 5, 2, 7, 2, 2)
A(18, 7) (46, 135, 90) (18, 18, 6, 3, 1) (04, 3, 3, 0, 6, 6)
A(18, 8) (47, 138, 92) (16, 22, 6, 2, 1) (04, 6, 0, 7, 4, 1)
19 A(19, 0) (19, 54, 36) (18, 015, 1) (18, 015, 1) R(0)
A(19, 1) (49, 144, 96) (21, 18, 6, 0, 4) (04, 4, 0, 15)
A(19, 2) (51, 150, 100) (21, 18, 6, 6) (04, 1, 8, 6, 0, 4)
A(19, 3) (49, 144, 96) (24, 12, 6, 6, 1) (04, 4, 0, 15)
A(19, 4) (51, 150, 100) (20, 20, 6, 4, 1) (04, 4, 4, 4, 4, 3)
A(19, 5) (51, 150, 100) (20, 20, 6, 4, 1) (04, 4, 4, 4, 4, 3)
A(19, 6) (51, 150, 100) (20, 20, 6, 4, 1) (04, 6, 0, 6, 4, 3)
A(19, 7) (52, 153, 102) (21, 15, 15, 0, 1) (04, 4, 3, 3, 6, 3)
20 A(20, 0) (20, 57, 38) (19, 016, 1) (19, 016, 1) R(0)
A(20, 1) (56, 165, 110) (10, 45, 06, 1) (04, 5, 5, 0, 0, 10) R(1)
A(20, 2) (56, 165, 110) (25, 15, 10, 6) (05, 5, 10, 0, 5)
A(20, 3) (56, 165, 110) (21, 24, 6, 4, 0, 1) (04, 4, 2, 4, 6, 3, 1)
A(20, 4) (56, 165, 110) (23, 20, 7, 5, 1) (04, 5, 1, 4, 4, 6)
A(20, 5) (55, 162, 108) (20, 26, 4, 4, 0, 0, 1) (0, 0, 0, 2, 2, 0, 4, 12)
21 A(21, 0) (21, 60, 40) (20, 017, 1) (20, 017, 1) R(0)
A(21, 1) (61, 180, 120) (15, 40, 5, 05, 1) (04, 5, 0, 5, 0, 11) R(2)
A(21, 2) (61, 180, 120) (30, 10, 15, 6) (06, 15, 0, 6)
A(21, 3) (61, 180, 120) (24, 24, 9, 0, 4) (04, 6, 0, 3, 0, 12)
A(21, 4) (61, 180, 120) (22, 28, 6, 4, 0, 0, 1) (04, 4, 0, 4, 8, 4, 0, 1) M
A(21, 5) (61, 180, 120) (26, 20, 9, 4, 2) (04, 5, 0, 3, 4, 9)
A(21, 6) (63, 186, 124) (25, 20, 15, 2, 1) (04, 1, 0, 11, 0, 8, 0, 1) M
A(21, 7) (64, 189, 126) (24, 22, 15, 3) (06, 12, 0, 6, 3) M
22 A(22, 0) (22, 63, 42) (21, 018, 1) (21, 018, 1) R(0)
A(22, 1) (67, 198, 132) (11, 55, 07, 1) (05, 11, 0, 0, 0, 11) R(1)
A(22, 2) (70, 207, 138) (24, 30, 12, 3, 1) (04, 1, 0, 6, 3, 9, 0, 3)
Continued on next page
8 Ars Math. Contemp. 2 (2009) 1–25
n A(n, 0) f = (f0, f1, f2) t = (t2, t3, t4, . . . ) r = (r2, r3, r4, . . . ) Notes
A(22, 3) (67, 198, 132) (27, 28, 0, 12) (06, 12, 0, 9, 0, 1)
A(22, 4) (67, 198, 132) (27, 25, 9, 3, 3) (04, 4, 0, 6, 0, 6, 6)
23 A(23, 0) (23, 66, 44) (22, 019, 1) (22, 019, 1) R(0)
A(23, 1) (75, 222, 148) (27, 32, 10, 4, 2) (04, 1, 0, 6, 2, 7, 4, 3)
24 A(24, 0) (24, 69, 46) (23, 020, 1) (23, 020, 1) R(0)
A(24, 1) (79, 234, 156) (12, 66, 08, 1) (05, 6, 6, 0, 0, 0, 12) R(1)
A(24, 2) (77, 228, 152) (32, 32, 0, 12, 0, 0, 1) (05, 4, 0, 0, 20) m
A(24, 3) (80, 237, 158) (31, 32, 9, 5, 3) (04, 1, 0, 6, 1, 6, 6, 4)
25 A(25, 0) (25, 72, 48) (24, 021, 1) (24, 021, 1) R(0)
A(25, 1) (85, 252, 168) (18, 60, 6, 07, 1) (06, 12, 0, 0, 0, 13) R(2)
A(25, 2) (85, 252, 168) (36, 28, 15, 0, 6) (04, 4, 0, 3, 0, 6, 0, 12) M
A(25, 3) (91, 270, 180) (30, 40, 15, 6) (08, 15, 0, 10)
A(25, 4) (85, 252, 168) (36, 30, 9, 6, 4) (04, 1, 0, 9, 0, 3, 0, 12)
A(25, 5) (81, 240, 160) (36, 32, 0, 8, 4, 0, 1) (06, 5, 0, 20) M
A(25, 6) (85, 252, 168) (36, 30, 9, 6, 4) (04, 1, 0, 6, 0, 6, 6, 6)
A(25, 7) (85, 252, 168) (33, 34, 12, 2, 3, 0, 1) (04, 2, 0, 4, 4, 4, 0, 1)
26 A(26, 0) (26, 75, 50) (25, 022, 1) (25, 022, 1) R(0)
A(26, 1) (92, 273, 182) (13, 78, 09, 1) (06, 13, 04, 13) R(1)
A(26, 2) (96, 285, 190) (35, 40, 10, 11) (08, 11, 5, 10)
A(26, 3) (92, 273, 182) (37, 36, 9, 6, 3, 1) (04, 1, 0, 7, 2, 2, 1, 8, 4, 1)
A(26, 4) (92, 273, 182) (35, 39, 10, 4, 3, 0, 1) (04, 1, 1, 4, 4, 2, 2, 7, 4, 1)
27 A(27, 0) (27, 78, 52) (26, 023, 1) (26, 023, 1) R(0)
A(27, 1) (101, 300, 200) (40, 40, 6, 14, 1) (08, 8, 8, 11)
A(27, 2) (99, 294, 196) (39, 40, 10, 6, 2, 2) (04, 1, 0, 5, 4, 1, 2, 4, 8, 2)
A(27, 3) (99, 294, 196) (39, 40, 10, 6, 2, 2) (04, 1, 0, 6, 2, 2, 2, 5, 6, 3)
A(27, 4) (99, 294, 196) (38, 42, 9, 6, 3, 0, 1) (04, 1, 0, 5, 4, 2, 0, 7, 4, 4)
28 A(28, 0) (28, 81, 54) (27, 024, 1) (27, 024, 1) R(0)
A(28, 1) (106, 315, 210) (14, 91, 010, 1) (06, 7, 7, 04, 14) R(1)
A(28, 2) (106, 315, 210) (45, 40, 3, 15, 3) (08, 6, 9, 13)
A(28, 3) (106, 315, 210) (45, 40, 3, 15, 3) (08, 6, 9, 13)
A(28, 4) (106, 315, 210) (41, 44, 11, 6, 2, 1, 1) (04, 1, 0, 4, 4, 2, 1, 4, 6, 6)
A(28, 5) (106, 315, 210) (42, 42, 12, 6, 1, 3) (04, 1, 0, 4, 4, 1, 3, 1, 10, 4)
A(28, 6) (106, 315, 210) (42, 42, 12, 6, 1, 3) (04, 1, 0, 6, 0, 3, 3, 3, 6, 6)
29 A(29, 0) (29, 84, 56) (28, 025, 1) (28, 025, 1) R(0)
A(29, 1) (113, 336, 224) (21, 84, 7, 09, 1) (06, 7, 0, 7, 0, 0, 0, 15) R(2)
A(29, 2) (113, 336, 224) (50, 40, 1, 14, 6) (08, 5, 8, 16)
A(29, 3) (113, 336, 224) (44, 46, 13, 6, 2, 0, 2) (04, 1, 0, 3, 4, 3, 0, 4, 4, 10)
A(29, 4) (113, 336, 224) (45, 44, 14, 6, 1, 2, 1) (04, 1, 0, 3, 4, 2, 2, 1, 8, 8)
A(29, 5) (113, 336, 224) (45, 44, 14, 6, 1, 2, 1) (04, 1, 0, 4, 2, 3, 2, 2, 6, 9)
30 A(30, 0) (30, 87, 58) (29, 026, 1) (29, 026, 1) R(0)
A(30, 1) (121, 360, 240) (15, 105, 011, 1) (07, 15, 05, 15) R(1)
A(30, 2) (116, 345, 230) (55, 40, 0, 11, 10) (08, 5, 5, 20)
A(30, 3) (120, 357, 238) (49, 44, 17, 6, 1, 1, 2) (04, 1, 0, 3, 2, 4, 1, 2, 4, 13)
31 A(31, 0) (31, 90, 60) (30, 027, 1) (30, 027, 1) R(0)
A(31, 1) (121, 360, 240) (60, 40, 0, 6, 15) (08, 6, 0, 25) M
Continued on next page
B. Grünbaum: A catalogue of simplicial arrangements in the real projective plane 9
n A(n, 0) f = (f0, f1, f2) t = (t2, t3, t4, . . . ) r = (r2, r3, r4, . . . ) Notes
A(31, 2) (127, 378, 252) (54, 42, 21, 6, 1, 0, 3) (04, 1, 0, 0, 0, 9, 0, 6, 0, 15) M
A(31, 3) (127, 378, 252) (54, 42, 21, 6, 1, 0, 3) (04, 1, 0, 3, 0, 6, 0, 3, 0, 18) M
32 A(32, 0) (32, 93, 62) (31, 028, 1) (31, 028, 1) R(0)
A(32, 1) (137, 408, 272) (16, 120, 012, 1) (07, 8, 8, 05, 16) R(1)
33 A(33, 0) (33, 96, 63) (32, 029, 1) (32, 029, 1) R(0)
A(33, 1) (145, 432, 288) (24, 112, 8, 011, 1) (08, 16, 05, 17) R(2)
34 A(34, 0) (34, 99, 66) (33, 030, 1) (33, 030, 1) R(0)
A(34, 1) (154, 459, 306) (17, 136, 013, 1) (08, 17, 06, 17) R(1)
A(34, 2) (154, 459, 306) (60, 63, 18, 6, 4, 0, 3) (06, 3, 3, 3, 0, 4, 0, 6, 0, 9, 6)
35 A(35, 0) (35, 102, 68) (34, 031, 1) (34, 031, 1) R(0)
36 A(36, 0) (36, 105, 70) (35, 032, 1) (35, 032, 1) R(0)
A(36, 1) (172, 513, 342) (18, 153, 014, 1) (08, 9, 9, 06, 18) R(1)
37 A(37, 0) (37, 108, 72) (36, 033, 1) (36, 033, 1) R(0)
A(37, 1) (181, 540, 360) (27, 144, 9, 013, 1) (08, 9, 0, 9, 05, 19) R(2)
A(37, 2) (181, 540, 360) (72, 72, 12, 24, 06, 1) (010, 13, 0, 0, 0, 24) m, M
A(37, 3) (181, 540, 360) (72, 72, 24, 0, 10, 0, 3) (06, 3, 0, 6, 0, 4, 03, 12, 0, 12) M
3 Illustrations of selected arrangements
∞
A(6, 1)
The above are four different presentations of the same simplicial arrangementA(6, 1). Ad-
ditional ones could be added, but it seems that the ones shown here are sufficient to illustrate
the variety of forms in which isomorphic simplicial arrangements may appear. Naturally,
in most of the other such arrangements the number of possible appearances would be even
greater, making the catalog unwieldy. That is the reason why only one or two possible
presentations are shown for most of the other simplicial arrangements. In most cases the
form shown is the one with greatest symmetry.
10 Ars Math. Contemp. 2 (2009) 1–25
∞ ∞
A(7, 1)
A(8, 1) A(9, 1)
∞ ∞
A(10, 2)
∞ ∞
A(10, 3)
B. Grünbaum: A catalogue of simplicial arrangements in the real projective plane 11
∞ ∞
A(11, 1)
A(12, 1)
∞ ∞
A(12, 2)
A(12, 3)
∞
A(13, 1)
12 Ars Math. Contemp. 2 (2009) 1–25
∞ ∞
A(13, 2)
∞
A(13, 3)
∞ ∞
A(13, 4)
∞
A(14, 2)
∞
A(14, 3)
B. Grünbaum: A catalogue of simplicial arrangements in the real projective plane 13
∞
A(14, 4)
A(15, 1)
oo
A(15, 2) A(15, 3)
∞
A(15, 4) A(15, 5)
14 Ars Math. Contemp. 2 (2009) 1–25
A(16, 1)
∞
A(16, 2)
∞
A(16, 3)
∞
A(16, 4)
A(16, 5)
∞
A(16, 6)
A(16, 7)
∞
A(17, 1)
B. Grünbaum: A catalogue of simplicial arrangements in the real projective plane 15
∞
A(17, 2)
∞
A(17, 3)
∞
A(17, 4)
A(17, 4) has two lines with four quadruple points each, while A(17, 2) has no such line.
∞
A(17, 5)
∞
A(17, 6)
∞
A(17, 7)
16 Ars Math. Contemp. 2 (2009) 1–25
∞
A(17, 8) A(18, 2)
∞ ∞
A(18, 3)
A(18, 4) A(18, 5)
Each of A(18, 4) and A(18, 5) contains three quadruple points that determine three lines.
These lines determine 4 triangles. In A(18, 4) there is a triangle that contains three of the
quintuple points, while no such triangle exists in A(18, 5).
∞
A(18, 6)
∞
A(18, 7)
B. Grünbaum: A catalogue of simplicial arrangements in the real projective plane 17
∞
A(18, 8)
∞
A(19, 1)
∞
A(19, 2)
∞
A(19, 3)
∞
A(19, 4)
∞
A(19, 5)
A(19, 4) andA(19, 5) differ by the order of the points at-infinity of different multiplicities.
∞ ∞
A(19, 6)
18 Ars Math. Contemp. 2 (2009) 1–25
∞
A(19, 7) A(20, 2)
∞
A(20, 3)
∞
A(20, 4)
A(20, 5)
∞
A(21, 2)
∞ ∞
A(21, 3)
B. Grünbaum: A catalogue of simplicial arrangements in the real projective plane 19
oo
A(21, 4)
∞ ∞
A(21, 5)
∞
A(21, 6) A(21, 7)
∞
A(22, 2)
∞
A(22, 3)
20 Ars Math. Contemp. 2 (2009) 1–25
oo
A(22, 4)
∞
A(23, 1)
A(24, 1) A(24, 2)
∞
A(24, 3)
∞
A(25, 1)
∞ ∞
A(25, 2)
B. Grünbaum: A catalogue of simplicial arrangements in the real projective plane 21
A(25, 3)
oo
A(25, 4)
∞
A(25, 5)
∞
A(25, 6)
oo
A(25, 7)
∞
A(26, 2)
oo
A(26, 3)
oo
A(26, 4)
22 Ars Math. Contemp. 2 (2009) 1–25
A(27, 1)
oo
A(27, 2)
oo
A(27, 3)
oo
A(27, 4)
A(28, 2) A(28, 3)
In A(28, 3) one of the triangles determined by the 3 sextuple points contains no quintuple
point. In A(28, 2) there is no such triangle.
oo
A(28, 4)
oo
A(28, 5)
B. Grünbaum: A catalogue of simplicial arrangements in the real projective plane 23
oo
A(28, 6)
∞
A(29, 2)
oo
A(29, 3)
oo
A(29, 4)
oo
A(29, 5) A(30, 2)
oo
A(30, 3)
∞
A(31, 1)
24 Ars Math. Contemp. 2 (2009) 1–25
oo
A(31, 2)
oo
A(31, 3)
oo
A(34, 2)
∞
A(37, 2)
oo
A(37, 3)
References
[1] G. Barthel, F. Hirzebruch and T. Höfer, Geradenkonfigurationen und Algebraische Flächen,
Vieweg, Braunschweig, 1987.
[2] P. Erdös and G. Purdy, Extremal problems in combinatorial geometry, in: R. L. Graham, M.
Grötschel and L. Lovász (eds.), Handbook of Combinatorics, Elsevier, New York, 1995, 809–
874.
[3] S. Felsner, Geometric Graphs and Arrangements. Vieweg, Wiesbaden, 2004.
[4] J. E. Goodman, Pseudoline arrangements, in: J. E. Goodman and J. O’Rourke (eds.), Handbook
of Discrete and Computational Geometry, 2nd ed., Chapman & Hall/CRC, New York, 2004,
97–128.
[5] B. Grünbaum, Arrangements of hyperplanes, in: R. C. Mullin et al. (eds.), Proc. Second
Louisiana Conf. on Combinatorics, Graph Theory and Computing, Louisiana State University,
Baton Rouge, 41–106.
B. Grünbaum: A catalogue of simplicial arrangements in the real projective plane 25
[6] B. Grünbaum, Arrangements and Spreads, CBMS Regional Conference Series in Mathematics,
No. 10., AMS, 1972, reprinted 1980.
[7] F. Hirzebruch, Singularities of algebraic surfaces and characteristic numbers, in: Proc. Lefschetz
Centenial Conf., Mexico City 1984, Part I, Contemporary Mathematics 58, AMS, 1986, 141–
155.
[8] E. Melchior, Über Vielseite der projektiven Ebene, Deutsche Mathematik 5 (1941), 461–475.