SMARANDACHE ANTI-GEOMETRY
                                  
                               by         
                            
        Sandy P. Chimienti                       Mihaly Bencze     
        Mathematics and Science Department       6, Hatmanului Street
        University of New Mexico                 2212 Sacele 3
        Gallup, NM  87301, USA                   Jud. Brasov, Romania


Abstract:  This is an experimental geometry.  All Hilbert's 20 axioms of the Euclidean 
  GGeometry are denied in this vanguardist geometry of the real chaos!  What is even more 
  intriguing?  F.Smarandache[5] has even found in 1969 a model of it!

Key Words:  Hilbert's Axioms, Euclidean Geometry, Non-Euclidean Geometry, 
            Smarandache Geometries, Geometrical Model

Introduction:
   Here it is exposed the Smarandache Anti-Geometry:
   It is possible to entirely de-formalize Hilbert's groups of axioms
of the Euclidean Geometry, and to construct a model such that none of
his fixed axioms holds.
    Let's consider the following things: 
      - a set of <points>:  A, B, C, ... 
      - a set of <lines>:  h, k, l, ...
      - a set of <planes>:  alpha, beta, gamma, ...
    and 
      - a set of relationships among these elements:  "are situated",
           "between", "parallel", "congruent", "continuous", etc.
Then, we can deny all Hilbert's twenty axioms [see David Hilbert, 
"Foundations of Geometry", translated by E. J. Townsend, 1950;
and Roberto Bonola, "Non-Euclidean Geometry", 1938].
There exist cases, within a geometric model, when the same axiom is 
verified by certain points/lines/planes and denied by others. 
      

      GROUP I.  ANTI-AXIOMS OF CONNECTION:
         
         I.1.  Two distinct points A and B do not always completely
                determine a line.
               
                Let's consider the following model MD:
                get an ordinary plane delta, but with an infinite 
                hole in of the following shape:









                                    P
   p                              .
     semi-plane delta1   . .           
   l                              .   .
                                 .     .       curve f1 (frontier)
   a    .........................       ...........................
                                N            
   n                                                 
                                                .I         
   e                                                   .J     
                                                            
                                                              
                                                   .K            
   d                                                           
                                                          
   e                                                      
                                                        
   l                                                    
        .........................       ...........................                     
   t                             .     .        curve f2 (frontier)
                                  .   .                            
   a  semi-plane delta2  . .
                                    .
                                    Q


                Plane delta is a reunion of two disjoint planar
                semi-planes;
                f1 lies in MD, but f2 does not;      
                P, Q are two extreme points on f that belong to MD.

                  One defines a LINE l as a geodesic curve:  if two 
                points A, B that belong to MD lie in l, then
                the shortest curve lied in MD between A and B
                lies in l also.
                If a line passes two times through the same point, then 
                it is called double point (KNOT).
                  

                  One defines a PLANE alpha as a surface such that for 
                any two points A, B that lie in alpha and belong to 
                MD there is a geodesic which passes through A, B and 
                lies in alpha also.
                
                Now, let's have two strings of the same length:
                one ties P and Q with the first string s1 such that
                the curve s1 is folded in two or more different 
                planes and s1 is under the plane delta;
                next, do the same with string s2, tie Q with P, but 
                over the plane delta and such that s2 has a different
                form from s1;
                and a third string s3, from P to Q, much longer than s1.
                s1, s2, s3 belong to MD. 

                Let I, J, K be three isolated points -- as some islands,
                i.e. not joined with any other point of MD,
                exterior to the plane delta.
                  
                  This model has a measure, because the (pseudo-)line is
                the shortest way (lenth) to go from a point to another
                (when possible).

Question 37:
                Of course, this model is not perfect, and is far from 
                the best.  Readers are asked to improve it, or to make up
                a new one that is better.

               (Let A, B be two distinct points in delta1-f1.  P and Q are
                two points on s1, but they do not completely determine a
                line, referring to the first axiom of Hilbert,
                because A-P-s1-Q are different from B-P-s1-Q.)


         I.2.  There is at least a line l and at least two distinct 
                 points A and B of l, such that A and B do not 
                 completely determine the line l. 
               
               (Line A-P-s1-Q are not completely determined by P and Q
                in the previous construction, because B-P-s1-Q is another
                line passing through P and Q too.)


         I.3.  Three points A, B, C not situated in the same line do 
                 not always completely determine a plane alpha
               
               (Let A, B be two distinct points in delta1-f1, such that
                A, B, P are not co-linear.  There are many planes 
                containing these three points: delta1 extended with any 
                surface s containing s1, but not cutting s2 in between
                P and Q, for example.)


         I.4.  There is at least a plane, alpha, and at least three 
                 points A, B, C in it not lying in the same line, such
                 that A, B, C do not completely determine the plane        
                 alpha.

                (See the previous example.)


         I.5.  If two points A, B of a line l lie in a plane alpha, 
                doesn't mean that every point of l lies in alpha.

               (Let A be a point in delta1-f1, and B another point on
                s1 in between P and Q.  Let alpha be the following plane: 
                delta1 extended with a surface s containing s1, but not 
                cutting s2 in between P and Q, and tangent to delta2 on
                a line QC, where C is a point in delta2-f2.
                Let D be point in delta2-f2, not lying on the line QC.
                Now, A, B, D are lying on the same line A-P-s1-Q-D, 
                A, B are in the plane alpha, but D do not.)


         I.6.  If two planes alpha, beta have a point A in common,
                doesn't mean they have at least a second point in
                common.

               (Construct the following plane alpha: a closed surface 
                containing s1 and s2, and intersecting delta1 in one point
                only, P.  Then alpha and delta1 have a single point in 
                common.)


         I.7.  There exist lines where lies only one point, 
                or planes where lie only two points,
                or space where lie only three points.

               (Hilbert's I.7 axiom may be contradicted if the model
                has discontinuities.  
                Let's consider the isolated points area. 
                  The point I may be regarded as a line, because it's not 
                possible to add any new point to I to form a line.
                  One constructs a surface that intersects the model only 
                in the points I and J.)  
                  

      
      GROUP II.  ANTI-AXIOMS OF ORDER:
         
        II.1.  If A, B, C are points of a line and B lies between A 
                 and C, doesn't mean that always B lies also between 
                 C and A.

                 [Let T lie in s1, and V lie in s2, both of them
                  closer to Q, but different from it.  Then:
                  P, T, V are points on the line P-s1-Q-s2-P 
                  ( i.e. the closed curve that starts from the point P 
                    and lies in s1 and passes through the point Q and
                    lies back to s2 and ends in P ),
                  and T lies between P and V 
                    -- because PT and TV are both geodesics --,
                  but T doesn't lie between V and P
                    -- because from V the line goes to P and then to T,
                       therefore P lies between V and T.]

               [By definition: a segment AB is a system of points
                lying upon a line between A and B (the extremes are
                included).
                
                Warning:  AB may be different from BA;
                for example: 
                  the segment PQ formed by the system of points 
                  starting with P, ending with Q, and lying in s1, 
                  is different from the segment QP formed by the
                  system of points starting with Q, ending with P,
                  but belonging to s2.
                Worse, AB may be sometimes different from AB;
                for example:
                  the segment PQ formed by the system of points 
                  starting with P, ending with Q, and lying in s1, 
                  is different from the segment PQ formed by the
                  system of points starting with P, ending with Q,
                  but belonging to s2.]


        II.2.  If A and C are two points of a line, then: 
                 there does not always exist a point B lying between A 
                 and C, 
               or there does not always exist a point D such that C lies 
                 between A and D.

               [For example:
                let F be a point on f1, F different from P, 
                and G a point in delta1, G doesn't belong to f1;
                draw the line l which passes through G and F;
                then: 
                there exists a point B lying between G and F
                  -- because GF is an obvious segment --,
                but there is no point D such that F lies between
                G and D -- because GF is right bounded in F 
                ( GF may not be extended to the other side of F,
                 because otherwise the line will not remain a 
                 geodesic anymore ).]
                

        II.3.  There exist at least three points situated on 
                 a line such that: 
                 one point lies between the other two,
                 and another point lies also between the other two.

               [For example:
                let R, T be two distinct points, different 
                from P and Q, situated on the line P-s1-Q-s2-P, 
                such that the lenghts PR, RT, TP are all equal;
                then:
                  R lies between P and T,
                  and T lies between R and P;
                  also P lies between T and R.]


        II.4.  Four points A, B, C, D of a line can not always be
                 arranged: 
                 such that B lies between A and C and also 
                 between A and D, 
                 and such that C lies between A and D and also between
                 B and D.

               [For examples:
                - let R, T be two distinct points, different 
                from P and Q, situated on the line P-s1-Q-s2-P such
                that the lenghts PR, RQ, QT, TP are all equal,
                therefore R belongs to s1, and T belongs to s2;
                then P, R, Q, T are situated on the same line:
                  such that R lies between P and Q, but not between
                   P and T 
                    -- because the geodesic PT does not pass through
                        R --,
                  and such that Q does not lie between P and T
                    -- because the geodesic PT does not pass through
                        Q --,
                   but lies between R and T;
                 - let A, B be two points in delta2-f2 such that A, Q, B
                 are colinear, and C, D two points on s1, s2 respectively,
                 all of the four points being different from P and Q;
                 then A, B, C, D are points situated on the same line
                 A-Q-s1-P-s2-Q-B, which is the same with line 
                 A-Q-s2-P-s1-Q-B, therefore we may have two different
                 orders of these four points in the same time:
                 A, C, D, B and A, D, C, B.]


        II.5.  Let A, B, C be three points not lying in the same 
                line, and l a line lying in the same plane ABC and
                not passing through any of the points A, B, C.  
                Then, if the line l passes through a point of the 
                segment AB, it doesn't mean that always the line l 
                will pass through either a point of the segment BC 
                or a point of the segment AC.

               [For example:
                let AB be a segment passing through P in the 
                semi-plane delta1, and C a point lying in delta1
                too on the left side of the line AB; 
                thus A, B, C do not lie on the same line;
                now, consider the line Q-s2-P-s1-Q-D, where D is
                a point lying in the semi-plane delta2 not on f2:
                therefore this line passes through the point P of
                the segment AB, but do not pass through any point
                of the segment BC, nor through any point of the 
                segment AC.]
      
      
      GROUP III.  ANTI-AXIOM OF PARALLELS.

               In a plane alpha there can be drawn through a point
               A, lying outside of a line l, either no line,
               or only one line, or a finite number of lines,
               or an infinite number of lines which do not intersect
               the line l.  (At least two of these situations should occur.)
               The line(s) is (are) called the parallel(s) to l
               through the given point A.

               [ For examples:
                - let l0 be the line N-P-s1-Q-R, where N is a point 
                   lying in delta1 not on f1, and R is a similar
                   point lying in delta2 not on f2, 
                   and let A be a point lying on s2,
                   then:  no parallel to l0 can be drawn through A
                   (because any line passing through A, hence through
                   s2, will intersect s1, hence l0, in P and Q);
                - if the line l1 lies in delta1 such that l1 does 
                   not intersect the frontier f1, then:
                   through any point lying on the left side of l1 
                   one and only one parallel will pass;
                - let B be a point lying in f1, different from P,
                   and another point C lying in delta1, not on f1;
                   let A be a point lying in delta1 outside of BC;
                   then:  an infinite number of parallels to the 
                   line BC can be drawn through the point A.
          

                 Theorem.  There are at least two lines l1, l2 of a 
                   plane, which do not meet a third line l3 of the 
                   same plane, but they meet each other, 
                   ( i.e. if l1 is parallel to l3, and l2 is parallel 
                     to l3, and all of them are in the same plane, 
                     it's not necessary that l1 is parallel to l2 ).
                 [ For example: 
                    consider three points A, B, C lying in f1, and
                    different from P, and D a point in delta1 not on 
                    f1;  draw the lines AD, BE and CE such that
                    E is a point in delta1 not on f1 and both BE
                    and CE do not intersect AD;
                    then:  BE is parallel to AD, CE is also parallel
                    to AD, but BE is not parallel to CE because the
                    point E belong to both of them. ]


      GROUP IV.  ANTI-AXIOMS OF CONGRUENCE           
                                  
               IV.1. If A, B are two points on a line l, and A' is a
                        point upon the same or another line l', then:
                        upon a given side of A' on the line l', we can 
                        not always find only one point B' so that the
                        segment AB is congruent to the segment A'B'.

                   [ For examples:
                    - let AB be segment lying in delta1 and having no 
                      point in common with f1, and construct the line 
                      C-P-s1-Q-s2-P (noted by l') which is the same 
                      with C-P-s2-Q-s1-P, where C is a point lying in 
                      delta1 not on f1 nor on AB; 
                      take a point A' on l', in between C and P, such 
                      that A'P is smaller than AB;
                      now, there exist two distinct points B1' on s1 
                      and B2'on s2, such that A'B1' is congruent to AB
                      and A'B2' is congruent to AB,
                      with A'B1' different from A'B2';
                    - but if we consider a line l' lying in delta1 and
                      limited by the frontier f1 on the right side 
                      (the limit point being noted by M),
                      and take a point A' on l', close to M, such that
                      A'M is less than A'B', then:  there is no point
                      B' on the right side of l' so that A'B' is 
                      congruent to AB. ]

                      A segment may not be congruent to itself!

                   [ For example:
                      - let A be a point on s1, closer to P,
                        and B a point on s2, closer to P also;
                        A and B are lying on the same line A-Q-B-P-A
                        which is the same with line A-P-B-Q-A,
                        but AB meseared on the first representation
                        of the line is strictly greater than AB
                        meseared on the second representation of
                        their line. ]

               IV.2.  If a segment AB is congruent to the segment
                        A'B' and also to the segment A''B'', then 
                        not always the segment A'B' is congruent to
                        the segment A''B''.

                    [ For example:                 
                    - let AB be a segment lying in delta1-f1, and
                      consider the line C-P-s1-Q-s2-P-D, where C, D are
                      two distinct points in delta1-f1 such that C, P, D
                      are colinear.  Suppose that the segment AB is
                      congruent to the segment CD (i.e. C-P-s1-Q-s2-P-D).
                      Get also an obvious segment A'B' in delta1-f1,
                      different from the preceding ones, but congruent 
                      to AB.
                      Then the segment A'B' is not congruent to the segment
                      CD (considered as C-P-D, i.e. not passing through Q.)
                

               
               IV.3.  If AB, BC are two segments of the same line l
                        which have no points in common aside from the
                        point B,
                        and A'B', B'C' are two segments of the same line
                        or of another line l' having no point other than
                       B' in common, such that AB is congruent to A'B'
                       and BC is congruent to B'C', 
                       then not always the segment AC is congruent to
                       A'C'.

                    [ For example:
                       let l be a line lying in delta1, not on f1,
                       and A, B, C three distinct points on l, such 
                       that AC is greater than s1;
                       let l' be the following line: A'-P-s1-Q-s2-P
                       where A' lies in delta1, not on f1,
                       and get B' on s1 such that A'B' is congruent
                       to AB, get C' on s2 such that BC is congruent
                       to B'C' (the points A, B, C are thus chosen);
                       then:  the segment A'C' which is first seen as
                       A'-P-B'-Q-C' is not congruent to AC, 
                       because A'C' is the geodesic A'-P-C' (the 
                       shortest way from A' to C' does not pass 
                       through B') which is strictly less than AC. ]

              Definitions.    Let h, k be two lines having a point O
                            in common.  Then the system (h, O, k) is
                            called the angle of the lines h and k in
                            the point O.
                            ( Because some of our lines are curves,
                              we take the angle of the tangents to 
                              the curves in their common point. )
                              
                              The angle formed by the lines h and k
                            situated in the same plane, noted by 
                            <(h, k), is equal to the arithmetic mean 
                            of the angles formed by h and k in all
                            their common points.


               IV.4.  Let an angle (h, k) be given in the plane alpha,
                      and let a line h' be given in the plane beta.
                      Suppose that in the plane beta a definite side
                      of the line h' be assigned, and a point O'.
                      Then in the plane beta there are one, or more, 
                      or even no half-line(s) k' emanating from the 
                      point O' such that the angle (h, k) is 
                      congruent to the angle (h', k'), 
                      and at the same time the interior points of
                      the angle (h', k') lie upon one or both sides
                      of h'.

                   [ Examples:
                    - Let A be a point in delta1-f1, and B, C two
                      distinct points in delta2-f2;
                      let h be the line A-P-s1-Q-B, and k be the
                      line A-P-s2-Q-C;  because h and k intersect
                      in an infinite number of points (the segment 
                      AP), where they normally coincide -- i.e. in
                      each such point their angle is congruent to
                      zero, the angle (h, k) is congruent to zero.
                      Now, let A' be a point in delta1-f1, different 
                      from A, and B' a point in delta2-f2, different
                      from B, and draw the line h' as A'-P-s1-Q-B';
                      there exist an infinite number of lines k', of
                      the form A'-P-s2-Q-C' (where C' is any point in
                      delta2-f2, not on the line QB'), such that the
                      angle (h, k) is congruent to (h', K'),
                      because (h', k') is also congruent to zero, and
                      the line A'-P-s2-Q-C' is different from the line
                      A'-P-s2-Q-D' if D' is not on the line QC'.
                    - If h, k, and h' are three lines in delta1-P, 
                      which intersect the frontier f1 in at most one 
                      point, then there exist only one line k' on a 
                      given part of h' such that the angle (h, k) is 
                      congruent to the angle (h', k').
                    - *Is there any case when, with these hypotheses,                     
                      no k' exists ?                           
                    - Not every angle is congruent to itself;                             
                      for example: 
                        <(s1, s2) is not congruent to <(s1, s2)
                      [because one can construct two distinct lines:
                       P-s1-Q-A and P-s2-Q-A, where A is a point in 
                       delta2-f2, for the first angle, which becomes equal
                       to zero;
                       and P-s1-Q-A and P-s2-Q-B, where B is another point
                       in delta2-f2, B different from A, for the second
                       angle, which becomes strictly greater than zero!].


                                           
               IV. 5. If the angle (h, k) is congruent to the angle 
                      (h', k',) and the angle (h'', k''), then the 
                      angle (h', k') is not always congruent to the
                      angle (h'', k'').

                      (A similar construction to the previous one.)


               IV. 6. Let ABC and A'B'C' be two triangles such that
                        AB is congruent to A'B', 
                        AC is congruent to A'C',
                        <BAC is congruent to <B'A'C'.
                      Then not always
                        <ABC is congruent to <A'B'C'
                        and <ACB is congruent to <A'C'B'.

                      [For example:
                       Let M, N be two distinct points in delta2-f2, thus 
                       obtaining the triangle PMN;
                       Now take three points R, M', N' in delta1-f1, such
                       that RM' is congruent to PM, RN' is congruent to RN,
                       and the angle (RM', RN') is congruent to the angle
                       (PM, PN).  RM'N' is an obvious triangle.
                       Of course, the two triangle are not congruent,
                       because for example PM and PN cut each other twice
                       -- in P and Q -- while RM' and RN' only once -- in 
                       R.
                       (These are geodesical triangles.)]

                Definitions: 

                  Two angles are called supplementary if they have the 
                  same vertex, one side in common, and the other sides
                  not common form a line.

                  A right angle is an angle congruent to its 
                  supplementary angle.

                  Two triangles are congruent if its angles are congruent
                  two by two, and its sides are congruent two by two.


                Propositions:

                  A right angle is not always congruent to another 
                  right angle.

                  For example:
                  Let A-P-s1-Q be a line, with A lying in delta1-f1,
                  and B-P-s1-Q another line, with B lying in 
                  delta1-f1 and B not lying in the line AP;
                  we consider the tangent t at s1 in P, and B chosen
                  in a way that <(AP, t) is not congruent to <(BP, t);
                  let A', B' be other points lying in delta1-f1
                  such that <APA' is congruent to <A'P-s1-Q,
                  and <BPB' is congruent to <B'P-s1-Q.
                  Then: 
                   - the angle APA' is right, because it is congruent 
                   to its supplementary (by construction);
                   - the angle BPB' is also right, because it is
                   congruent to its supplementary (by construction);
                   - but <APA' is not congruent to <BPB', 
                   because the first one is half of the angle A-P-s1-Q,
                   i.e. half of <(AP, t),
                   while the second one is half of the B-P-s1-Q,
                   i.e. half of <(BP, t).

                   The theorems of congruence for triangles 
                   [side, side, and angle in between;  angle, angle, and 
                   common side;  side, side, side] may not hold either
                   in the Critical Zone (s1, s2, f1, f2) of the Model.

                   Property:
                   The sum of the angles of a triangle can be:
                   - 180 degrees, if all its vertexes A, B, C are
                   lying, for example, in delta1-f1;
                   - strictly less than 180 degrees [ any value in the
                   interval (0, 180) ], 
                   for example:
                   let R, T be two points in delta2-f2 such that Q does
                   not l1e in RT, and S another point on s2;
                   then the triangle SRT has <(SR, ST) congruent to 0
                   because SR and ST have an infinite number of common
                   points (the segment SQ), and <QTR + <TRQ congruent
                   to 180-<TQR [ by construction we may vary <TQR in the
                   interval (0, 180) ];
                   - even 0 degree! 
                   let A be a point in delta1-f1, B a point in delta2-f2,
                   and C a point on s3, very close to P;
                   then ABC is a non-degenerate triangle (because its 
                   vertexes are non-colinear), but <(A-P-s1-Q-B, A-P-s3-C) 
                   = <(B-Q-s1-P-A, B-Q-s1-P-s3-C) = <(C-s3-P-A, 
                   C-s3-P-s1-Q-B) = 0
                   (one considers the lenth C-s3-P-s1-Q-B strictly less 
                   than C-s3-B);
                   the area of this triangle is also 0 !
                   - more than 180 degrees, 
                   for example:
                   let A, B be two points in delta1-f1, such that
                   <PAB + <PBA + <(s1, s2; in Q) is strictly greater
                   than 180 degrees;
                   then the triangle ABQ, formed by the intersection of 
                   the lines A-P-s2-Q, Q-s1-P-B, AB will have the sum of
                   its angles strictly greater than 180 degrees.

                   Definition:
                   A circle of center M is a totality of all points A 
                   for which the segments MA are congruent to one another.

                   For example, if the center is Q, and the lenth of the
                   segments MA is chosen greater than the lenth of s1, 
                   then the circle is formed by the arc of circle centered
                   in Q, of radius MA, and lying in delta2, plus another 
                   arc of circle centered in P, of radius MA-lenth of s1,
                   lying in delta1.


     GROUP V. ANTI-AXIOM OF CONTINUITY (ANTI-ARCHIMEDEAN AXIOM)
            
               Let A, B be two points. Take the points A1, A2, A3, A4, 
               ... so that A1 lies between A and A2, A2 lies between
               A1 and A3, A3 lies between A2 and A4, etc. and the 
               segments AA1, A1A2, A2A3, A3A4, ... are congruent to one
               another.
               Then, among this series of points, not always  there exists
               a certain point An such that B lies between A and An.

               For example:
               let A be a point in delta1-f1, and B a point on f1, B 
               different from P;
               on the line AB consider the points A1, A2, A3, A4, ...
               in between A and B, such that AA1, A1A2, A2A3, A3A4, etc.
               are congruent to one another; 
               then we find that there is no point behind B (considering 
               the direction from A to B), because B is a limit point 
               (the line AB ends in B).

               The Bolzano's (intermediate value) theorem may not hold in
               the Critical Zone of the Model.

 
Can you readers find a better model for this  anti-geometry?


References:
  [1] Ashbacher, Charles, "Smarandache Geometries", <Smarandache Notions
      Journal>, Vol. 8, No. 1-2-3, 212-215, 1997.
  [2] Chimienti, Sandy P., Bencze, Mihaly, "Smarandache Anti-Geometry",
      <Bulletin of Pure and Applied Sciences>, Delhi, India, Vol. 17E,
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