Delft University of Technology
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Onderafdeling der Wiskunde
en Informatica
Reports of the
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Onderafdeling der Wiskunde en Informatica Julianalaan 132, 2628 B L D e l f t
Postbus 3 5 6 , 2 6 0 0 AJ D e l f t
Department of Mathematics and Informatics Julianalaan 132, 2628 BL D e l f t
R E P O R T &k-k2 A n a u t o m a t i c p r o o f p r o c e d u r e f o r s e v e r a l g e o m e t r i e s ( r e p o r t 1) by T h . B r u y n and H . L . Claasen R e p o r t s of t h e D e p a r t m e n t of M a t h e m a t i c s and I n f o r m a t i c s n o . S^-'/Z D e l f t 198i|
T h i s r e p o r t i s t h e f i r s t o f a s e r i e s on t h e s u b j e c t I n d i c a t e d i n t h e t i t l e . We w i l l t r y t o d e v e l o p i n c o u r s e o f t i m e p r o c e d u r e s f o r p r o o f s o f t h e o r e m s i n s e v e r a l g e o m e t r i e s w h i c h c a n be e x e c u t e d i n a c o m p u t e r . U n d e r t h e t e r m " g e o m e t r y " we c e r t a i n l y u n d e r s t a n d any p r o j e c t i v e g e o m e t r y as d e f i n e d i n s e c t i o n 6, b u t a l s o we hope t o c o n s i d e r ( i n l a t e r r e p o r t s ) a f f i n e g e o m e t r i e s and e v e n p a r t s o f t h e E u c l i d i a n g e o m e t r y . E s p e c i a l l y t h e s e t t i n g o f t h e l a t t e r g o a l i s f o r t h e t i m e b e i n g o n l y a " d e c l a r a t i o n o f i n t e n t i o n " more t h a n a f i r m p r o g r a m t h e l i n e o f w h i c h i s v e r y c l e a r t o u s . I n 1960 C a r t o n p u b l i s h e d h i s p a p e r 2 . H e r e a m e t h o d seemed t o be o u t l i n e d t o u s e a c o m p u t e r t o p e r f o r m p r o o f s and c o n s t r u c t i o n s w i t h i n t h e E u c l i d i c a n g e o m e t r y . H o w e v e r , t h e a r t i c l e a p p e a r s t o be o v e r l y o p t i m i s t i c i n s e v e r a l a s p e c t s o f t h e d e s c r i b e d t h e o r y .
E.g. C a r t o n c o n s t r u c t s c o n f i g u r a t i o n s based on t h e t h e o r e m o f Pappus w h i c h a r e n o t r e a l i s a b l e i n t h e E u c l i d i a n g e o m e t r y . P e r h a p s t h e s e c o n f i g u r a t i o n s c o r r e s p o n d t o t r u e t h e o r e m s i n c e r t a i n P a p p l a n g e o m e t r i e s , b u t t h i s r e q u i r e s m o r e r e s e a r c h , t o s a y t h e l e a s t .
A l s o i t i s n o t c l e a r how good h i s p r o p o s e d p r o o f m e t h o d s a r e when we l o o k i n t o t h e c o m p l e x i t y o f h i s a l g o r i t h m s ( I f he p r o p o s e s a n y , t h a t i s ) . On t h e o t h e r hand C a r t o n ' s a r t i c l e h a s been f o r us a b e g i n n i n g o f a l o n g r o a d h o p e f u l l y l e a d i n g t o u s e f u l p r o o f p r o c e d u r e s i n w h i c h t h e c o m p u t e r p l a y s a c e n t r a l r o l e . I n t h i s f i r s t r e p o r t we s h a l l c o n s i d e r t h e p r o j e c t i v e g e o m e t r y o v e r t h e r e a l s PG(2,R) and g i v e some o u t l o o k t o " n e i g h b o u r i n g " g e o m e t r i e s ( t h e a f f i n e g e o m e t r y o v e r t h e r e a l s , f o r e x a m p l e ) .
t h a t t h i s f i r s t r e p o r t o n l y g i v e s t h e r e a d e r some i k l i n g o f o u r d e d u c t i o n m e t h o d s . T h e r e i s s t i l l a l o n g way t o go a f t e r t h i s r e p o r t and many t h i n g s e v e n l i e b e y o n d o u r h o r i z o n a t t h i s moment.
I n t h e s e c t i o n s 2,3 and 4 we l a y t h e m a t h e m a t i c a l f o u n d a t i o n f o r t h i s and s u b s e q u e n t r e p o r t s i n t r o d u c i n g a " t o r r e n t " o f d e f i n i t i o n s . I t m u s t be p o i n t e d o u t ( a s C a r t o n d i d ) t h a t t h e r u l e o f d e n y i n g t h e c o n s e q u e n t seems d i f f i c u l t t o t r a n s l a t e i n t o c o m p u t e r l a n g u a g e . So we s e t o u t t o s t a t e t h e d e f i n i t i o n s , t h e a x i o m s and t h e t h e o r e m s i n t h e a f f i r m a t i v e . I n t h e f i r s t s e c t i o n s as w e l l as i n s e c t i o n 5 we s u p p r e s s t h e g e o m e t r i c a l m e a n i n g o f t h e t h e o r y w h i c h i s d e v e l o p e d , a l t h o u g h , o f c o u r s e , i t i s t h e k e y t o t h e why o f t h e s e s e c t i o n s as i s shown i n s e c t i o n 6 w h e r e we f i t a p a r t o f t h e p u z z l e t o g e t h e r a p p l y i n g t h e t h e o r y t o P G ( 2 , R ) . I n s e c t i o n 7 we show t h a t i n t h i s r e p o r t we a r e d e a l i n g w i t h a w e l l - d e f i n e d p a r t o f P G ( 2 , R ) ; v i z . t h a t p a r t o f PG(2,R) i n w h i c h t h e c o n s t r u c t i o n c a n be p e r f o r m e d by a r u l e r a l o n e . We a l s o g i v e some o u t l o o k t o r u l e r and compass c o n s t r u c t i o n s .
Our m e t h o d d e p e n d s h e a v i l y on m a n i p u l a t i o n s w i t h c e r t a i n g r a p h s c o n n e c t e d w i t h t h e o r e m s i n a g e o m e t r y . T h e r e f o r e , i n s e c t i o n 8, we p u t t h e g r a p h t h e o r e t i c a l a s p e c t s o f t h e m e t h o d i n t o p e r s p e c t i v e . I n s e c t i o n 9 we d i s c u s s a p a r t o f o u r d e d u c t i o n m e t h o d and g i v e some e x a m p l e s . As a n i m p o r t a n t r e s u l t we p r o v e d t h e t h e o r e m o f D e s a r g u e s f r o m t h e t h e o r e m o f P a p p u s . Our p r o o f p r o c e d u r e i s i n e s s e n c e a t h e o r e m g e n e r a t i n g o n e . To p u t i t i n t h e n e g a t i v e : f o r t h e moment we a r e n o t a b l e t o p u t t o a t e s t a g i v e n ( u n p r o v e d ) h y p o t h e s i s b u t a r e " o n l y " a b l e t o " g e n e r a t e " new t h e o r e m s f r o m g i v e n t h e o r e m s , " N i c e " r e s u l t s o n l y come by c h a n c e . T h a t c h a n c e , h o w e v e r , c a n be s t e e r e d by t h e b r u t e f o r c e o f t h e c o m p u t e r and i n t e l l i g e n t l y d e v i s e d a l g o r i t h m s . l i n e , f o r e x a m p l e ) by a d i r e c t e d e x t e n s i v e s e a r c h o f t h e c o m p u t e r i n t o a l l p o s s i b i l i t i e s w h i c h a r e o p e n a t a c e r t a i n moment. Because o u r m e t h o d s a r e n o t c o m p l e t e l y d i s c u s s e d i n t h i s r e p o r t i t seemed b e t t e r t o r e f r a i n f r o m g i v i n g c o m p u t e r p r o g r a m s h e r e . S e c t i o n 11 o u t l i n e s s e v e r a l o u t l o o k s f o r f u r t h e r d e v e l o p m e n t s . Most o f t h e t h e o r e m s p r o v e d i n t h i s r e p o r t a r e i n f a c t m e t a - t h e o r e m s ; t h e o r e m s on PG(2,R) m o s t l y . O n l y i n s e c t i o n 9 we p r o v e some t h e o r e m s ( i n f a c t S ( 2 ) , S ( 3 ) and S ( 4 ) ) w h i c h a r e t h e o r e m s o f PG(2,R) i t s e l f . I n t h i s r e p o r t N d e n o t e s t h e s e t o f t h e n a t u r a l n u m b e r s , R and C t h e f i e l d o f t h e r e a l and t h e c o m p l e x n u m b e r s , r e s p e c t i v l l y .
2 . SOME DEFINITIONS
2 . 1 . D e f i n i t i o n : L e t P and L be two d i s j o i n t s e t s and l e t P x L be t h e
C a r t h e s i a n p r o d u c t of P and L . L e t I c P x L . The t r i p l e ( P , L , I ) i s c a l l e d an i n c i d e n c e s t r u c t u r e . I f both P and L a r e empty, then t h e i n c i d e n c e s t r u c t u r e i s c a l l e d empty. I f n e P and m 6 L and (n,m) € I t h e n n and m a r e c a l l e d
i n c i d e n t . An i n c i d e n c e s t r u c t u r e B = ( P * , L * , I * ) i s c a l l e d a s u b s t r u c t u r e of A i f P* c P, L * c L and I * = ( P * X L * )
n
I . 2.2. D e f i n i t i o n : L e t A = ( P , L , I ) be an i n c i d e n c e s t r u c t u r e . I t i s s a i d t h a t A i s a g e o m e t r i c a l i n c i d e n c e s t r u c t u r e ( a b b r e v i a t e d t o G I S ) i f 1 ) two d i f f e r e n t e l e m e n t s of P a r e i n c i d e n t w i t h a t most one element o f L; 2) two d i f f e r e n t e l e m e n t s of L a r e i n c i d e n t w i t h a t most one element of P. I s A empty t h e n , by d e f i n i t i o n , A i s a G I S . 2.3. As p o i n t e d out i n t h e i n t r o d u c t i o n i t seems d i f f i c u l t t o u s e t h e " r u l e of d e n y i n g t h e c o n s e q u e n t " i n our d e v i s e d a u t o m a t i c p r o o f p r o c e d u r e . I f one c o u l d u s e t h i s r u l e t h e n , o f c o u r s e , one of t h e two c o n d i t i o n s i n d e f i n i t i o n ( 2 . 2 ) i s s u p e r f l u o u s .2 . 4 . I n t h i s r e p o r t we s h a l l o n l y c o n s i d e r g e o m e t r i c a l i n c i d e n c e s t r u c t u r e s .
L e t A = ( P , L , I ) be a G I S . The e l e m e n t s of P a r e c a l l e d p o i n t s , t h e e l e m e n t s o f L a r e c a l l e d l i n e s and the e l e m e n t s o f I a r e c a l l e d i n c i d e n c e s .
I s n € P and m € L and (n,m) € I then i t i s s a i d t h a t n(m) i s i n c i d e n t w i t h m ( n ) . Moreover i t i s s a i d t h a t m ( n ) b e l o n g s t o t h e i n c i d e n c e
I f n^ e P, n2 € P ( n ^ ^ n 2 ) and m € L such t h a t (n^,m) € I and (n2,m) € I than m I s c a l l e d a l i n e c o n n e c t i n g n^ and n2.
I f n € P and m^ € L and m2 € L (m^ 5^ m2) such t h a t (n,m^) 6 I
and (n,m2) e I then i t i s s a i d t h a t n i s t h e p o i n t o f i n t e r s e c t i o n o f
m^^ and m2.
2 . 5 . I n most p a r t s of t h i s r e p o r t t h e I n c i d e n c e s of a G I S p l a y an I m p o r t a n t
r o l e . I n t h e s e I n s t a n c e s i t i s e a s y t o have a common name f o r t h e p o i n t s and t h e l i n e s o f t h e G I S c o n s i d e r e d .
I f A = ( P , L , I ) then K = P u L and t h e e l e m e n t s o f K a r e c a l l e d nodes.
I f n € P and m € L t h e n t h e f a c t t h a t n and m a r e i n c i d e n t i s , by
d e f i n i t i o n , denoted by (n,m) € I . Here (n,m) i s an o r d e r e d p a i r . O f t e n , however, we s h a l l denoted t h e same I n c i d e n c e by t h e u n o r d e r e d p a i r
n,m] = [m,n]. We s t i l l s a y t h a t [n,m] e I . 2 . 6 . L e t A be a G I S . P o i n t s o f A i n c i d e n t w i t h t h e same l i n e a r e c a l l e d c o l l i n e a r ; l i n e s of A i n c i d e n t w i t h t h e same p o i n t a r e c a l l e d c o n c u r r e n t . A number of p o i n t s ( l i n e s ) o f A a r e s a i d t o be l y i n g f r e e i f no t h r e e of t h e p o i n t s ( l i n e s ) a r e c o l l i n e a r ( c o n c u r r e n t ) . 2 . 7 . L e t F = ( P Q , L Q , I Q ) be a G I S . I t i s s a i d t h a t F i s a f i g u r e i f PQ and L Q a r e f i n i t e s e t s . A s u b f i g u r e o f a G I S A = ( P , L , I ) i s a f i g u r e w h i c h i s a s u b s t r u c t u r e of A. A s u b f i g u r e of a G I S A i s sometimes c a l l e d a f i g u r e I n A.
2.8. L e t F be a f i g u r e and l e t n be a node of F . The number v ( F , n ) = I { m € K | [n,m ] € l } | i s c a l l e d t h e v a l e n c y of n ( i n F ) . A f i g u r e F i s s a i d t o be c l o s e d i f e v e r y node of F h a s a v a l e n c y i n F of at l e a s t 3. 2.9. D e f i n i t i o n : L e t F be a f i g u r e , l e t n be a node of F and l e t [n,m] be an i n c i d e n c e of F . I t i s s a i d t h a t i ) n i s s i n g u l a r ( i n F ) i f v ( F , n ) > 3; i i ) n i s normal ( i n F ) i f v ( F , n ) = 3; i i i ) n i s abnormal ( i n F ) i f v ( F , n ) < 3; i v ) n,m i s s i n g u l a r ( i n F ) i f both nodes a r e s i n g u l a r i n F;
v ) [n,m] i s mixed ( i n F ) i f one of the nodes i s s i n g u l a r and
t h e o t h e r o n e i s normal i n F;
v i ) [n,m] i s normal ( i n F ) i f both nodes a r e normal i n F; v i i ) [n,m] i s abnormal ( i n F ) i f a t l e a s t one of the nodes
i s abnormal i n F. 2.10. D e f i n i t i o n : A c o n f i g u r a t i o n ( p ^ . q ^ ) i s a c l o s e d f i g u r e F = ( P , L , I ) s u c h t h a t |P| = p and | L | = q and e v e r y l i n e of F i s i n c i d e n t w i t h r p o i n t s of F and e v e r y p o i n t of F i s i n c i d e n t w i t h s l i n e s of F . 2.11. D e f i n i t i o n ; L e t A = ( P , L , I ) and B = ( P Q , L P , I Q ) be two i n c i d e n c e s t r u c t u r e s . A and B a r e s a i d t o be i s o m o r p h i c (A = B ) i f t h e r e a r e b i j e c t i v e mappings of P on PQ and of L on L g s u c h t h a t i f n € P i s mapped on np € PQ and m € L I s mapped on mQ e L Q t h e n (n,m) 6 I i f and o n l y i f (nQ.mg) € I Q .
3. THE COMPONENTS OF A FIGURE
3.1. I n s e c t i o n 3 F i s a l w a y s f i g u r e . F = ( P , L , I ) , |P| = p and | L | = q; pq 5^ 0.
3'2. D e f i n i t i o n : A ( q x p ) m a t r i x M i s c a l l l e d an i n c i d e n c e m a t r i x of F i f t h e columns of M a r e l a b e l e d by t h e p o i n t s of F and the rows of M a r e l a b e l e d by the l i n e s of F, such t h a t the e n t r y (m,n)
of M i s 1 i f (n,m) € I and o t h e r w i s e i t i s 0. 3.3. B e c a u s e F i s a GIS t h e r e a r e i n an i n c i d e n c e m a t r i x of F no r e c t a n g u l a r p a t t e r n s of A o n e s . 3.4. N o t a t i o n s : L e t x,y € N \ {O} t h e n € i s t h e a l l - o n e T v e c t o r of l e n g t h x; t h a t i s i = ( 1 , 1 , . . . 1 ) . F u r t h e r m o r e O^^y i s t h e x by y a l l z e r o m a t r i x . I n R'^ ( , ) i s t h e o r d i n a r y i n n e r p r o d u c t of v e c t o r s of R^. 3.5. I n the n o t a t i o n of ( 3 . 2 ) .
Remark t h a t t h e element i n M^j^ w i t h l a b e l n € L has t h e v a l u e v ( F , n ) and t h a t the element i n M'^j^ w i t h l a b e l m € P has the v a l u e v ( F , m ) .
L e s s f o r m a l ; t h e sum of a row ( c o l u m n ) of M i s the v a l a n c y of t h e c o r r e s p o n d i n g node of F .
C l e a r l y ( M j p , j ^ ) = ( M ^ j ^ . j ^ ) = | l | .
3.6. D e f i n i t i o n : The L e v i g r a p h A ( F ) i s the g r a p h ( K , I )
T h a t i s : The node s e t of A ( F ) a r e the nodes of F and the edge s e t of A ( F ) a r e the i n c i d e n c e s of F: two nodes of A ( F ) a r e c o n n e c t e d i f and o n l y i f t h e two nodes i n F a r e i n c i d e n t .
The edges of A ( F ) a r e c a l l e d i n c i d e n c e s and we use t h e same
c h a r a c t e r i s a t i o n s f o r the nodes and the I n c i d e n c e s of A ( F ) a s we d i d f o r the nodes and t h e i n c i d e n c e s of F i n ( 2 . 9 ) .
3.7. The L e v i graph has been i n t r o d u c e d by L e v i [ 6 ] . I t e n a b l e s the u s e r to s t u d y , f o r example, t h e automorphism of a g i v e n f i g u r e . E.g. see C o x e t e r [ 4 ] .
I n our method a n o t h e r graph i s of more i m p o r t a n c e : the d e r i v e d graph ( s e e 3 . 1 2 ) .
3.8. Lemma: A ( F ) i s a graph on (p+q) n o d e s . E v e r y c i r c u i t of A ( F ) has an even number of edges and no c i r c u i t of A ( F ) has a l e n g t h l e s s t h a n 6 ( t h e g i r t h of A ( F ) i s a t l e a s t 6 ) .
F i s c l o s e d i f and o n l y i f e v e r y node i n A ( F ) has a v a l e n c y of a t l e a s t 3.
P r o o f : The f i r s t a s s e r t i o n i s e v i d e n t .
I f two nodes of A ( F ) a r e c o n n e c t e d t h e n one node
c o r r e s p o n d s to a l i n e and one node c o r r e s p o n d s t o a p o i n t of F . T h i s , of c o u r s e , i m p l i e s t h a t e v e r y c i r c u i t i n A ( F ) has even l e n g t h . B e c a u s e F I s a GIS no c i r c u i t of A ( F ) can have l e n g t h 4.
The l a s t a s s e r t i o n f o l l o w s b e c a u s e t h e v a l e n c y i n A ( F ) of a node e q u a l s the v a l e n c y i n F of the same node .
3.9. L e t M be an i n c i d e n c e - m a t r i x of F and A ( F ) t h e L e v i graph of F. L e t X be the f o l l o w i n g b l o c k - m a t r i x : X =
!
' 0 q,q M T M 0 p ,n Then, of c o u r s e , t h e ( p + q ) x ( p + q ) m a t r i x X i s an a d j a c e n c y m a t r i x of A ( F ) . 3.10. The v e c t o r y_ = X j ^ ^ ^ i s s a i d t o be a v a l e n c y v e c t o r of F . 3.11. We now s h a l l i n t r o d u c e an, i n t h i s r e p o r t , u s e f u l " t r i c k " , c a l l e d t h e " s p l i t t i n g of ( a ) n o d e ( s ) " , w h i c h can be a p p l i e d to any g r a p h . L e t r= ( V , E ) be a s i m p l e g r a p h . L e t n be a node of Y of v a l e n c y v. L e t [n,mj^] (1 = 1,2...v) be t h e edges to w h i c h n b e l o n g s .Now r e p l a c e i n V the node n by v new nodes n j , n 2 , . . . n v and r e p l a c e i n E the I n c i d e n c e s [n,m^] by [n^,m^] ( 1 = 1 , 2 , . . . , v ) .
Then i t i s s a i d t h a t n i s s p l i t t e d . The graph
( n ) ^ ^
r = ( ( V \ { n } ) u { n ,...,n } , (E \ u [n,m ] ) u U [n ,m ] )
^ 1=1 1=1 ^ ^ i s c a l l e d the graph d e r i v e d from F by s p l i t t i n g n.
n i s s a i d to be the stem of t h e ( s p l i t t e d ) nodes n^,...,n^ of
r^^"*-
, n^..., n.^ a r e c a l l e d t h e r e p r e s e n t a t i v e s of ( t h e s t em) n.I f s u c c e s s i v e l y the nodes n^''^^ rS^^,.. . ,n^^^ of T
a r e s p l i t t e d then the r e s u l t i n g graph i s s a i d to be t h e graph d e r i v e d
from r by s p l i t t i n g n*^^). n ^ ^ ^ .. . ,n^*^). .
3.12. D e f i n i t i o n : L e t F be a c l o s e d f i g u r e and A ( F ) i t s L e v i g r a p h .
By r( F ) we denote the graph w h i c h i s d e r i v e d from A ( F ) by
s p l i t t i n g the s i n g u l a r nodes of A ( F ) .
r( F ) I s c a l l e d the d e r i v e d graph of F .
3.13. Most of the time we s h a l l c o n s i d e r i n t h i s r e p o r t f i g u r e s F w h i c h has t h e p r o p e r t y t h a t i t s L e v i graph i s a c o n n e c t e d g r a p h . I n g e n e r a l ,
however, the d e r i v e d graph r( F ) i s no l o n g e r c o n n e c t e d . E.g. i f [n,m
i s a s i n g u l a r i n c i d e n c e t h e n t h e r e i s i n r( F ) a component w h i c h has two ( s p l i t t e d ) nodes n^ and m^ (from the stem n and m r e s p e c t i v e l y ) and o n l y one i n c i d e n c e ^j^,™-^
The c a s e t h a t F h a s a c o n n e c t e d d e r i v e d g r a p h r( F ) I s I m p o r t a n t I n o u r t h e o r y . Hence t h e f o l l o w i n g d e f i n i t i o n . 3.14. D e f i n i t i o n : L e t F be a c l o s e d f i g u r e . F I s s a i d t o be a p r o p o s i t i o n f i g u r e I f r( F ) i s c o n n e c t e d . 3.15. Comment: L e t PG(2,R) be t h e p r o j e c t i v e p l a n e o v e r t h e r e a l s . (We s h a l l c o n s i d e r p r o j e c t i v e p l a n e s i n s e c t i o n 6). I n PG(2,R) t h e g e n e r a l t h e o r e m o f D e s a r g u e s i s v a l i d . The f i g u r e o f t h a t t h e o r e m i s a ( l O g . l O g ) - c o n f i g u r a t i o n and i s , i n o u r f o r m a l a t l o n , a p r o p o s i t i o n f i g u r e . I f we c o n s i d e r on t h e o t h e r hand a s p e c i a l i z a t i o n o f D e s a r g u e s ' t h e o r e m t h e n t h e c o r r e s p o n d i n g f i g u r e i s no l o n g e r a p r o p o s i t i o n f i g u r e . We g i v e a n e x a m p l e . L e t ( s t u ) r u n t h r o u g h t h e p e r m u t a t i o n s o f
(123),
l e t X^X2X2 be a t r i a n g l e , l e t 0 be a p o i n t i n s i d e X^X2X3 a n d l e t X'=X 0 n X^X . F u r t h e r m o r e l e t X X n X'X' = Y t h e n s s t u s t s t u a c c o r d i n g t h e D e s a r g u e s p r o p e r t y t h e p o i n t s Y^,Y2,Y3 a r e c o l l i n e a r . The f i g u r e F o f t h i s s p e c i a l i z a t i o n I s no l o n g e r a p r o p o s i t i o n f i g u r e . I n f a c t r( F ) h a s f o u r c o m p o n e n t s : 3 c o r r e s p o n d i n g t o s i n g u l a r i n c i d e n c e s and 1 I s o m o r p h i c t o t h e ( p r o p o s i t i o n ) f i g u r e c o r r e s p o n d i n g t o t h e g e n e r a l D e s a r g u i a n t h e o r e m . 3.16. G i v e n a c l o s e d f i g u r e F t h e t r a n s i t i o n o f F t o r( F ) i s d e f i n e d i n(3.12)
and
(3.11).
As I n 3.14 we s h a l l u s e t h e g r a p h r( F ) t o d e f i n e s e v e r a l p r o p e r t i e s o f F i t s e l f . So we s o m e t i m e s h a v e t o t r a n s l a t e p r o p e r t i e s o f r( F ) I n t o p r o p e r t i e s o f F. T h i s , h o w e v e r , comes n a t u r a l l y I f one adds a node n ' o f r( F ) t o t h e node n o f F i f n' i s a r e p r e s e n t a t i v e o f t h e ( s i n g u l a r ) node n and one a d d s n t o i t s e l f I f n i s n o r m a l .3.17. I t may seem cumbersome t o d e s c r i b e p r o p e r t i e s o f a f i g u r e F u s i n g t h e g r a p h r( F ) . However, i t t u r n s o u t t o be a u s e f u l m e t h o d , a l s o g i v i n g a c e r t a i n u n i t y o f f o r m u l a t i o n s . More s p e c i a l , see s e c t i o n 8. 3.18. L e t F be a c l o s e d f i g u r e a n d r( F ) i t s d e r i v e d g r a p h . The c o m p o n e n t s o f r( F ) d e f i n e a n e q u i v a l e n c e r e l a t i o n o n t h e i n c i d e n c e s o f r( F ) ; t h a t i s , on t h e I n c i d e n c e s o f I u s i n g t h e c o r r e s p o n d e n c e d e s c r i b e d i n
(3.16).
A n o n - e m p t y c l a s s o f t h i s e q u i v a l e n c e r e l a t i o n i s s a i d t o be a c e l l o f F ( s o a c e l l I s a s u b s e t o f I ) . We s a y t h a t a node n o f F b e l o n g s t o a c e l l C o f F i f n ( o r one o f i t s r e p r e s e n t a t i v e s ) i n r( F ) b e l o n g s t o t h e component o f r( F ) c o r r e s p o n d i n g t o C. 3.19. The number o f e l e m e n t s o f a c e l l i s e i t h e r 1 ( t h e c o r r e s p o n d i n g i n c i d e n c e i s s i n g u l a r ) o r a t l e a s t 3 ( t h e c o r r e s p o n d i n g component o f r( F ) c o n t a i n s a t l e a s t one node o f v a l e n c y 3). 3.20. D e f i n i t i o n : L e t C be a c e l l o f a c l o s e d f i g u r e F = ( P , L , I ) . L e t P^ be t h e s e t o f p o i n t s b e l o n g i n g t o C a n d L ^ t h e s e t o f l i n e s o f L b e l o n g i n g t o C t h e n a component o f F. A component F o f F i s s a i d t o be s i n g u l a r i f | c| = l and n o r m a l i f|c|>3.
3.21. A component o f a c l o s e d f i g u r e i s n o t n e c e s s a r i l y a c l o s e d f i g u r e I t s e l f . B e c a u s e o f t h e s t r u c t u r e o f r( F ) i t i s e v i d e n t t h a t a n o r m a l node o f F b e l o n g s t o one a n d o n l y component o f F. A s i n g u l a r node o f F, h o w e v e r , can b e l o n g t o s e v e r a l c o m p o n e n t s o f F ( i f t h e s p l i t t e d r e p r e s e n t a t i v e s o f t h a t s i n g u l a r node b e l o n g t o s e v e r a l c o m p o n e n t s o f r( F ) ) .Remark t h a t a s i n g u l a r component of F c o n s i s t s of one l i n e and one p o i n t o n l y , I n c i d e n t w i t h e a c h o t h e r .
I f , f o r e x a m p l e , F I s a p r o p o s i t i o n f i g u r e t h e n F I t s e l f I s the o n l y component of F . The component i s normal b e c a u s e F I s c l o s e d .
3.22. D e f i n i t i o n : L e t C be a c e l l of the c l o s e d f i g u r e F = ( P , L , I ) , l e t PQ be t h e s e t of l i n e s b e l o n g i n g t o I \ C and l e t L Q be t h e s e t of l i n e s b e l o n g i n g t o I \ C t h e n 7Q i s the f i g u r e F ( . = ( P ^ , L C , I \ C ) . FQ i s s a i d to be the complement of F ^ ( w i t h r e s p e c t to F ) .
3.23. By d e f i n i t i o n t h e r e i s a 1-1 c o r r e s p o n d e n c e between the components of
r( F ) and t h e components of F . L e t C be a c e l l of t h e c l o s e d f i g u r e F .
B e c a u s e F'-' i s not n e c e s s a r i l y c l o s e d r( F ' " ) i s I n g e n e r a l not
d e f i n e d . But e v e n i f F ^ I s c l o s e d i t s e l f the component of r( F ) c o r r e s p o n d i n g t o F*" i s i n g e n e r a l not e q u a l t o r( F * ^ ) , b e c a u s e F , as f i g u r e i n i t s own r i g h t , can c o n t a i n normal nodes w h i c h a r e i n F s i n g u l a r n o d e s .
The same r e m a r k s a p p l y t o F Q .
3.24. I n our t h e o r y i t i s i m p o r t a n t t o know the c o m p o n e n t - p a r t i t i o n of a g i v e n f i g u r e F. I n t h e o r y t h i s I s an e a s y m a t t e r .
1) C o n s t r u c t t h e L e v i graph of F . 2 ) C o n s t r u c t t h e d e r i v e d graph of F .
3) F i n d the components of r( F ) .
4 ) F i n d t h e c o r r e s p o n d i n g components of r( F ) I n F .
F o r s m a l l f i g u r e s t h i s k i n d of work can be done by hand, but f o r l a r g e r f i g u r e s we need t o have a computer.
I n a l a t e r r e p o r t we s h a l l t r e a t t h i s t o p i c I n f u l l l e n g t h ( s e e a l s o s e c t i o n 1 0 ) . 4. CONSTRUCTION SEQUENCES 4.1. I n s e c t i o n 4 F i s a l w a y s t h e c l o s e d f i g u r e F = ( P , L , I ) . 4.2. D e f i n i t i o n : The number (j)(F) = 2( | P | + 1 L | ) - 1 1 | i s s a i d to be t h e d e g r e e of freedom o f F. 4.3. D e f i n i t i o n : L e t F ( 0 ) = ( P Q , L Q . I Q ) and F = ( P , L , I ) be f i g u r e s t h e n i t i s s a i d t h a t a ) F can be e l e m e n t a r y c o n s t r u c t e d from F ( 0 ) i f e i t h e r F = F ( 0 ) or a l ) t h e r e i s a node n s u c h t h a t ( P u L ) \ ( P Q U L Q ) = {n and
a2) t h e r e a r e a t most two nodes m^,m2 i n P Q U L Q such t h a t n , m j e I ( t = l , 2 ) ; b) F can be t r i v i a l l y c o n s t r u c t e d from F ( 0 ) i f t h e r e i s a f i n i t e s e q u e n c e of f i g u r e s F ( 0 ) , F ( l ) , . . . , F ( t ) s u c h t h a t b l ) F ( t ) = F and i f t>0 b2) F ( s + 1 ) can e l e m e n t a r y be c o n s t r u c t e d from F ( s ) ( s = 0 , l t - l ) ; c ) F i s a t r i v i a l f i g u r e i f F i s t r i v i a l l y c o n s t r u c t e d e i t h e r from t h e empty f i g u r e or from a t r i v i a l f i g u r e ;
d) the s e q u e n c e F ( 0 ) , F ( l ) F ( t ) i n b) i s a c o n s t r u c t i o n s e q u e n c e between F ( 0 ) and F;
e ) a s e q u e n c e o f f i g u r e s F ( 0 ) , F ( l ) , . . . , F ( t ) i s a c o n s t r u c t i o n s e q u e n c e ( f o r F ( t ) ) i f i t i s a c o n s t r u c t i o n s e q u e n c e b e t w e e n t h e empty f i g u r e F ( 0 ) and t h e ( t r i v i a l ) f i g u r e F ( t ) ; f ) F c a n be n o n - t r i v i a l l y c o n s t r u c t e d f r m F ( 0 ) i f t h e r e i s an i n c i d e n c e i i n I s u c h t h a t ( P , L , I \ { i } ) c a n be t r i v i a l l y c o n s t r u c t e d f r o m F ( 0 ) ; ( F i s c a l l e d a n o n -t r i v i a l f i g u r e ) g ) i f i n b ) o r f ) F ( 0 ) i s t h e empty f i g u r e we s a y t h a t F can be ( n o n - ) t r l v l a l l y c o n s t r u c t e d . 4.A. D e f i n i t i o n : L e t F ( 0 ) and F be f i g u r e s t h e n i t i s s a i d t h a t a ) F ( 0 ) c a n be e l e m e n t a r y r e d u c e d t o F i f F ( 0 ) c a n be e l e m e n t a r y c o n s t r u c t e d f r o m F; b ) F ( 0 ) c a n be ( n o n ) - t r l v i a l l y r e d u c e d t o F i f F ( 0 ) c a n be ( n o n ) - t r i v i a l l y c o n s t r u c t e d f r o m F; c ) t h e s e q u e n c e o f f i g u r e s F ( 0 ) , F ( l ) F ( t ) = F i s a_ r e d u c t i o n s e q u e n c e b e t w e e n F ( 0 ) and F i f F ( t ) , F ( t - l ) , F ( 0 ) i s a c o n s t r u c t i o n s e q u e n c e b e t w e e n F and F ( 0 ) ; d ) t h e s e q u e n c e o f f i g u r e s F ( 0 ) , F ( l ) F ( t ) i s a r e d u c t i o n s e q u e n c e ( f o r F ( 0 ) ) i f F ( t ) , F ( t - l ) F ( 0 ) i s a c o n s t r u c t i o n s e q u e n c e ( a n d t h e r e f o r e F ( t ) i s t h e e m p t y f i g u r e ) . 4.5. D e f i n i t i o n : The c o m p o s i t i o n o f a f i g u r e F f r o m a f i g u r e F ( 0 ) w h i c h c a n be ( n o n - ) t r i v i a l c o n s t r u c t e d f r o m F ( 0 ) i s s o m e t i m e s c a l l e d a ( ( n o n - ) t r i v i a l ) c o n s t r u c t i o n ( o f F f r o m F ( 0 ) ; o f F ( i f F ( 0 ) i s e m p t y ) ) . I f F i s e l e m e n t a r y c o n s t r u c t e d f r o m F ( 0 ) a n d F ^ F ( 0 ) t h e n t h e c o n s t r u c t i o n o f F f r o m F ( 0 ) i s s o m e t i m e s c a l l e d a c o n s t r u c t i o n s t e p . 4.6. Remark t h a t i f t h e f i g u r e s G ( l ) a n d G ( 2 ) b o t h a r e t r i v i a l l y c o n s t r u c t e d f r o m t h e same f i g u r e F u s i n g t h e same c o n s t r u c t i o n s e q u e n c e t h e n we o n l y known t h a t G ( l ) = G ( 2 ) . B u t t h i s i s s u f f i c i e n t f o r o u r p u r p o s e . I n g e n e r a l we s h a l l n o t s t a t e e x p l i c l t y t h a t a g i v e n c o n s t r u c t i o n d e t e r m i n e a f i g u r e o n l y up t o i s o m o r p h i s m s . 4.7. I f F i s t r i v i a l l y c o n s t r u c t e d f r o m F ( 0 ) t h e n 4>(F) > 4 > ( F ( 0 ) ) . I f F i s e l e m e n t a r y c o n s t r u c t e d f o r m G and $ ( F ) = $ ( G ) t h e n o n l y one o f two t h i n g s c a n have h a p p e n e d . E i t h e r t w o l i n e s o f G, n o t i n t e r s e c t i o n i n G, a r e now i n t e r s e c t i o n i n F i n a new, u n a m b i g u o u s l y d e f i n e d , p o i n t o r ( d u a l l y ) t w o p o i n t s o f G, n o t b e l o n g i n g t o t h e same l i n e i n G, a r e now b o t h l y i n g on a new,
u n a m b i g u o u s l y d e f i n e d , l i n e . So i n t h i s case ( * ( F ) = <I>(G)) t h e f i g u r e G i s i n d e e d u n i q u e l y d e t e r m i n e d by F and t h e c o n s t r u c t i o n s t e p c o n s t r u c t i n g G f r o m F. One c o u l d s a y t h a t G i s " f i x e d " by F and t h e c o n s t r u c t i o n s t e p u n d e r c o n s i d e r a t i o n . 4.8. T h e o r e m : L e t F = ( P , L , I ) be a p r o p o s i t i o n f i g u r e a n d i a n a r b i t r a r i l y c h o s e n i n c i d e n c e o f F. L e t F^ = ( P , L , l \ { i } ) t h e n F j i s a t r i v i a l f i g u r e . F i s a n o n - t r i v i a l f i g u r e . P r o o f : The w h o l e p r o o f c a n p e r h a p s b e s t been f o l l o w e d i n r( F ) . L e t 1 = n , n Q j . W i t h o u t l o s s o f g e n e r a l i t y we c a n s u p p o s e t h a t n i s n o r m a l .
We compose a r e d u c t i o n s e q u e n c e f o r F^. Once t h i s i s done we c a n c o n s i d e r t h e " i n v e r s e " o f t h i s s e q u e n c e as a c o n s t r u c t i o n s e q u e n c e f o r F J w h i c h c o n s t r u c t s F^ ( u p t o i s o m o r p h i s m s , o f c o u r s e ) . The a d d i t i o n o f 1 t o I \ { l } c o m p l e t e s t h e n t h e n o n - t r i v i a l c o n s t r u c t i o n o f F; so F i s n o n - t r i v i a l . The r e d u c t i o n o f F^ c o n s i s t s o f r e d u c t i o n s t e p s o f t h e f o l l o w i n g f o r m : a ) l e a v e o u t an a b n o r m a l node p; b ) l e a v e o u t t h e i n c i d e n c e s ( i f t h e r e a r e a n y ) t o w h i c h p b e l o n g s .
We k e e p t h i s p r o c e s s g o i n g on as l o n g as t h e r e a r e s t i l l a b n o r m a l n o d e s .
Because Fj. i s no l o n g e r c l o s e d t h e r e i s a t l e a s t one a b n o r m a l node ( e . g . n ) . So t h e p r o c e s s can s t a r t . E v e r y r e d u c t i o n s t e p " c r e a t e s " a t l e a s t one new a b n o r m a l n o d e , b e c a u s e e v e r y i n c i d e n c e i s e i t h e r n o r m a l o r m i x e d . L e t m be any n o r m a l node o f F. L e t n,n2 , n^ ,n2 .'^s ' " s + l .»•••» .^t»™ ] ^ p a t h i n r( F ) b e t w e e n n and m ( s u c h a p a t h e x i s t s b e c a u s e r( F ) i s c o n n e c t e d ) . The s t e p w h i c h l e a v e s o u t n ( w h i c h i s c e r t a i n l y a b n o r m a l ) makes n^ a b n o r m a l and t h e r e f o r e n^ i s l e f t o u t d u r i n g t h e r e d u c t i o n p r o c e s s . The same a p p l i e s t o e v e r y node n^ o f t h i s p a t h . Hence m becomes e v e n t u a l l y a b n o r m a l i t s e l f and w i l l be l e f t o u t b e f o r e t h e r e d u c t i o n p r o c e s s s t o p s . I f m i s a s i n g u l a r node o f F w i t h v a l e n c y v t h e n t h e r e a r e v p a t h s i n r( F ) b e t w e e n n and t h e v r e p r e s e n t a t i v e s o f m. E v e r y r e p r e s e n t a t i v e o f m i s I n c i d e n t w i t h a n o r m a l node w h i c h becomes a b n o r m a l d u r i n g t h e r e d u c t i o n p r o c e s s , h e n c e d u r i n g t h a t p r o c e s s t h e v a l e n c y o f m d e c r e a s e s f r o m v t o a t m o s t 2, b u t t h e n m i t s e l f has become a b n o r m a l and w i l l a l s o be l e f t o u t , b e f o r e t h e r e d u c t i o n p r o c e s s s t o p s . So, c l e a r l y t h e a b o v e m e n t i o n e d r e d u c t i o n p r o c e s s o n l y s t o p s when no nodes a r e l e f t . T h i s i m p l i e s t h e t h e o r e m . 4 . 9 . I n t h e o r e m ( 4 . 8 ) one can c h o o s e t h e i n c i d e n c e 1 a r b i t a r i l y . I f we no l o n g e r r e q u i r e t h a t F i s a p r o p o s i t i o n f i g u r e , t h e n t h e s i t u a t i o n i s o p e n . I t s t i l l can h o l d t h a t f o r e v e r y 1 t h e f i g u r e F^ i s t r i v i a l b u t t h e r e a r e c e r t a i n l y c o u n t e r e x a m p l e s now.
E.g. l e t i n ( 4 . 8 ) F be a c l o s e d f i g u r e w i t h 2 c e l l s C ( l ) and C ( 2 ) and l e t C ( 2 ) be a s i n g u l a r c e l l .
C( 1) Choose f o r 1 now t h e o n l y i n c i d e n c e o f C ( 2 ) . T h e n F j = F , w h i c h i s a p r o p o s i t i o n f i g u r e , and so F. ,
1 i s n o n - t r i v i a l .
GENERATING NEW FIGURES
I n t h i s s e c t i o n we s h a l l d i s c u s s a m e t h o d t o " g e n e r a t e " new f i g u r e s o u t o f g i v e n o r a l r e a d y g e n e r a t e d f i g u r e s . H e r e we s h a l l n o t g i v e t h e g e o m e t r i c a l b a c k g r o u n d b u t p o s t p o n e t h a t t o s e c t i o n 9. So, s p e a k i n g , i n p o p u l a r t e r m s , we s i m p l y p l a y a "game" a c c o r d i n g t o g i v e n r u l e s .
The d e f i n i t i o n ( 5 . 3 ) i s more p r o m i s i n g t h a n we s h a l l make t r u e i n t h i s r e p o r t , b e c a u s e we s h a l l o n l y d i s c u s s h e r e t h e m e t h o d t o g e n e r a t e f i g u r e s w h i c h a r e t h e c o n s e q u e n t o f o n l y one f i g u r e and o f t h a t m e t h o d , we s h a l l o n l y g i v e some s p e c i a l c a s e s . I n a s u b s e q u e n t r e p o r t we s h a l l d i s c u s s t h e m e t h o d i n f u l l . M o r e o v e r t h e d e f i n i t i o n ( 5 . 3 ) may seem a w k w a r d , b u t t h e m e t h o d i s i n f a c t q u i t e e l e g a n t as we hope t o d e m o n s t r a t e t o t h e r e a d e r . Remember t h a t i f A i s a c l o s e d f i g u r e and C i s a c e l l o f A t h e n i s t h e c o m p l e m e n t o f t h e f i g u r e o f t h e c e l l C ( s e e 3 . 2 2 ) . 5. 3.3. D e f i n i t i o n : L e t A be a c l o s e d f i g u r e w i t h a t l e a s t 3 c e l l s . L e t t h e r e be two c e l l s C ( l ) and C ( 2 ) and two p r o p o s i t i o n f i g u r e s F ( l ) and F ( 2 ) s u c h t h a t A^^^.^ = F ( t ) ( t = l , 2 ) . L e t C ( 0 ) ( / C ( l ) , C ( 2 ) ) be a c e l l o f A t h e n A^^^Q^ i s s a i d t o be e l e m e n t a r y g e n e r a t e d by F ( l ) and F ( 2 ) . I f F ( l ) = F ( 2 ) t h e n A^^Q^ i s s a i d t o be e l e m e n t a r y g e n e r a t e d by F ( l ) . A i t s e l f i s s a i d t o be a t r a n s f o r m a t i o n f i g u r e . 5 . 4 . D e f i n i t i o n : I n t h e n o t a t i o n o f ( 5 . 3 ) . The t r a n s f o r m a t i o n f i g u r e A i s s a i d t o be a d e d u c t i o n f i g u r e i f t h e r e i s i n A a c e l l C ?'C(1), C ( 2 ) s u c h t h a t A^, i s a p r o p o s i t i o n f i g u r e .
5.5. We need a c l o s e r l o o k I n t o the s t r u c t u r e of a t r a n s f o r m a t i o n f i g u r e A.
F i r s t , f o r any i n c i d e n c e i of A we have e i t h e r i € A^ o r i e A(-, f o r any c e l l C of A (by d e f i n i t i o n of c o u r s e ) .
Second, the normal and t h e s p l i t t e d nodes of A^^^^ ( i n t h e
n o t a t i o n of ( 5 . 3 ) ) a l l are nodes of A^^i) (and a l s o t h e nodes of '' are nodes of A^^j))»
T h i r d , A can be n o n - t r i v i a l l y c o n s t r u c t e d from A^ f o r any c e l l C such t h a t i s a p r o p o s i t i o n f i g u r e . T h i s can be seen as f o l l o w s .
Consider r( A ) , l e t C(s) ( s = 0 , l , . . . , t ) be the c e l l s of A and l e t r( s ) be the component of A c o r r e s p o n d i n g t o C(s) ( s = 0 , l , . . . , t ) . Let C(0)=C.
I f we leave out r( 0 ) then the r e s t of r( A ) becomes the connected graph r(A(-.^Qp. There i s a t l e a s t one ( s i n g u l a r ) node n b e l o n g i n g t o
r( 0 ) as w e l l as t o u r( s ) , f o r A has at l e a s t 3 c e l l s and s = l
^^'^C(O)^ i s a connected graph.
Because r( 0 ) i s connected t h e r e i s a t r i v i a l c o n s t r u c t i n g sequence Z, s t a r t i n g w i t h n, which c o n s t r u c t s A^^*^^ ( a p a r t from perhaps one
i n c i d e n c e , i f A'^^°^ i t s e l f i s c l o s e d ) .
Now l e t us b e g i n w i t h A^^^Q'J and l e t us augment
A(-.^Q^
u s i n gthe c o n s t r u c t i n g sequences L. Because A i t s e l f i s c l o s e d but r( 0 ) i s connected we s t i l l need e x a c t l y one n o n - t r i v i a l c o n s t r u c t i o n step t o complete A. T h i s can be shown by a r e a s o n i n g which i s s i m i l a r t o the one we used i n t h e p r o o f of 4.8.
We see t h e r e f o r e t h a t indeed A can be n o n - t r i v i a l l y c o n s t r u c t e d from
^C(O)-5.6. D e f i n i t i o n : Let t € N \ { 0,1,2}.
L e t ( F ( l ) , F ( 2 ) , . . . , F ( t ) ) be an o r d e r e d t - t u p l e of p r o p o s i t i o n f i g u r e s (some of which may be i s o m o r p h i c ) w i t h t h e f o l l o w i n g p r o p e r t i e s :
1) F ( 3 ) i s e l e m e n t a r y generated by F ( l ) and F ( 2 ) ; 2) i f t > 3 , f o r 4<s<t t h e r e are an s^ and an s^,
l<S2^<S2<;t-l, such t h a t F ( s ) i s elementary generated by
F(s^) and F ( S 2 ) ;
Then F ( t ) i s s a i d t o be generated by F ( l ) and F ( 2 ) ( o r by F ( l ) i f F ( 1 ) 2 F ( 2 ) ) .
5.7. The c r u x of the g e n e r a t i n g method d e f i n e d i n 5.6 i s , of c o u r s e , the c o m p o s i t i o n of d e d u c t i o n f i g u r e s which " t r a n s f o r m " f o r s=3,4,..,t F ( s ^ ) and F ( s 2 ) i n t o F ( s ) .
5.8. We s h a l l d i s c u s s a method t o compose at l e a s t a t r a n s f o r m a t i o n f i g u r e s t a r t i n g from a g i v e n p r o p o s i t i o n f i g u r e F. T h i s method w i l l g i v e ( a ) f l g u r e ( s ) elementary generated by F. The second q u e s t i o n then i s whether those generated f i g u r e s are p r o p o s i t i o n f i g u r e s or n o t . I n g e n e r a l t h i s q u e s t i o n has t o be d e a l t w i t h by the computer (see s e c t i o n 10 and a l a t e r r e p o r t ) .
5.9. The method f o r the c o m p o s i t i o n of a t r a n s f o r m a t i o n f i g u r e s t a r t i n g from a g i v e n p r o p o s i t i o n f i g u r e b o i l s down t o f i n d i n g a d i s c o n n e c t i n g set Q (see W i l s o n [ 7 ] ) , " c u t t i n g l o o s e " one of the components of the graph determined by and " c o p y i n g " t h a t component. The whole method i s I l l u s t r a t e d i n s e c t i o n 9 ( f r o m (9.11) o n ) .
I n t h i s r e p o r t we s h a l l o n l y d e a l w i t h one s p e c i a l case: every node i n t h e component which i s t o be copied has t o belong t o a c i r c u i t .
Otherwise the r e s u l t i n g t r a n s f o r m a t i o n f i g u r e would have a c i r c u i t of l e n g t h 4 which i s f o r b i d d e n , of course.
I n a l a t e r r e p o r t we s h a l l d e a l w i t h the g e n e r a l case.
5.10. Let F = ( P , L , I ) be a p r o p o s i t i o n f i g u r e and l e t r ( F ) be i t s d e r i v e d graph.
L e t V = Ü V 2 be a p a r t i t i o n of the node set V of r ( F ) i n t o non-empty d i s j o i n t s u b s e t s . I f i s a non-empty subset of i n c i d e n c e s , which have one node i n and one node I n V 2 , then we say, t h a t i s a
The next d e f i n i t i o n I s awkward but u s e f u l .
5.11. D e f i n i t i o n : L e t fi be a d i s c o n n e c t i n g set I n r ( F ) c o n t a i n i n g o n l y normal
I n c i d e n c e s w i t h t h e f o l l o w i n g p r o p e r t i e s :
1) a f t e r t h e removal o f il t h e r e s u l t i n g graph has two
components;
2) t h e node set o f a t l e a s t one o f t h e components c o n t a i n s
o n l y nodes which a r e normal I n r ( F ) .
Let r ( l ) be such a component and l e t
U = { [ n g . m g ] , s = l , 2 . . . t } , t h e nodes ng
b e l o n g i n g t o r ( l ) .
Let A be t h e graph the node set o f which I s the u n i o n o f
m i , m2,..., and the node set o f r ( l ) and t h e
I n c i d e n c e set o f which i s t h e u n i o n o f fi and t h e i n c i d e n c e
set o f r ( l ) .
I t i s s a i d t h a t the graph A i s a s t a r t e r ( f o r r ( F ) ) . t h a t
the nodes mg ( s = l , 2 , . . . , t ) a r e t h e boundery nodes o f A
and t h a t fi i s t h e d i s c o n n e c t i n g s e t f o r A.
5.12. Remark t h a t a s t a r t e r A i s determined once t h e d i s c o n n e c t i n g s e t fi and
the boundery nodes a r e g i v e n .
Note a l s o t h a t the v a l e n c y o f every node i n A i s e i t h e r 3 o r 1 and
t h a t t h e nodes o f v a l e n c y 1 p r e c i s e l y correspond t o t h e boundery nodes
of A.
5.13. As an example c o n s i d e r i n (9.14) (see a l s o ( 9 . 1 6 ) ) t h e s t a r t e r A
determined by t h e d i s c o n n e c t i n g set { [ l 3 , 1 8 ] , [ 4 , 1 5 ] , [ 9 , 1 0 ] , [ l 6 , l l ] ,
1,12], [ 6 , 1 7 ] } and t h e set o f boundery nodes [ 13, 4, 9, 16, 1 , 6 }.
A i t s e l f i s g i v e n i n (9.17) as t h e graph marked A ( l ) .
5.14. L e t A be a s t a r t e r f o r r ( F ) . L e t r ^ ( F ) be t h e graph d e r i v e d from
r ( F ) by s p l i t t i n g t h e boundery nodes o f A.
5.15.
^A(r')
^las a t l e a s t two components; one component I s A.
5.16. L e t A be a graph i s o m o r p h i c t o A w i t h nodes Pg and
which do not belong t o P. The node p^ correspond 1-1 t o t h e normal
nodes o f A and t h e nodes q^ correspond 1-1 t o t h e nodes o f v a l e n c y
1 i n A. For u = l , 2 , . . t t h e node q^ i s c o n s i d e r e d as a
r e p r e s e n t a t i v e o f t h e stem m^. The graph A* I s c a l l e d a copy o f
A-5.17. I n (9.19) t h e graph A * ( l ) a t t h e bottom i s a copy o f A ( l ) .
5.18. Now c o n s i d e r the f i g u r e A w i t h the d e r i v e d graph r ^ ( F ) u A*. The
f i g u r e A i s w e l l - d e f i n e d , as one e a s i l y checkes.
5.19. A has compared w i t h F t new s i n g u l a r nodes, namely t h e nodes m
s
which have each I n A the v a l e n c y 4, and a number o f new normal nodes
( v i z . t h e nodes p„).
s
5.20. r ( A ) and t h e r e f o r e A has a t l e a s t t h r e e components.
Let { C ( 0 ) , C ( l ) , C ( 2 ) , . . . } be t h e c e l l - p a r t i t i o n o f A.
Suppose
A C ( I )corresponds t o A and
A C ( 2 )to A*. C l e a r l y
^C(2) ^^^^ complement o f A^^^'^ w i t h r e s p e c t t o A; see ( 3 . 2 2 ) )
i s F, a p r o p o s i t i o n f i g u r e , and A ( . Q ^ i s a f i g u r e F ( l ) i s o m o r p h i c
to F and t h e r e f o r e a l s o a p r o p o s i t i o n f i g u r e .
T h e r e f o r e A i s a t r a n s f o r m a t i o n f i g u r e .
Now t h e r e i s another c e l l , C(0) say. I n t h e f o r m u l a t i o n o f ( 5 . 3 ) and
(5.6) t h e f i g u r e A^^Q^ i s ( e l e m e n t a r y ) generated by F.
I t s t i l l remains t o be determined whether or not Aj-,^Q^ i s a
p r o p o s i t i o n f i g u r e .
5.21. I f i n (9.19) we take f o r C(0) t h e c e l l connected w i t h t h e f i r s t graph
from t h e t o p t h e r e s u l t i n g f i g u r e H(2) i s a p r o p o s i t i o n f i g u r e indeed
as i s shown i n ( 9 . 2 2 ) .
6. P G ( 2 . R ) AND I T S PROPOSITIONS
6.1. I n o r d e r to be a b l e to g l u e the p i e c e s of t h i s r e p o r t t o g e t h e r ( w h i c h s h a l l be done i n t h e s e c t i o n s 7 and 9 ) , we s t i l l have t o d i s c u s s the p r o j e c t i v e p l a n e s i n b r i e f . 6.2. D e f i n i t i o n : A p r o j e c t i v e p l a n e PG i s a g e o m e t r i c a l i n c i d e n c e s t r u c t u r e P G = ( P , L , I ) s u c h t h a t 1) f o r a l l n,m 6 P t h e r e i s a p € L s u c h t h a t ( n , p ) € I and (m,p)
e
I ; 2) f o r a l l n,m € L t h e r e i s a p € P s u c h t h a t ( p , n ) € I and (p,m)e
I ; 3 ) t h e r e a r e f o u r n o n - c o l l i n e a r p o i n t s i n P. 6.3. F o r r e l e v a n t r e f e r e n c e s on p r o j e c t i v e p l a n e s s e e f o r example H a l l [s], pg. 346-420.6.4. As i s known ( s e e H a l l ' s book) e v e r y p r o j e c t i v e p l a n e can be c o o r d i n i z e d by a s o - c a l l e d t e r n a r y r i n g . T h i s t e r n a r y r i n g u n i q u e l y d e t e r m i n e s t h e p l a n e .
As one adds p r o p e r t i e s to t h e t e r n a r y r i n g the p r o j e c t i v e p l a n e g e t s a r i c h e r s t r u c t u r e .
I n t h i s r e p o r t we c o n s i d e r the p r o j e c t i v e p l a n e t h e t e r n a r y r i n g of w h i c h i s the r e a l f i e l d R. I n o t h e r words we c o n s i d e r the o r d i n a r y 2-d i m e n s i o n a l p r o j e c t i v e geometry o v e r the r e a l s . We 2-denote t h i s p r o j e c t i v e geometry by P G ( 2 , R ) . 6.5. Of c o u r s e , t h i s s y n t h e t i c d e f i n i t i o n of PG(2,R) I m p l i e s the f o l l o w i n g a n a l y t i c d e s c r i p t i o n of P G ( 2 , R ) = ( P , L , I ) . A p o i n t ( o f P ) i s a c l a s s of p r o p o r t i o n a l ( 3 x 1 ) - m a t r i c e s o v e r R X=(x^ ,X2,X3)'^ ^ ( 0 , 0 , 0 ) ' ^ and a l i n e ( o f L ) i s a c l a s s of p r o p o r t i o n a l ( l x 3 ) - m a t r i c e s o v e r R U = ( u ^ , U 2 , U 3 ) ? ^ ( 0 , 0 , 0 ) . (X,U) € I i f and o n l y i f UX=u^x^ + U2X2 + U3X3 = 0.
I n P G ( 2 , R ) the theorem of Pappus and t h e p r i n c i p l e of d u a l i t y h o l d .
6.6. I n t h i s r e p o r t we not o n l y r e s t r i c t o u r s e l v e s to PG(2,R) but moreover t o the s o - c a l l e d t h e o r y of i n t e r s e c t i o n s i n P G ( 2 . R ) .
6.7. The t h e o r y of i n t e r s e c t i o n s i n a p r o j e c t i v e p l a n e PG b e a r s upon
a x i o m e s , d e f i n i t i o n s and theorems w h i c h a r e a s s e r t i o n s on s u b s t r u c t u r e s of PG and upon p r o o f s and c o n s t r u c t i o n s w h i c h a r e ( n o n ) - t r i v i a l
c o n s t r u c t i o n s ( i n the s e n s e of s e c t i o n 3 ) .
6.8. E . g . the axiom t h a t two d i f f e r e n t p o i n t s a r e i n c i d e n t w i t h a t most one l i n e i s , i n t h i s c o n t e x t , an a d m i s s i b l e axiom and so i s t h e a s s e r t i o n t h a t k d i f f e r e n t p o i n t s a r e l y i n g f r e e ; they form, by d e f i n i t i o n , a k - a r c .
However, the s t a t e m e n t t h a t s e v e r a l p o i n t s b e l o n g to the same c o n i c i s not a d m i s s i b l e : a c o n i c not b e i n g a s u b s t r u c t u r e of PG, as d e f i n e d i n ( 2 . 1 ) .
6.9. R e l a t e d to PG(2,R) one can s a y ( i n a n a l y t i c t e r m s ) t h a t i n the t h e o r y of i n t e r s e c t i o n s one c o n s i d e r s s e t s of l i n e a r e q u a t i o n s and t h e i r s o l u t i o n s . A l s o , a s remarked i n ( 7 . 3 ) , i n the t h e o r y of i n t e r s e c t i o n s of P G ( 2 , R ) e v e r y c o n s t r u c t i o n can be p e r f o r m e d by t h e u s e of a r u l e r a l o n e . 6.10. From now on we o n l y c o n s i d e r P G ( 2 , R ) i n t h i s r e p o r t , u n l e s s o t h e r w i s e s t a t e d . :
6 . 1 1 . L e t us c o n s i d e r a theorem ( o r a x i o m ) T of the t h e o r y of i n t e r s e c t i o n s and suppose t h a t T i s an a s s e r t i o n on a f i n i t e s e t of p o i n t s PQ, a f i n i t e s e t of l i n e s LQ and a f i n i t e s e t of i n c i d e n t s I Q c PQ x L Q . Then F ( 0 ) = ( P Q , L Q , I Q ) i s a f i g u r e i n t h e s e n s e of s e c t i o n 2 . I t i s s a i d t h a t T i s a f i n i t e theorem ( a x i o m ) . F ( 0 ) i s s a i d to be the f i g u r e of T. I f F ( 0 ) i s a c l o s e d s u b s t r u c t u r e then i t i s s a i d t h a t T I s a c l o s e d theorem ( a x i c m ) .
The L e v i g r a p h A ( F Q ) i s s a i d to be the L e v i g r a p h of T and the d e r i v e d graph r ( F ( 0 ) ) i s s a i d to be the d e r i v e d g r a p h o f T.
6 . 1 2 . Most of the time we c o n s i d e r f i n i t e t h e o r e m s . L e t F be a f i g u r e of a f i n i t e theorem T. By v i r t u e of the p r o p e r t i e s of a theorem t h e r e i s a t l e a s t one s u b f i g u r e F ( l ) of F w h i c h i s a n o n - t r i v i a l f i g u r e . F o r the c o n s e q u e n t of a theorem i n c l u d e s a s t a t e m e n t on a n o n - t r i v i a l c o n s t r u c t i o n s t e p . The same a p p l i e s t o a x i o m s . 6 . 1 3 . T h e r e f o r e i f T i s a f i n i t e c l o s e d theorem ( a x i o m ) t h e n the s i m p l e s t c a s e w h i c h can o c c u r i s t h a t t h e f i g u r e of T I s a p r o p o s i t i o n f i g u r e . B e c a u s e i n t h i s r e p o r t t h i s k i n d of theorems p l a y a c e n t r a l r o l e we s h a l l c a l l t h i s k i n d of theorems p r o p o s i t i o n s . I n t h i s r e p o r t t h e r e f o r e we make a d i s t i n c t i o n between theorems and p r o p o s i t i o n s .
Hence t h e f o l l o w i n g d e f i n i t i o n .
6 . 1 4 . D e f i n i t i o n ; A theorem or axiom i n the t h e o r y of i n t e r s e c t i o n s i s s a i d t o be a p r o p o s i t i o n i f t h e f i g u r e of the theorem or axiom I s a p r o p o s i t i o n f i g u r e .
A theorem or axiom i s s a i d to be a s p e c i a l i z a t i o n of a p r o p o s i t i o n P i f the f i g u r e of t h e theorem o r axiom c o n s i s t s of a p r o p o s i t i o n f i g u r e of P and one or more f i g u r e s the c e l l s of w h i c h a r e s i n g u l a r .
6 . 1 5 . The s p e c i a l i z a t i o n of D e s a r g u e s ' theorem d e s c r i b e d i n ( 3 . 1 5 ) i s a s p e c i a l i z a t i o n of the p r o p o s i t i o n of D e s a r g u e s i n the s e n s e of ( 6 . 1 4 ) .
6 . 1 6 . We s h a l l now s t u d y the c o n c e p t p r o p o s i t i o n , i t s f i g u r e and i t s d e r i v e d g r a p h a l i t t l e more c l o s e l y .
6 . 1 7 . A p r o p o s i t i o n (and e v e r y theorem and axiom f o r t h a t m a t t e r ) i s m o s t l y d e s c r i b e d I n words ( w h i c h i s t h e most c o n v e n t i o n a l way and i s found I n most t e x t b o o k s ) , but I n t h i s r e p o r t we have c r e a t e d some new methods
t o d e s c r i b e a p r o p o s i t i o n . 6 . 1 8 . L e t T be a t r u e p r o p o s i t i o n i n P G ( 2 , R ) , By v i r t u e of the p r o p e r t i e s of a p r o p o s i t i o n t h i s p r o p o s i t i o n T c o n s i s t s of two p a r t s . 1) the a n t e c e d e n t : a s t a t e m e n t on c e r t a i n c o n d i t i o n s r e l a t e d t o p o i n t s , l i n e s and t h e i r i n c i d e n c e s ; 2 ) the c o n s e q u e n t : a s t a t e m e n t t h a t i f the c o n d i t i o n s of t h e a n t e c e d e n t a r e f u l f i l l e d t h e n a c e r t a i n l i n e and a c e r t a i n p o i n t a r e i n c i d e n t . 6 . 1 9 . U s i n g a ( p r o p o s i t i o n ) f i g u r e of T we can d e r i v e from ( 4 . 8 ) t h a t t h e a n t e c e d e n t of T can be f o r m u l a t e d as a t r i v i a l c o n s t r u c t i o n s e q u e n c e and t h a t t h e c o n s e q u e n t of T can be f o r m u l a t e d a s a n o n - t r i v i a l c o n s t r u c t i o n .
6 . 2 0 . L e t F be a f i g u r e of T t h e n the c o n s e q u e n t of T b e a r s upon one p a r t i c u l a r i n c i d e n c e 1 of F . More p r e c i s e : L e t F j = ( P , L , I \ { i } ) , 1 = [nQ,n^] and G a f i g u r e i s o m o r p h i c to F^ t h e n i n G t h e nodes nJ, and nj:, (nJ, c o r r e s p o n d i n g to nQ and nJ c o r r e s p o n d i n g to n-^) a r e I n c i d e n t . 6 . 2 1 . The i d e n t i f i c a t i o n p r o c e s s : I t i s i n t h i s way t h a t we s h a l l I d e n t i f y a p r o p o s i t i o n T w i t h i t s f i g u r e F and a l s o w i t h i t s L e v i g r a p h A ( F ) and I t s d e r i v e d g r a p h r ( F ) . ( S e e , however, ( 6 . 2 5 ) and ( 6 . 2 6 ) ) .
6.22. I n t h i s way we have added to e v e r y p r o p o s i t i o n a p r o p o s i t i o n f i g u r e . However, t h e r e a r e p r o p o s i t i o n f i g u r e s w h i c h do not c o r r e s p o n d to a p r o p o s i t i o n , w h i c h i s t r u e i n P G ( 2 , R ) . ( T h i s goes a l m o s t w i t h o u t s a y i n g , but s t i l l . ) E . g . the w e l l - k n o w n ( 7 3 , 7 3 ) - c o n f i g u r a t i o n c o r r e s p o n d i n g t o the p r o j e c t i v e p l a n e o v e r G F ( 2 ) i s a p r o p o s i t i o n f i g u r e but t h e r e i s not a c o r r e s p o n d i n g t r u e theorem i n P G ( 2 , R ) .
6.23. L e t j be any i n c i d e n c e of F = ( P , L , I ) and Fj = ( P , L , I \ { j } ) . L e t Z be a t r i v i a l c o n s t r u c t i o n s e q u e n c e c o n s t r u c t i n g F j . Then E can be
c o n s i d e r e d as t h e a n t e c e d e n t of a p r o p o s i t i o n T* and t h e n o n - t r i v i a l c o n s t r u c t i o n w h i c h adds j to F j as the c o n s e q u e n t of T*.
Of c o u r s e , the q u e s t i o n i s , whether T* i s a t r u e theorem w i t h i n the t h e o r y of i n t e r s e c t i o n s of PG(2,R) whenever T i s t r u e .
I n f a c t we run h e r e i n t o one of the e s s e n t i a l s of p r o p o s i t i o n s , b e c a u s e t h e f o l l o w i n g theorem h o l d s .
6.24 Theorem: I n t h e n o t a t i o n of ( 6 . 1 8 ) t h r o u g h ( 6 . 2 3 ) . T and T* a r e e q u i v a l e n t p r o p o s i t i o n s of P G ( 2 , R ) .
P r o o f ; L e t T be a t r u e p r o p o s i t i o n . i = [ n Q , n ^ ] and l e t j = [ n , . _ i , n t ] ( t € N \ { 0 l ) .
Suppose n-^ and n j . _ j ^ a r e normal ( t h i s can be assumed w i t h o u t l o s s of g e n e r a l i t y ) .
T h e r e i s p a t h II i n r ( F ) between n^ and n^. of the f o l l o w i n g form:
[ " O » " l ] » [ " i ] » • ' • » [ " t - l ' ^ t J *
n-^ I s I n c i d e n t w i t h nQ,n2 and a n o t h e r node m, s a y .
L e t F ^ ^ ^ = ( P , L , I \ { [ n ^ , n 2 ] } ) and l e t E ^ ^ ^ be a t r i v i a l c o n s t r u c t i o n s e q u e n c e c o n s t r u c t i n g F ^ ^ ^ .
Then z ( l ) c a n be c o n s i d e r e d a s t h e a n t e c e d e n t of a p r o p o s i t i o n T^^^ w h i c h has to have a s a c o n s e q u e n t t h e c o n c l u s i o n t h a t n^ and n , a r e i n c i d e n t .
However, t h e ( p o s s i b l y new) node m^, i n c i d e n t w i t h n2 and m ( a e l e m e n t a r y c o n s t r u c t i o n s t e p ) has to be i n c i d e n t w i t h n^, a c c o r d i n g to T ( s e e a l s o ( 6 . 2 0 ) ) . But t h e n mj^,nQ 61 and m ,m € l , so mj^ = n^ w h i c h y i e l d s n;^,n2]el and T^-"-^ i s t r u e
Remark t h a t i n t h i s p a r t of the p r o o f we need not to demand t h a t e i t h e r n^ or n2 i s n o r m a l .
I n d u c t i v e l y we can p r o c e e d t h i s way a l o n g t h e p a t h II w h i c h i n the end l e a d s to the c o n c l u s i o n t h a t T* i s t r u e .
Hence T I m p l i e s T*. Now r e v e r s i n g the p r o c e s s and we can c o n c l u d e t h a t T* a l s o i m p l i e s T w h i c h p r o v e s the theorem . 6.25. We now t u r n the m a t t e r a r o u n d . L e t F = ( P , L , I ) be a p r o p o s i t i o n f i g u r e of a t r u e p r o p o s i t i o n T of P G ( 2 , R ) . Then, a c c o r d i n g to ( 6 . 2 4 ) w i t h e v e r y i n c i d e n c e j of F t h e r e c o r r e s p o n d s a t r u e p r o p o s i t i o n T* i n t h e s e n s e of ( 6 . 2 0 ) . I n d e e d T* i s e q u i v a l e n t to T, but i t s w o r d i n g c o u l d be q u i t e d i f f e r e n t . So, i n p r i n c i p l e , we have | l | p r o p o s i t i o n s c o n n e c t e d w i t h one p r o p o s i t i o n f i g u r e F and we see t h a t we can c h a r a c t e r i z e t h e c l a s s of t h e s e e q u i v a l e n t theorems by one f i g u r e F and a l s o by the g r a p h s
A(F)
andr ( F ) .
Hence the f o l l o w i n g d e f i n i t i o n . 6.26. D e f i n i t i o n : I n the n o t a t i o n of ( 6 . 2 5 ) . The c l a s s of e q u i v a l e n t p r o p o s i t i o n s c h a r a c t e r i z e d by t h e same graph r ( F ) i s s a i d to be a g r a p h - c l a s s ( o f p r o p o s i t i o n s ) ( c o n n e c t e d w i t h r ( F ) .I t i s s a i d t h a t two p r o p o s i t i o n s of the same g r a p h - c l a s s a r e g r a p h - e q u i v a l e n t .
6.27. L a t e r i n t h i s r e p o r t we s h a l l " g e n e r a t e " p r o p o s i t i o n s of w h i c h I t I s a l m o s t I m p o s s i b l e to g i v e an a d e q u a t e w o r d i n g ( f o r example p r o p o s i t i o n S ( 4 ) I n ( 9 . 3 7 ) ) . T h e r e f o r e we s h a l l , i n s e c t i o n 8, d i s c u s s the
p o s s i b i l i t y to c h a r a c t e r i z e a p r o p o s i t i o n ( o r b e t t e r a c l a s s of g r a p h -e q u i v a l -e n t p r o p o s i t i o n s ) b y i t s g r a p h .
6.28. With r e g a r d to ( 6 . 2 5 ) we must remark t h a t i t l o o k s more p r o m i s i n g t h a n i t a c t u a l l y i s .
F o r example, c o n s i d e r the theorem of D e s a r g u e s S ( 3 ) . A f i g u r e of S ( 3 ) has 30 i n c i d e n c e s , but i t i s w e l l - k n o w n t h a t by an a p p r o p r i a t e l y c h o s e n p e r m u t a t i o n of the p o i n t s and the l i n e s the w o r d i n g of S ( 3 ) a l w a y s can be c h o s e n i n the same way. F o r the theorem of Pappus we have a s i m i l a r s i t u a t i o n .
A l s o i t i s an open q u e s t i o n f o r the moment w h e t h e r the p r o p o s i t i o n s a r e t h e o n l y theorems w i t h a p r o p e r t y a n a l o g o u s to t h e one g i v e n i n ( 6 . 2 4 ) . As i n the c a s e of theorem ( 4 . 8 ) i t i s e a s y to f i n d c o u n t e r e x a m p l e s . F o r example, f o r e v e r y theorem w h i c h i s a s p e c i a l i z a t i o n of a
p r o p o s i t i o n t h e r e a r e i n c i d e n c e s which a r e not the c o n s e q u e n t of t h e theorem, namely t h e s i n g u l a r i n c i d e n c e s .
On t h e o t h e r hand the above d i s c u s s e d t h e o r y i s not l i m i t e d t o P G ( 2 , R ) , but i t seems t h a t f o r any p r o j e c t i v e p l a n e a s i m i l a r t h e o r y can be d e v e l o p e d .
We i n t e n d to r e t u r n to t h e s e problems i n l a t e r r e p o r t s .
7. THE RULER AND COMPASS CONSTRUCTIONS IN P G ( 2 . R )
7.1. W i t h a r u l e r a l o n e one can i n PG(2,R) perform t h e f o l l o w i n g c o n s t r u c t i o n s : 1) c h o o s i n g a p o i n t or a l i n e i n the p l a n e ; 2 ) c h o o s i n g a p o i n t on a g i v e n l i n e or c h o o s i n g a l i n e t h r o u g h a g i v e n p o i n t ; 3 ) c h o o s i n g a p o i n t on two g i v e n l i n e s ( t h e i n t e r s e c t i o n of t h e s e l i n e s ) or c h o o s i n g a l i n e t h r o u g h two g i v e n p o i n t s . 7.2. However, t h e s e a r e p r e c i s e l y the e l e m e n t a r y c o n s t r u c t i o n s d e s c r i b e d i n ( 4 . 3 ) . B e c a u s e of t h i s the f o l l o w i n g h o l d s . 7.3. Theorem: E v e r y t r i v i a l c o n s t r u c t i o n i n PG(2,R) ( i n the s e n s e of s e c t i o n 4) can be performed by r u l e r c o n s t r u c t i o n s a l o n e and the o t h e r way a r o u n d .
7.4. Theorem ( 7 . 3 ) c h a r a c t e r i z e s the t h e o r y of i n t e r s e c t i o n s of PG(2,R) a s t h a t p a r t of the p r o j e c t i v e geometry o v e r R t h e c o n s t r u c t i o n s of w h i c h can be performed by a r u l e r a l o n e . 7.5. B e c a u s e any r u l e r c o n s t r u c t i o n s i n PG(2,R) can be r e p r e s e n t e d by a t r i v i a l c o n s t r u c t i o n ( s e q u e n c e ) we c o n c l u d e t h a t e v e r y r u l e r c o n s t r u c t i o n i n PG(2,R) can be s i m u l a t e d by a computer ( s e e s e c t i o n 1 0 ) .
7.6. T h i s i s even more g e n e r a l . Suppose t h e r e I s g i v e n a c i r c l e and i t s c e n t r e . (Of c o u r s e , we o v e r s t e p h e r e the b o u d e r i e s of t h e c o n c e p t of a p r o j e c t i v e p l a n e b e c a u s e a c i r c l e i s not a p r o j e c t i v e e n t i t y . So, t h a t problem has t o be overcome t o o ) .
7.7. A c c o r d i n g t o C o u r a n t [ 3 ] once a c i r c l e and i t s c e n t r e has been g i v e n one c a n p e r f o r m a l l c o n s t r u c t i o n s by r u l e r and compass by u s i n g a r u l e r a l o n e .
7.8. C o n s e q u e n t l y , once t h e " f o r e i g n " e n t i t y c i r c l e and i t s c e n t r e has been g i v e n a l l t h e c o n s t r u c t i o n s w i t h r u l e r and compass o f PG(2,R) f a l l w i t h i n t h e b o u n d e r i e s o f t h e t h e o r y o f I n t e r s e c t i o n s o f P G ( 2 , R ) . 7.9. A l s o we n o t e t h a t i f i t w e r e p o s s i b l e t o r e p r e s e n t a c i r c l e and i t s c e n t r e i n a c o m p u t e r t h e n a l l t h e r u l e r and compass c o n s t r u c t i o n s i n PG(2,R) c o u l d be s i m i l a t e d i n t h e c o m p u t e r . 7.10. C a r t o n i n h i s p a p e r [ 2 ] seems t o open p e r s p e c t i v e s t o s o l v e t h i s . H o w e v e r , t h i s k i n d o f p r o b l e m s w i l l n o t be d e a l t w i t h i n t h i s r e p o r t .
8. THE CLASSIFICATION OF PROPOSITIONS OF PG(2.R)
8 . 1 . As a l r e a d y r e m a r k e d i n ( 6 . 2 7 ) some p r o p o s i t i o n s o f PG(2,R) a r e h a r d t o f o r m u l a t e . 8.2. A l s o i t i s i m p o r t a n t i n o u r m e t h o d s t o d e t e r m i n e w h i c h p r o p o s i t i o n ( w h i c h g r a p h - c l a s s o f p r o p o s i t i o n s ) we a r e d e a l i n g w i t h o n t h e b a s i s o f t h e k n o w l e d g e a g i v e n d e r i v e d g r a p h o n l y . 8.3. T h e r e f o r e we s e t o u t i n s u b s e q u e n t r e p o r t s t o d e s c r i b e p r o p o s i t i o n s o f PG(2,R) by t h e p r o p e r t i e s o f t h e c o r r e s p o n d i n g d e r i v e d g r a p h s . As f a r as t h i s p r o b l e m i s c o n c e r n e d we make I n t h i s r e p o r t o n l y some p r e l i m i n a r y r e m a r k s on t h i s s u b j e c t .
8.4. F i r s t some t r i v i a l r e m a r k s . The v a l e n c y o f a node i n a d e r i v e d g r a p h I s e i t h e r 1 o r 3. I f a l l n o d e s o f t h e f i g u r e F c o n s i d e r e d a r e n o r m a l t h e n r( F ) = A ( F ) and r( F ) i s a t r l v a l e n t r e g u l a r g r a p h . I n any c a s e t h e g i r t h o f t h e c o m p o n e n t s o f t h e d e r i v e d g r a p h i s a t l e a s t 6. 8.5. A c c o r d i n g t o C o x e t e r [ A ] t h e g r a p h o f t h e t h e o r e m o f Pappus i s a t r l v a l e n t 3 - t r a n s l t l v e g r a p h on 18 nodes and t h e g r a p h o f t h e t h e o r e m o f D e s a r g u e s i s a t r l v a l e n t 3 - t r a n s l t l v e g r a p h on 20 nodes ( s e e a l s o B i g g s [ l ] ) . 8.6. The r e s u l t s o f ( 8 . 5 ) a r e f a r f r o m t r i v i a l . A l s o b e c a u s e t h e p a p e r o f C o x e t e r has n o t b e e n s t u d i e d i n a l l d e p t h s by t h e a u t h o r s a t t h e moment i t i s n o t c l e a r w h e t h e r i t i s p o s s i b l e t o c h a r a c t e r i z e p r o p o s i t i o n s o f PG(2,R) by t h e s e m e t h o d s u n a m b i g u o u s l y .
