Combinatorial Intricacies of Labeled Fano Planes

Abstract

Given a seven-element set $X=\{1,2,3,4,5,6,7\}$, there are 30 ways to define a Fano plane on it. Let us call a line of such a Fano plane—that is to say an unordered triple from X—ordinary or defective, according to whether the sum of two smaller integers from the triple is or is not equal to the remaining one, respectively. A point of the labeled Fano plane is said to be of the order s, $0\le s\le 3$, if there are s defective lines passing through it. With such structural refinement in mind, the 30 Fano planes are shown to fall into eight distinct types. Out of the total of 35 lines, nine ordinary lines are of five different kinds, whereas the remaining 26 defective lines yield as many as ten distinct types. It is shown that no labeled Fano plane can have all points of zero-th order, or feature just one point of order two. A connection with prominent configurations in Steiner triple systems is also pointed out.

1. Introduction

The Fano plane is the smallest projective plane. It consists of seven lines and seven points, with three points on a line and, dually, three lines per point, where every pair of points is connected by a line, every line intersects every other line, and there are four points such that no line contains more than two of them. It is well known [1,2,3] that there are thirty different Fano planes on a given seven-element set. Slightly rephrased, there are thirty different ways to label the points of the Fano plane by integers from 1 to 7, two labeled Fano planes having zero, one or three lines in common, and each line occurring in six Fano planes. The set of thirty labeled Fano planes can be uniquely partitioned into two sets (say A and B) of fifteen elements each, such that any two labeled Fanos in the same set have just one line in common. Following the labeling adopted by Polster [4], the two sets read

$$A_1:\{124,136,157,\overline{235},267,\overline{347},456\},B_1:\{127,136,\overline{145},\overline{235},\overline{246},\overline{347},567\},$$

$$A_2:\{127,136,\overline{145},234,256,357,467\},B_2:\{125,136,147,234,267,357,456\},$$

$$A_3:\{125,136,147,237,\overline{246},345,567\},B_3:\{124,136,157,237,256,345,467\},$$

$$A_4:\{125,\overline{134},\overline{167},236,247,357,456\},B_4:\{127,\overline{134},\overline{156},236,245,357,467\},$$

$$A_5:\{127,135,146,236,245,\overline{347},567\},B_5:\{124,135,\overline{167},236,\overline{257},\overline{347},456\},$$

$$A_6:\{124,137,\overline{156},236,\overline{257},345,467\},B_6:\{125,137,146,236,247,345,567\},$$

$$A_7:\{\overline{123},147,\overline{156},245,267,346,357\},B_7:\{\overline{123},\overline{145},\overline{167},247,256,346,357\},$$

$$A_8:\{124,135,\overline{167},237,256,346,457\},B_8:\{126,135,147,237,245,346,567\},$$

$$A_9:\{126,137,\overline{145},\overline{235},247,346,567\},B_9:\{124,137,\overline{156},\overline{235},267,346,457\},$$

$$A_{10}:\{\overline{123},\overline{145},\overline{167},\overline{246},\overline{257},\overline{347},356\},B_{10}:\{\overline{123},146,157,245,267,\overline{347},356\},$$

$$A_{11}:\{126,\overline{134},157,237,245,356,467\},B_{11}:\{125,\overline{134},\overline{167},237,\overline{246},356,457\},$$

$$A_{12}:\{125,137,146,234,267,356,457\},B_{12}:\{126,137,\overline{145},234,\overline{257},356,467\},$$

$$A_{13}:\{\overline{123},146,157,247,256,345,367\},B_{13}:\{\overline{123},147,\overline{156},\overline{246},\overline{257},345,367\},$$

$$A_{14}:\{127,\overline{134},\overline{156},\overline{235},\overline{246},367,457\},B_{14}:\{126,\overline{134},157,\overline{235},247,367,456\},$$

$$A_{15}:\{126,135,147,234,\overline{257},367,456\},B_{15}:\{127,135,146,234,256,367,457\},$$

where for triples of integers we use a shorthand notation $\{a,b,c\}=abc$.

2. Refined Structure of Numbered Fano Planes

Given a line $abc$, where, without loss of generality, we can take $1\le a<b<c\le 7$, we shall distinguish between the cases when $a+b=c$ and $a+b\ne c$ and call the former/latter ordinary/defective [5]; for the sake of convenience, in the above-given sets A and B, all ordinary lines are denoted by overbars. Furthermore, a point of the labeled Fano plane is said to be of order s, $0\le s\le 3$ if there are s defective lines passing through it; hence, in addition to two different kinds of lines, a labeled Fano plane can potentially feature up to four distinct types of points. A detailed inspection of each of the 30 labeled Fano planes given above shows that they fall, in terms of such structural refinement, into eight different types, as portrayed in Figure 1 and summarized in

Table 1. The eight distinct types of labeled Fano planes, each being uniquely characterized by the number of points of every particular order. To better understand the symmetry pattern, we arrange complementary types into pairs and also list the two non-existent types α and γ.

Type	Points of Order
	0	1	2	3
(α)	(7)	(0)	(0)	(0)
${\alpha }^{\prime }$	0	0	0	7
β	4	3	0	0
${\beta }^{\prime }$	0	0	3	4
(γ)	(2)	(4)	(1)	(0)
${\gamma }^{\prime }$	0	1	4	2
${\delta }_1$	1	3	3	0
${\delta }_1^{\prime }$	0	3	3	1
${\delta }_2$	0	6	0	1
${\delta }_2^{\prime }$	1	0	6	0

. It is also interesting to compare the distributions of types within each 15-element set, given in

Table 2. Types of Fano planes in set A.

Plane	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15
Type	${\gamma }^{\prime }$	${\beta }^{\prime }$	${\beta }^{\prime }$	${\gamma }^{\prime }$	${\beta }^{\prime }$	${\gamma }^{\prime }$	${\gamma }^{\prime }$	${\beta }^{\prime }$	${\gamma }^{\prime }$	β	${\beta }^{\prime }$	${\alpha }^{\prime }$	${\beta }^{\prime }$	${\delta }_2$	${\beta }^{\prime }$

and

Table 3. Types of Fano planes in set B.

Plane	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15
Type	${\delta }_1$	${\alpha }^{\prime }$	${\alpha }^{\prime }$	${\gamma }^{\prime }$	${\delta }_2^{\prime }$	${\alpha }^{\prime }$	${\delta }_2^{\prime }$	${\alpha }^{\prime }$	${\gamma }^{\prime }$	${\gamma }^{\prime }$	${\delta }_1^{\prime }$	${\gamma }^{\prime }$	${\delta }_1$	${\gamma }^{\prime }$	${\alpha }^{\prime }$

, as well as cardinalities of individual types, listed in

Table 4. Cardinalities of individual types of Fano planes.

Type	(α)	β	(γ)	${\delta }_1$	${\delta }_2$	${\alpha }^{\prime }$	${\beta }^{\prime }$	${\gamma }^{\prime }$	${\delta }_1^{\prime }$	${\delta }_2^{\prime }$
Set $\mathit{A}$	-	1	-	0	1	1	7	5	0	0
Set $\mathit{B}$	-	0	-	2	0	5	0	5	1	2
Total	-	1	-	2	1	6	7	10	1	2

; note a pronounced asymmetry between sets A and B.

As easily discernible from Figure 1, the different types of Fano planes—and the dual pairs formed by them—are more readily ascertained by the number of defective lines they contain; the only exception is the case when there are three defective lines, where one has to distinguish whether these are concurrent or not. From this perspective, the lack of a plane of type γ becomes even more interesting. We can see through exhaustion that there is no such plane, but it would be really interesting to see if there are some geometric explanations for why it is impossible.

In addition to classifying planes, we can also classify lines in terms of the types of six planes passing through each of them, the corresponding findings being given in

Table 5. The types of lines; nine ordinary lines go first. Lines belong to a given type if they possess the same string of parameters.

Line	(α)	β	(γ)	${\delta }_1$	${\delta }_2$	${\alpha }^{\prime }$	${\beta }^{\prime }$	${\gamma }^{\prime }$	${\delta }_1^{\prime }$	${\delta }_2^{\prime }$
123	-	1	-	1	0	0	1	2	0	1
145	-	1	-	1	0	0	1	2	0	1
257	-	1	-	1	0	0	1	2	0	1
347	-	1	-	1	0	0	1	2	0	1
156	-	0	-	1	1	0	0	4	0	0
235	-	0	-	1	1	0	0	4	0	0
246	-	1	-	2	1	0	1	0	1	0
167	-	1	-	0	0	0	1	1	1	2
134	-	0	-	0	1	0	1	3	1	0
124	-	0	-	0	0	1	1	3	0	1
236	-	0	-	0	0	1	1	3	0	1
247	-	0	-	0	0	1	1	3	0	1
346	-	0	-	0	0	1	1	3	0	1
357	-	0	-	0	0	1	1	3	0	1
456	-	0	-	0	0	1	1	3	0	1
136	-	0	-	1	0	2	2	1	0	0
147	-	0	-	1	0	2	2	1	0	0
345	-	0	-	1	0	2	2	1	0	0
567	-	0	-	1	0	2	2	1	0	0
126	-	0	-	0	0	1	2	3	0	0
157	-	0	-	0	0	1	2	3	0	0
245	-	0	-	0	0	1	2	3	0	0
467	-	0	-	0	0	1	2	3	0	0
135	-	0	-	0	0	2	3	0	0	1
237	-	0	-	0	0	2	3	0	0	1
256	-	0	-	0	0	2	3	0	0	1
127	-	0	-	1	1	1	2	1	0	0
367	-	0	-	1	1	1	2	1	0	0
146	-	0	-	0	0	3	2	1	0	0
234	-	0	-	0	0	3	2	1	0	0
137	-	0	-	0	0	2	0	4	0	0
267	-	0	-	0	0	2	0	4	0	0
356	-	1	-	0	0	1	1	2	1	0
457	-	0	-	0	1	2	1	1	1	0
125	-	0	-	0	0	3	1	1	1	0

. We find that nine ordinary lines are of five different kinds, whereas the remaining 26 defective lines yield as many as ten distinct types.

To conclude the paper, let us have a look at Figure 1 and focus on that sub-geometry in each labeled Fano plane representative which is formed by defective lines and points lying on them. We shall find the Fano plane itself and, using the language of [6], all its derivatives depicted in Figure 2 that play a crucial role in classifying Steiner triple systems (e.g., [7,8]). As readily discerned from Figure 1, the semihead lies in a type-${\beta }^{\prime }$ plane, the mia in a type-${\gamma }^{\prime }$ plane, a sail is hosted by the type-${\delta }_1^{\prime }$ plane, the Pasch configuration is located in a type-${\delta }_2^{\prime }$ plane, the triangle in a type-${\delta }_1$ plane, and the Fano plane itself coincides with a type-${\alpha }^{\prime }$ plane. It should also be pointed out that if one instead considers sub-geometries formed by ordinary lines, then both the Fano plane and its derivate mia are absent.

3. Conclusions

The Fano plane occurs in algebraic geometry and geometric algebra in a number of disguises [9,10], providing a link between such important mathematical concepts as design theory, error-correcting codes, Latin squares, skew-Hadamard matrices, Klein’s quartic curve, and Leech’s eight-dimensional minimal sphere-packing lattice, being perhaps most recognized as a “gadget” completely describing the algebra structure of the octonions [11]. The above-described properties of labeled Fano planes clearly demonstrate that there is still much that this prominent object of discrete mathematics is likely to teach us. Although some readers may find the adopted labeling of Fano planes rather arbitrary or unnatural, there is already a piece of solid motivation for it stemming from its successful use in discovering an interesting sequence of nested finite geometries associated with Cayley–Dickson algebras [5]. An even more intriguing question is whether a similar strategy could also be applied to get deeper insight into so-called q-analogs of the Fano plane, where there are still many open problems, even for $q=2,$ to be properly addressed (for a recent review, see [12]).

Obviously, any finite geometry (point–line incidence structure) with lines of size three can be looked at this way. We have already performed the corresponding examination of the Möbius–Kantor $\left(8_3\right)$-configuration, the Pappus $\left(9_3\right)$-configuration, as well as the Desargues $\left({10}_3\right)$-configuration, and plan to publish its outcome in a separate paper.