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 : A, B, C, ...
- a set of : h, k, l, ...
- a set of : 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',
, Vol. 8, No. 1-2-3, 212-215, 1997.
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