Annals of Discrete Matliematics 30 (1986) 203216 0 Elsevier Science Publishers B.V. (Nortlr.Hollatid)
PASCALIAN CONFIGURATIONS I N PROJECTIVE PLANES Giorgio Faina D i p a r t i m e n t o d i Matematica Universita d i Perugia 06100 P e r u g i a ITALIA
INTRODUCTION Following T i t s [19],
l e t P he a p r o j e c t i v e Oval of a p r o j e c t i v e
p l a n e TI and l e t P ( d ) be t h e f i g u r e formed by a i l o f t h e s e c a n t s o r t a n g e n t s t o 5: which a r e p a s c a l i a n l i n e s w i t h r e s p e c t t o R (see [4,p.
3701).
The f i g u r e s P ( 2 ) a r e c a l l e d ?  p a s C a l i a n
configurations
of 2. The p r o b l e m o f d e t e r m i n i n g t h e c o n f i g u r a t i o n P(G) was i n t r o d u c e d by F. E u e k e n h o u t i n [ 4 ] a n d , by u s i n g t h e s e c o n f i g u r a t i o n s , i t i s
p o s s i b l e t o produce an i n t e r e s t i n g c l a s s i f i c a t i o n f o r t h e p r o j e c t i v e o v a l s (see, a l s o , [ 7 ] ) . The p u r p o s e of t h i s p a p e r i s t c p r o v e t h e following r e s u l t :
I t seems e x t r e m e l y i n t e r e s t i n g t o m e n t i o n t h a t i n a G a l o i s p l a n e
h P G ( 2 , q ) , q = p , t h e p r o b l e m of d e t e r m i n i n g t h e nonempty R  p a s c a l i a n configurations i s completely resolved. I f o f S e g r e [18],
q
i s o d d , by t h e t h e o r e m
t h e o v a l s a r e c o n i c s , f o r which e a c h l i n e i s p a s c a l i a n
(see [4, p. 3 7 2 1 ) .
I f instead
q
i s e v e n t h e n t h e r e is, o t h e r t h a n
G. Fairia
204
the class of conics, another class of ovals (called L k a n . & U v n ow& [lo]) which have a single nonexterior pascalian line and such a line is a tangent (see [lo]). We also note that there are projective ovals having nonexterior pascalian lines in nondesarguesian planes, but in this setting the problem is very far from being resolved (see [4, p. 3801 , [21] 1 . At present two other Rpascalian configurations were found: a) If G is the Wagner's oval [21]
then the Rpascalian configuration
contains a unique line (see [7] : Buekenhout also discovered that this line is a tangent to S7 (see 1 4 1 ) b ) If R is the Tits ovoide 2i
.
&abt&ldon [19],
set of tangents to R (see [4, p. 3821 and
then P(Q) coincides with the
171).
So we see that in 2 ) and 3) of our Theoran we have exhibited
new types of
2pascalian configurations.
1. DEFINITIONS
PRELlMINARy RESllLTs
For definitions of the terms projectibe plane, aollineation, translation plare, desarguesian plane and nearfield see, for exanple, Segre [MI. A phojective u u d is defined (see [19]) as a set of pints R of a projective
plane I such that
M)
three are collinear and through each there passes one and
only ore line (the tangent) that contains IY) other pints of R.
is a 6ple (a a a b b b ) of p i n t s of R , m t recessarily 1' 2' 3' 1' 2' 3 distinct, such that : ai,bj # aj,bi, ai#aj and b #b for i#j and ai#bi i j (i,j=lr2,3), where a , h is a w e n t of R when a.=b Hexagons have three 11i j' distinct hiagorule p o i a t h a.,b,n (ifj) and are called pcrlc&aii when
A hexagon of
c)
_ I
1
_ I .
3
c,% J i
these three points are collinear. The €amus theorem of Euekenhout
143 m y
be
stated as: Id each i a c h i b e d lwxagot1 in a pm jcc..Citive v u d R 0 p a c d i a n then R in a t u n i c i n a p o jeotiwe pappian ptane.
Let R be a projective oval ii1 a projective plane TI. A line R of
R
205
Pusculiatr Corij~gurutiorzs111 Projective Planes
i s c a l l e d RpuncaeLun i f e a c h hexagon i n s c r i b e d i n R which h a s t w o d i a g o n a l p o i n t s on L a l s o h a s t h e t h i r d d i a g o n a l p o i n t on 1. The f i g u r e P ( Q ) formed by a l l o f t h e p a s c a l i a n l i n e s o f
II which a r e
s e c a n t o r t a n g e n t t o R i s c a l l e d R  p a d c a L h n c u n d i g u f i a t i o n o f II. Oval d o u b l e l o o p s
1.1.
I t h a s b e e n o b s e r v e d by many a u t h o r s ( [4]
, [ 6 ], [7] 1 ,
that a projecti
ve o v a l may be u s e d t o d e f i n e o p e r a t i o n s o f a d d i t i o n @ and m u l t i on i t s p o i n t s . W e w i l l d e s c r i b e a method f o r d o i n g t h i s
plication
which i s a s l i g h t m o d i f i c a t i o n o f B u e k e n h o u t ' s p r o c e d u r e [ 4 , p.
3731
w i l l c a l l e d a n o v a l dou6Le
The r e s u l t i n g a l g e b r a i c s y s t e m (Q ; @ , O ) T
Loop. Let
R be a p r o j e c t i v e oval i n a p r o j e c t i v e p l a n e ll. A r b i t r a r i l y
s e l e c t t h r e e p o i n t s on R and l a b e l them
ylo,i
p o i n t of i n t e r s e c t i o n of t h e t a n g e n t s a t p o i n t s of R, o t h e r than
y
and t h e n l a b e l t h e
o
and
i s a s s i g n e d a and
o
is assigned 1 .
i
I f a and b a r e t h e symbols a s s i g n e d t o t h e p o i n t s
if 0;
in
},
a#o if
x . The
y , a r e t h e n a r b i t r a r i l y a s s i g n e d symbols
with the r e s t r i c t i o n t h a t
Q\&
with
a
and
b
of
we d e f i n e t h e sum
[email protected] i n t h e f o l l o w i n g way:

then the l i n e a=o
then l e t
a point
z
w i l l m e e t R i n a point
a,x
a'=o. If
o t h e r than
bfo
x; i f
p o i n t o f i n t e r s e c t i o n of t h e l i n e
a ' ,z n R } \ { a ? =pl
then
c=a'
then t h e l i n e b=o
then l e t
a',z
o,b
other than
meets
z=x. L e t
x,y
C be t h e
w i t h QN{a'j; i f
~
. Letting
a'

c be t h e symbol a s s i g n e d t o
t h i s p o i n t we d e f i n e u616=c. If
a r b E R \ { o l y ] , w e d e f i n e t h e p r o d u c t a o b as f o l l o w s :
the tangent a t then the l i n e
i
b,j
__
meets t h e s e c a n t
o,y
i n a point
w i l l m e e t R i n another point
j; i f
b'#i;
if
a,i
b#i b=i
then l e t b ' = i . L e t t h e i n t e r s e c t i o n of t h e l i n e with t h e l i n e o , y be h. L e t c be t h e p o i n t o f i n t e r s e c t i o n of t h e l i n e

h,b'

with t h e s e t R\{b'] ; i f { h,b'nR}
N
jb'} =$
then
c=b'. Letting
c be t h e symbol a s s i g n e d t o t h i s p o i n t w e d e f i n e a 0 6 = c . I f w e c a l l t h e s e t of symbols u s e d
Q , then i t i s e a s i l y seen t h a t
.
206
0
G.Fuiria together with the operations defined above is a double loop which
is denoted by where
which is also called an o v a l double l o o p ,
(Q,;@,')
{o,y,i}. This leads us to
S:=
LEMMA 1.[6] The loop (Q
S

the line
,@)
is an abelian group if, and only if, i
is a Rpascalian line; the loop (Q,,')
x,y
group if, and only if, the line
+ Qs:=Qx{o}.
cy
is a Rpascalian line, where
It is not difficult to verify that every point fied with a involutorial permutation
is an abelian
I(p)
p€II\R can be identi
of the points of R as
follows: if
p ~ u \ R ,two points of R are a pair in the involutorial permutation if they lie on the same line through
I(p)
LEMMA 2 . [ 4 ] 
p.
For a line l of a projective plane containing a pro
jective oval R , the following are equivalent: 1)
.i? is
Rpascalian, and
2) for each triple of involutions I(p) , I ( q ) , I (r) with centers
on l, the composition
I(p)I(q)I(r)
is also an involution
with center on 1. An a u t o m o 4 p h i n m of a projective oval R is a permutation $ of the points of R which preserves the involutorial permutations I(p), where
v
p~ll\R,that is to say:
PErI\R,
3!
q&n\R
: $ I(p)lp=I(q).
The automorphisms of R form a group. Denote this group by
AutR.The
following is easily proven: each c u L l k n e a t i o n mirkph44tn 0 6
06
ll t h a t pekmuteh R i n t o i t b e t 6 i n d u c e s an a u t o 
0.
We also have the following result. LEMMA 3.[4]
Let 52 be a projective oval of a projective plane Il and
let a be a collineation of TI that permutes R into itself. The line
207
Pascaliaii Coiifigirrarioiisiir Projwtive PIaiies _.
x,y
,
where
x,y~:R,is a Rpascalian line if, and only if, the line
(x),w.(y) is a (1pascalian line.
1.2. The nearfield of order nine (Andrb [l]) Let
2
x =1
an irriducible quadratic over GF(3). Let
of all elements of the form where we assume
on K
a+bi
as
a
and
b
K
be the set
vary over GF(3),
2
i =1. We wish to define an addition and a product
in such a way that, using the field GF(3) addition, K will
be a nearfield. We define the addition as follows (a+bi)+ (c+di):= (a+c)+ (b+d)i f for all
a,b,c,deGF (3).
We define the product in the following way: ai=ia, for all
a€GF(3)
a(btc)=ab+ac, for all ab+ba=O, for all
a,b,c~K
a,bEK\GF(3), where
afb
and
a+b#O.
It is evident that (K,+) is an abelian group and that K\{O} is a group. G’ven
(a+b#O), there is a unique
a,b,cEK
XEK
such that
ax+bx+c=O. 2
Finally, a =1
for all
acK\GF(3).
1.3. The nondesarguesian translation plane of order 9 (Andr6 [l] From the Rearfield of order 9 K
)
we may now construct a transla0
tion affine plane of order nine, denoted by T
,
as follows (see,
for example, [1] :

points are the pairs (x,y) for all
x,y~K;
lines are defined as sets of points (x,y) whose coordinates x,y satisfy an equation of one of the forms (aEK),
(1) x=a
(2)
y=ax+b
(a,bcK).
There is, up to isomorphism, a unique projective plane that
0
T =T\{d}
infinity of same
TO.
To
for a line d of
T
such
T , where d is called the line at
and its points are called points at infinity of the
208
If
C.Faitia p
is the point at infinity of
y=ax+b
is the point at infinity of
by (a). If
p
denoted by
(m).
then it will be denoted x=a
then it will be
It has been shown by Denniston [5] and Nizette [14] that in the translation nondesarguesian plane of order nine dual
T
(and in its
T I ) the ovals fall into a single transitivity class under the
collineation group. The selfduality property make it unnecessary to study
and T'
T
jective oval in
Rodriguez )6] the group oval of
T
separately; so the following example of pro
T will suffices:
discovered the oval
AutR
of
32
collineations that leaves invariant an
and proved that AutR
I J
x' =ixiy y' =ix+iy
and Nizette [14] has studied
R
have generators
x ' =x
x'=x
Y"Y
with
io=ir a € Aut K.
1.4.
The Hughes plane of order nine (Zappa [22])
From the nearfield of order plane, denoted by
H
9 K
we may now construct a projective
as follows:
 the points of H are the triplets
( x ,x ,x 1 , where
x.EK, other 1 2 3 than ( O , O , O ) with the identification (x1,x2,x3)=(kxl,kx rkx3) for 2 all nonzero k in K;
 the lines of H will now be the sets of points satisfy an equation of the form any automorphism of
K
(x,y,z) which
x+yt+z=O, teK, such that if cf is
then the mapping
x'=a xa+b yo+c zu 1 1 1
z'=a xO+b yO+c z(J 3 3 3 with
(VaitbircicGF3), i=1,2,3)
det(a,,b,,c,)#O, is a collineation of
H.
209
Pascaliun Configurations in Projective Planes
Denniston [ 5 ] and Nizette [14] have discovered that in the Hughes plane
H, ovals fall into two transitivity classes under the colli
neation group of 48
riant under
H. An oval,
D , in one of these classes is inva
collineations, as against
16
collineations for
the other class. So the following examples of nonisomorphic projective ovals of
H
will suffice:
N={(l,i,O), (1,il,O),(l,l,i+l),(l,l,il), (O,i,l),(O,i,l), (l,~,il),(l,~,i+l), (l,l,i),(l,l,i)l. In [14], Nizette proved also that
AutN have generators
a
a
x'=xy
x =x
x'=x +y
y 1 =y
y'=x+y
a a y'=x  y
Z'=Z
Z'=Z
1
0
(with i'=il).
2. PASCALIAN CONFIGURATIONS JJ Let
R
T
AND IN
T'
be the Rodriguez oval of the nondesarguesian translation
plane of order nine T. First we show that each nonexterior line (to R ) through the point ( 0 , O ) is a Rpascalian line. Let the tangent
y=ix
(O,l)=i, (0,i)=i,
(i,i)=2i+l, (i,i)=2+i, (i,i)=2+2i, (i,i)=l+i Q
denote
and label the points of R in the following way:
(i)=O, (i)=, (l,O)=l, ( 1 , 0 ) = 2 ,
Letting
.t
be the symbol assigned to the set Rx{(i)l
coinciding with the set of elements of the nearfield the following triplet of points on R :
.
(i.e. Q K),
is
we select
G. Fabra
310
By
1.1, the algebraic system (QS;@,O) is an oval double loop. Now, with a straightforward proof which we omit for shortness, we can to check that
[email protected]=a+b, for all
a,bEQ (1.e. for all
a,bEK).
Since (K,+) is an abelian group, we have that (QS ,@)
is an abelian
group. Therefore, by Lemma 1, we have that the tangent at
(i)=
is a Rpascalian line. Also, s h c e AutR {
is a transitive permutation group on the set
,
(i), (i)I C R (see [ 4 ] )
is
by Lemma 3 , we have that the tangent y=ix
Rpascalian.
NOW, in order to show that the secant
x=O
it is only necessary to prove that, for + o
loop ( Q s ,
)
, where
+ QS=R\{ (0,l), (0,1)1 ,
is a
S=I (0,l), (O,l), (i)1 , the is an abelian group. A very
long, but straightforward computation, shows it. AutR
that (see [ 1 4 ] )
fixes the point
tive on the points of R*{(i),(i)}. all nonexterior lines through
Rpascalian line,
It is well known
( 0 , O ) and that it is transi
Thus, from Lemma 3, we have that
( 0 , O ) are Rpascalian.
Now we will prove the nonexistence of nonexterior Rpascalian (0,O).The points of R may, for
lines not passing through the point shortness, be denoted by digits from
0
to
9
as follows:
!i)=o, (i)=1, (1,0)=2, (1,O)= 3 , (i,i)=4 ,(i ,i)=5 ,(1,i)=6 ,(i,I)=7 , (0,1)=8 and (0,1)=9. By [4, p. 3831 and [20, table 32/34],
it follows that, if we denote
by G(8) the group of all elements in AutR
then
IG(8)
1=4
and that
G(8)={f
f
f
which fix the point
8,
f 1 , where
1’ 2 ’ 3 ’ 4
Finally, since AutR acts transitively on R\{ (i), (i)1 , by Lemma 3, the only thing remaining to be shown is that the lines
    8r8
I
5,8
r
O,8
x e not Rpascalian.
i
218
r
0,l
21 I
Pascalinn Configurations in Projective Planes
First of all, consider the following points of ll\R: __
~


~

__
p =1,9r18,8, p =1,6n8,8, p =5,8nO,O, p =5,8nOI2, p =0,8fll,l, 1 2 3 4 5

~~
____
p =0,8ni,2, p =2,8no,o, p =2,8noI3,p =0,1n2,2, p =0,1n2,4, 6 7 8 9 10 
pl1=0,1fl2,6. Now, without giving the proofs (which are straightforward but timeconsuming) we remark that

p1,p2,p3~8,8 but
I(p1)I(p2)I(p3) 
not a involutory permutation of R with center on Lemma 2, it follows that the line
__
Repeating this process, replacing gives that the line
~
5'8
Similarly: I (p5)I (P,) I (P,)
8,8
is
8,8; thus, by
is not Rpascalian.
I (p,) I (p,) I (p,)
by I (p,) I (p4)I(p,)
,
I (P,) I (P,) I (p7)I I (P,) I (pl0)I (pll) are
not involutory permutations of R with center in

~
0,8, 2'8, 0,1,
respectively. Hence these lines are not Rpascalian. The R'pascalian configuration of the dual
T'
of
T
is again of
the same type and we omit the analogous proof.
3. PASCALIAN CONFIGURATIONS
IN
H
In [8], Hughes reproduces the plane
H
in the useful following way:
 the points are the symbols A i , B i ,C.,D ,E.,FifGifi=O,1,..., 12; i i i seven of the lines are the following sets of points 1) IAO,A1 'A3 I Ag,Bo tCo I Do I Eo I Fo 'Go 1 2) IAotB1rB8tD3 'Dll
,
is not Rpascalian.
tE2 tE5 ,E6,G7,Gg1

G.
217
G){A
0'
C
7'
C,D 9
D
2'
FUiflO
D E E F,F) 5' 6' 3' 11' 1 8
7) {AOiB3rB11 r C 2 rC5rC6 r D7 i D g rG1 rG8);
 the remaining lines are found by successively adding one to the subscripts, reducing modulo 13. In this notation, we remark that (see [5] and [ 1 4 ] )
the ovals D und
N of section 1.4 are the following sets of symbols:
B0rE0rC 6 r D 6 r C 7 rD 7 I
P'{A4'A5'All'A12'
N={B
0'
c0' c 4 ,G4' c 6 ' D 6' B 7' F 7' B ll'E1l'
'
We first show that the Dpascalian configuration is the empty set. The suggestive term JLeaL is used for the points A
of H I then in D i there are four real points and six i m a g i n a h y points. It is well
known that (see [5]
, [14])
IAutPj=48 and we note further important
properties: 1) Auto is generated by: (A11A12)(BoEo)( C 6 C 7 ) ( D 6 D 7 )
2) AutD is transitive on the set of real points of 0; 3 ) AutD is transitive on the set of imaginary points of
D;
4)
I (AutD)xl=12
5)
(AutD)x is transitive on the set of imaginary points of D for
for all real point
XED;
all real point xrD; 6 ) if
is a real point of D , then
x
set of real points of 7 ) if
y
(AutD)x is transitive on the
D.{xl;
is a imaginary point of D then
8) (AutD) =(AutD) : BO EO 9) (AutD) BO
is transitive on the set {C
I (Autp)
Y
1=8;
D C D I. 6' 6 ' 7' 7
We omit the proof which is very long, but not difficult. By the above properties of AutP and Lemma 3 , the only thing remaining
213
Pascalian Configurations in Projective Planes
to be shown is that no one of the lines A4rA4r A12iA12
   I
A4iA5
r
BOrEO
I
BOrC6
I
A4rA12
is Dpascalian. As in the proof of section 2, it is sufficient to exhibit some appropriate involutorial permutation of the points of
D
. First of
all, consider the following points of i l \ P
:
Now we remark that the following permutations
are not involutory permutations of type in

I(p) with the centers p
~

A4,A4, A12,A12, A4,A5, A4,A12, B O I E O ~BO'C6

respective1.y. Hence, by Lemma 2, these lines are not Dpascalian. Finally, we must show that the Npascalian configuration is the empty set too. Also in this case, it is well known that (see [5] and [14]) 1AutNI=16 and it is not difficult to check that:
1) AutN is generated by
X=(C C F E G D B B
4 6 7 1 1 4 6 7 11
and
2) AutN fixes the set {Bo,Co}; 3 ) AutN is transitive on the sets {B
0'
C
0
1
and I=N\$ 0 ,C0 1 respecti
vely; 4) (AutN) is transitive on 1; BO 5 ) (AutN) =(AutN) : BO cO =(AutN) =(Id,ul G4
.
Hence, in order to prove that P(N)=gf, it is sufficient to show that no one of the following lines is Npascalian:
G. Faina
214
A repetitionofthe arguments used in the earlier proof of this CtiOn
98
shows that the permutations
are not involutory permutations of the points of N
with center in
the above mentioned lines, respectively, wile we have that:

C41G4, E 8' E 0EC 4 ' C 6' C 5' G0EC 4' D 6' E 2' A 1EC 4IB7' Hence, by Lemma 2, no one of these lines is a Npascalian line.
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