Pascalian Configurations in Projective Planes

Pascalian Configurations in Projective Planes

Annals of Discrete Matliematics 30 (1986) 203-216 0 Elsevier Science Publishers B.V. (Nortlr.Hollatid) PASCALIAN CONFIGURATIONS I N PROJECTIVE PLANES...

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Annals of Discrete Matliematics 30 (1986) 203-216 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 non-empty 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 non-exterior pascalian line and such a line is a tangent (see [lo]). We also note that there are projective ovals having non-exterior pascalian lines in non-desarguesian planes, but in this setting the problem is very far from being resolved (see [4, p. 3801 , [21] 1 . At present two other R-pascalian configurations were found: a) If G is the Wagner's oval [21]

then the R-pascalian 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

2-pascalian configurations.

1. DEFINITIONS

PRELlMINARy RESllLTs

For definitions of the terms projectibe plane, aollineation, translation plare, desarguesian plane and near-field 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 6-ple (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 R-puncae-Lun 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 R-pascalian line; the loop (Q,,')

x,y

group if, and only if, the line

+ Qs:=Qx{o}.

cy

is a R-pascalian 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

R-pascalian, 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 R-pascalian line if, and only if, the line

(x),w.(y) is a (1-pascalian line.

1.2. The near-field 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 near-field. 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 non-desarguesian translation plane of order 9 (Andr6 [l] From the Rear-field 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 non-desarguesian plane of order nine dual

T

(and in its

T I ) the ovals fall into a single transitivity class under the

collineation group. The self-duality 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' =ix-iy 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 near-field 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 non-zero 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 non-isomorphic projective ovals of

H

will suffice:

N={(l,i,O), (1,-i-l,O),(l,-l,i+l),(l,-l,-i-l), (O,i,l),(O,i,-l), (l,~,-i-l),(l,~,i+l), (l,l,i),(l,l,-i)l. In [14], Nizette proved also that

AutN have generators

a

a

x'=x-y

x =-x

x'=x +y

y 1 =-y

y'=x+y

a a y'=x - y

Z'=Z

Z'=Z

1

0

(with i'=-i-l).

2. PASCALIAN CONFIGURATIONS JJ Let

R

T

AND IN

T'

be the Rodriguez oval of the non-desarguesian translation

plane of order nine T. First we show that each non-exterior line (to R ) through the point ( 0 , O ) is a R-pascalian 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 near-field 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 R-pascalian 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

R-pascalian.

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 non-exterior lines through

R-pascalian line,

It is well known

( 0 , O ) and that it is transi-

Thus, from Lemma 3, we have that

( 0 , O ) are R-pascalian.

Now we will prove the non-existence of non-exterior R-pascalian (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 R-pascalian.

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 R-pascalian.

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 R-pascalian. 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 R-pascalian.

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 sub-scripts, 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 D-pascalian 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 D-pascalian. 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 D-pascalian. Finally, we must show that the N-pascalian 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 N-pascalian:

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 N-pascalian line.

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2. A. BARLOTTI , Un'oodehvatione i & V h n O Le Matematiche 21 (1966) 23-29. 3. U. BARTOCCI, Condidekazioni d d h Universita di Rcana (1967). 4.

F. BUEKENHOUT, 333-393.

5. R.H.F.

ad un ,teahema di B. Sqhe &O&a

d&e

ova&

6u.i

q-mcki,

Tesi di Laurea,

Etude imkin6Eque d u o v a t u , Renl. Mat. (5) 25 (19661,

DENNISTON, On

Math. 4 (19711, 61-89.

~MCAi n

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