                   GENERALISED GEOMETRY
INTRODUCTION Generalised Geometry, which we will discuss in the chapters that follow, is a «New Geometry» that emerged from the need to broaden certain parts of Euclidean Geometry, such as the definition of parallelism and other mathematical concepts, as these are laid out in Euclidean Geometry. The key points of Generalised Geometry as the following: 1.
In Generalised Geometry, the definition of parallelism (as defined by Euclidean Geometry) is extended to include any point set (point, shape, etc). Therefore, some of the weak points of Euclidean Geometry are: Let¢s assume that we have a point A on the plane, and another point M outside of it; then, according to Euclidean Geometry, there is no way of drawing a parallel line from point M to point A. Let¢s
assume that we have a triangle ABC on the plane and a point M outside of it; then, according to Euclidean Geometry, there is no way of drawing a parallel line from point M to triangle ABC. Let¢s assume that we have a rectilinear segment AB on the plane and a point M outside of it.
In this case, according to Euclidean Geometry, thee can only be only parallel line from point M to the rectilinear segment AB (the wellknown
Parallel Postulate). Conversely, however, according to Generalised Geometry, an infinite number of parallel lines can be drawn from point M to the rectilinear segment AB. Apart from the above examples (a), (b) and (c), I could cite several more weaknesses in Euclidean Geometry, with regards to the concept of parallelism. Subsequently, as will be demonstrated below, the concept of parallelism, as defined by Euclidean Geometry, is only a partial definition of
parallelism, as defined in Generalised Geometry. 2. As it is widely known, in Euclidean Geometry the vertices of various geometric shapes (e.g. triangles, quadrilaterals, etc) are geometric points (i.e. unit point sets). Conversely, in Generalised Geometry, the vertices of various geometric shapes can be a number of different point sets (straight lines, circle circumferences, rectilinear segments, etc). These geometric shapes will be known as
generalised geometric shapes, e.g. generalised triangles, generalised squares, etc. Subsequently, geometric shapes, as we know them through Euclidean Geometry, are only a part of the generalised geometric shapes of Generalised Geometry. 3. As will be demonstrated below, Generalised Geometry expands on our wellknown Euclidean and nonEuclidean spaces. Subsequently, Euclidean and nonEuclidean spaces, as we know them, are only a part of
Generalised Euclidean and nonEuclidean spaces of Generalised Geometry. GENERALISED GEOMETRY OF EUCLIDEAN SPACE GENERALISED GEOMETRY OF THE PLANE DEFINITIONS – BASIC CONCEPTS As mentioned above, the cornerstone of Generalised Geometry is the new definition of the concept of parallelism. This definition is as follows: DEFINITION: Any two point sets A and B (on the
plane or in space) and in any space (Euclidean or nonEuclidean) are parallel when, and only when, each point of the point set A is at an equal distance d from the point set B, and, reversely, when each point of point set B is at the same distance d from the point set A. The point sets A and B may be of any kind (unit, multisegment, connective, nonconnective, open, closed, solid, etc). This new definition of parallelism bridges numerous «gaps» of Euclidean Geometry and plays a
fundamental role in the overall structure of Generalised Geometry. POINT SET CATEGORIES According to the above definition, point sets are divided into the following three categories: Parallel point sets. Intersecting point sets. Incompatible point sets.
Definition É: Two point sets Á and Â are parallel when the
definition of parallel point sets, as above, can be applied to them. Definition II: Two point sets Á and Â are intersecting when they share at least one point. (Note: According to this definition, tangential point sets are intersecting point sets in Generalised Geometry). Definition III: Two point sets Á and Â are incompatible, when they share no points and are not parallel. EXAMPLES
Á. Parallel point sets are: 1. Two points A and B on the plane, fig. 1.      
 Fig. 1 2. The ends A and B of a rectilinear segment AB, fig. 2         Fig. 2 3. The centre Á of a circle, and its circumference Â, fig. 3         Fig. 3 4. Opposite sides AB and CD of a square ABCD, fig. 4  
      Fig. 4 5. Circumferences Á and Â of two concentric circles, fig. 5      
   Fig. 5 6. Surfaces Á and B of two concentric spheres, fig. 6 
                 
                     Fig. 6 7. Two parallel straight lines Á and Â on the plane, fig. 7         Fig. 7 8. The three vertices Á, Â, C of an equilateral triangle ABC are parallel to one another, fig. 8         Fig. 8
9. In the figures below, the point sets Á and Â are parallel to one another, fig. 9         Fig. 9 And numerous other parallel point sets Á and Â. Â. Intersecting point sets. 1. Two overlapping solid point sets Á and Â, fig. 10   
     Fig. 10 2.
The circumferences Á and Â of two intersecting or tangential circles, fig. 11         Fig. 11 3. Two intersecting straight lines Á and Â, fig. 12         Fig. 12 4. The cover C of a closed rectilinear segment [AB] and point Á, fig. 13         Fig. 13 5. The cover C
of a circle and its boundary (its circumference) Â, fig. 14         Fig. 14 6. The real axis ox, the closed interval Á = [1,10] the closed interval Â = [4,15], fig. 15        Fig. 15 And numerous other intersecting point sets. C. Incompatible point sets are: 1. A straight line Á
on the plane, and a point Â, outside of it, fig. 16         Fig. 16 2. On the plane, opposite sides AB and CD of a parallelogram ABCD, fig. 17. (Note: In Euclidean Geometry, sides AB and CD are considered parallel, while, according to the definition of parallelism on Generalised Geometry, sides AB and CD are not parallel).         Fig. 17
3. On the plane, the circumference Á on an ellipse and a focal point Â, fig. 18      
   Fig. 18 4. The boundary (the circumference) Á
of a circle, and its interior Â, fig. 19 
 
 fig. 20 6. Two identical triangles Á and Â, fig. 21 
          
             fig. 21 And numerous other examples of incompatible point sets. THE FUNDAMENTAL THEOREM OF GENERALISED GEOMETRY
The fundamental theorem of Generalised Geometry is as follows: THEOREM: From a point Ì, which is outside a point set Á, you can either draw no parallel lines (Í = 0) or you can only draw one (Í = 1) or you can draw an infinite number of parallel lines(Í=). Proof
Let¢s take the sum S of all point sets. Then, according to the definition of parallel point sets, as above: 1. There are points sets to which we can never draw a parallel line from a point Ì. (e.g. when point Ì is on the inside (curved) part of a corner
of the plane, and numerous other examples.) 2. There are points sets to which we can draw one, and only one, parallel line from a point Ì. (e.g. when, on the plane, point Ì is outside of a straight line xx¢ ,(Euclides¢s wellknown parallel postulate) and numerous other examples.) 3. There are points sets to which we can draw infinite parallel lines from a point Ì. (e.g. when, on the plane, point Ì
is outside of a point Á, in which case (as we know), the infinite number of arcs of the circle whose centre is point Á and whose radius is R = (MA), are parallel to point Á.) Subsequently, we may conclude from the above, that there are point sets to which no parallels can be drawn, (Í = 0), point sets to which one and only one parallel can be drawn (Í = 1) and, finally, point sets to which infinite parallels can be drawn (N=
). We will now prove that: When more than one parallel (Í > 1) can be drawn to a point set A from a point M outside of it, then an infinite number of parallels can be drawn (N=). Let¢s assume (fig. 22) that we have a point set Á, (e.g. a semicurve ox) on the plane, to which more than one parallel (Í > 1) can be drawn from a point Ì that is outside of it.        fig. 22 Given that more than one parallel (Í > 1) can be drawn to point set Á, it follows that at least two parallels can be drawn to it (Í = 2).
Thus, one parallel is the semicurve B_{1}x¢ and the second parallel is the semicurve B_{2}x¢ . In this case, as the two parallels B_{1}x¢ and B_{2}x¢ åare different (i.e. they are not the same), then one parallel is a subset of the other parallel. Which means there is a point set S = (B_{2}B_{1}) that belongs to parallel B_{2}x¢ êand not to parallel B_{1}x¢. But since the point set S
consists of infinite points that belong to parallel B_{2}x¢, it follows that infinite parallels can be drawn from point Ì to the point set Á, i.e.
Subsequently, when more than one parallel (Í > 1) can be drawn from point Ì, it follows that an infinite number of parallels can be drawn to point set Á. The above, then, proves the proposed theorem, i.e.: From a point Ì, which is outside a point set Á, you can either draw no parallel lines (Í = 0) or you can only draw one (Í = 1) or you can draw an infinite number of parallel lines (N=) . In addition, a direct consequence of this theorem is: THEOREM: We can never draw a specific number N of parallels, where N = a whole and positive number (Í > 1), from a point M that is outside of a point set A. Based on the above, we can now formulate the following theorems, which are easily proven:
THEOREM: From a point Ì, which is on the inside of a convex rectilinear shape (e.g. inside a pentagon), you can never draw a parallel (Í = 0) to its perimeter. THEOREM: On the plane, from a point Ì, which is outside (in the nonconvex part) of an angle you can only draw one (Í = 1) parallel Á towards it, fig. 23. 
      fig. 23 THEOREM: On the plane, we can draw infinite parallels (N=) to a rectilinear segment AB from a point M, which is outside of it, i.e. A_{1}x, A_{2}x, A_{3}x, A_{4}x,,
fig. 24. Of these parallels, only one, C, is a closed parallel, fig. 25. NOTE: This theorem clearly demonstrates the difference between Euclidean and Generalised Geometry because, according to Euclidean Geometry, you can only draw one parallel to rectilinear segment AB from a point M outside of it (the wellknown Parallel Postulate). This postulate does not apply in Generalised Geometry.         fig. 24         fig. 25 And numerous other theorems. EXAMPLES 1. We have, in space, a straight line xx¢ and a point Ì outside of it, which is located at a distance d from the straight line xx¢. In this case, we can draw infinite parallels (N=), from point Ì to the straight line xx¢. These infinite parallels (surfaces) are subsets of the closed surface of the cylinder, whose axis is the straight line xx¢ and whose radius is R = d. 2. From a point Ì, which is outside a conic (circle, ellipse, parabola, hyperbola), you may or may not be able to draw a parallel to the latter. It depends on the distance d of point Ì from their circumference
(their boundary). 3. From a point Ì that is inside a cube, pyramid, tetrahedron, etc, you can never draw a parallel to its surface. 4. On the plane, from a point Ì, which is outside of a solid point set (e.g. a disc S) you can never draw a parallel to its cover or its interior. In the case when a parallel can be drawn, only one, and always a closed parallel can be drawn from point M to the boundary C
of the point set, fig. 26.      
   fig. 26 5. In a regular tetrahedron ABCD, each of its vertices, is parallel to the point set of the remaining three vertices, fig. 26 (a). 
  
fig. 27 7. On the plane, two identical rectilinear shapes Á and Â can never be parallel to one another, e.g. two identical triangles Á and Â, fig. 28. 
   fig. 28 and numerous other examples. A very interesting problem of Generalised Geometry is the following: PROBLEM We have, on the plane, fig. 28 (a) a smooth and continuous curve C, with ends Á and Â. The object is to determine the maximum distance d_{max}
of a point Ì on the plane, which is outside of curve C, from which we may draw a closed parallel to curve C. 
                         fig. 28 (a) GENERALISED RECTILINEAR SHAPES Let¢s look, for the sake of simplicity, at Euclidean Geometry of the Plane. As it is well known, in Euclidean Geometry, a rectilinear shape (convex) is defined by its vertices A, B, C,
D ... N, which are points, i.e. unit point sets. Conversely, in Generalised Geometry, the vertices of generalised straight line geometric shapes may be any point set, e.g., unit, multisegment, connective, nonconnective, solid, nonsolid, etc. GENERALISED TRIANGLE Let¢s assume we have three point sets on the plane: an ellipse A, a rectilinear segment B and a parabola C
, which are incompatible to one another. In Generalised Geometry, those three point sets A, B and C create a generalised triangle ABC, with sides a,b,c fig. 29.         fig. 29 Fig. 30 shows various generalised triangles ABC and their sides a, b, c.                
fig. 30 FEATURES OF A GENERALISED TRIANGLE The features of a generalised triangle ABC correspond to those of a Euclidean triangle. In detail, these features are as follows: 1. Sides, fig. 31 The distance between point sets (vertices) Á and Â defines side c
. The distance between vertices B and C defines side a. The distance between vertices C and Á defines side b.
       fig. 31 2. Medians, fig. 32 The distance between vertex Á and the midpoint Ì_{1} of side a defines the median ì_{a}.
The distance between vertex Â and the midpoint Ì_{2} of side b defines the median ì_{b}. The distance between vertex C and the midpoint Ì_{3} of side c defines the median ì_{c}.
       fig. 32 3. Heights, fig. 33
The distance between vertex and the straight line of side a is height U_{a}. The distance between vertex B and the straight line of side b is height U_{b}. The distance between vertex C
and the straight line of side c is height U_{c}.
    
   fig. 33 4. Angles, fig. 34
The extensions of sides b and c define the angle of vertex . The extensions of sides c and a define the angle of vertex . The extensions of sides a and b define the angle of vertex .

           fig. 34 5. Bisectrices, fig. 35
The rectilinear segment that lies upon the bicectrix of angle and is contained between vertex Á and the straight line of side a is the bicectrix ä_{a}. The rectilinear segment that lies upon the bicectrix of angle
and is contained between vertex Â and the straight line of side b is the bicectrix ä_{b}. The rectilinear segment that lies upon the bicectrix of angle and is contained between vertex C and the straight line of side ac is the bicectrix ä_{c}.
   

   fig. 35 6. Area, fig. 36
The space between vertices Á, Â, C and sides a, b, c is the area Å. 
                       
fig. 36 7. Incircle, fig. 37 The circle whose circumference p is tangent to all three straight lines defined by sides a, b, c is the incircle.        fig. 37 8. Circumscribed circle, fig. 37 The circle whose circumference p¢, traverses the intersection points A_{1},B_{1},C_{1} of straight lines E_{1},E_{2},E_{3} which are defined, respectively, by sides a, b, c is the circumscribed circle. 9. Escribed circle, fig. 37 The circle whose circumference p¢¢, is tangent to the straight line Å_{1}, which is defined by side a, and the straight lines Å_{2}
_{ }and Å_{3}_{ }are tangent to it on the inside, is the escribed circle of side a. The above also applies to the escribed circle of side b and side c, respectively.
AN INTERESTING OBSERVATION According to the above, a generalised triangle ABC is: a.
Right, when one of its internal angles is a right angle/ b. Equilateral when all three of its sides a, b, c are equal, i.e. a = b = c. c. Isosceles when two of its sides are equal. As it is widely known, in Euclidean Geometry a triangle ABC has three sides, three heights, three medians, three bisectrices, etc Conversely, this is not always the case with generalised triangles ABC
and they may have more than three sides, heights, medians, bisectrices, etc. In Euclidean Geometry, the area E of a triangle ABC is always a positive number, E 0. Conversely, this is not always the case with generalised triangles and there may be a particular triangle ABC
whose sides are all zero, i.e. a = b = c = 0 and its area not zero, Å > 0, (e.g. three circle circumferences A, B, C which touch one another on the inside). The various theorems, conclusions, properties, etc, that apply to a Euclidean triangle ABC are only a part of the theorems, conclusions, properties, etc, that apply to a generalised triangle ABC. In other words, if the vertices A, B, C
of a generalised triangle ABC were to become points (i.e. unit point sets), then the theorems, conclusions, properties, etc that apply to generalised triangles would become the corresponding theorems, conclusions, properties, etc that apply to Euclidean triangles.
GENERALISED POLYGONS Diagonal: The distance between two adjacent sides is the diagonal of a Generalised polygon. Fig. 38
shows the diagonals of a quadrilateral ABCD and a pentagon ABCDE.        fig. 38 EQUAL GENERALISED RECTILINEAR SHAPES Definition: Two generalised rectilinear shapes (triangles, quadrilaters, pentagons, etc) are equal if, when one is placed over the other, all their features match. SIMILAR GENERALISED RECTILINEAR SHAPES Definition:
Two generalised rectilinear shapes (triangles, quadrilaters, pentagons, etc) are similar all their geometrical features are the same, and all corresponding angles are the same. REDUCIBLE AND NONREDUCIBLE POINT SETS Definition: Any point set A on the plane is known as reducible when there is no point M on the plane from which a parallel can be drawn to the point set A. Fig. 39 shows various reducible point sets Á.  
     
fig. 39 Definition: Any point set Á on the plane is known as nonreducible, when there is at least one pint M on the plane from which a parallel can be drawn to point set A. Fig. 40 shows various nonreducible point sets Á.         fig. 40 PARALLEL GENERALISED SHAPES Definition: On the plane, the closed parallel C_{o, }which_{ }is drawn from a point M that is
outside of a Euclidean rectilinear shape D_{o} will be known as a parallel generalised shape C_{o} of shape D_{o. }Figure 41 shows various parallel generalised shapes C_{o} of Euclidean shapes D_{o. }        fig. 41 Definition: The vertices of a generalised parallel shape C_{o}, are the curved segments of the closed parallel C_{o}, which correspond to the vertices of the Euclidean rectilinear shape D_{o},
e.g. the arches A¢,B¢,C¢ are the vertices of the parallel generalised triangleA¢B¢C¢, etc. Parallel generalised shapes are of particular interest in Generalised Geometry. Thus, for example, in the generalised triangle A¢B¢C¢, (C_{o}) of fig. 41, after working out all of its geometric features (e.g. sides, heights, medians, etc), we compare them to the corresponding geometric features of the Euclidean triangle ABC, (D_{o}
) and identify the relations between them. It is worth pointing out, at this stage, that parallel generalised shapes are the «bridge» between Euclidean and Generalised Geometry. Finally, a number of interesting theorems, conclusions, properties, etc, can be formulated in relation to parallel generalised shapes. GENERALISED GEOMETRY OF SPACE What we discussed above in relation to Generalised Geometry of the plane also applies to space.
Thus, for example, the centre Á of a sphere and its surface Â are parallel point sets, fig. 41 (a). Also, the surface Â of a «capsule» and its axis Â are parallel point sets, fig. 41 (a). The same applies to axis Á of a cylinder and its surface Â, fig. 41 (a).   
 
   fig. 41 (a) Also, fig. 41 (b) shows various generalised triangles ABC in space. 
   
   fig. 41 (b) ANALYTICAL AND GENERALISED GEOMETRY In this chapter we will look at certain examples that demonstrate how Analytical and Generalised Geometry may be combined. This is a very interesting field, as it allows us to formulate numerous significant theorems, conclusions, etc. It is clearly a field given to more indepth
mathematical research. EXAMPLES In the coordinate system we have: The circle A : x^{2} + y^{2} =4^{2}The herbola B : y=10x^{2}The point C : (15,0) The object is to work out the sides and heights of the generalised triangle ABC On the
real axis xx¢ work out the sides and the medians of generalised triangle ABC, whose vertices are the point sets: A : ( ,2] B : [3,8] C : [10,+ ] In the coordinate system xoy we have the ellipse: with focal points Â and C. Work out the sides a, b, c and the area Å of the generalised triangle ABC.
On the real axis xx¢ we have: Á:
The sum of terms of the sequence , Â: The sum of terms of the sequence, C: The sum of terms of the sequence,
Work out the sides of generalised triangle ABC.. In the coordinate system xoy we have the equation:

   where: 
 
   Work out the medians of generalised triangle ABC, where , are the roots of the above equation (1). On the complex plane, a generalised triangle
ABC has the vertices:
Á = 5 + 3i Â = 5 + 8i C = 5 – 10i Work out its medians. In the coordinate system xoy we have the straight lines:

   Work out the sides a, b, c and the area Å of the generalised triangle ABC. Â. GENERALISED GEOMETRY OF NONEUCLIDEAN SPACES The reasoning we applied to Generalised Geometry of Euclidean spaces in the preceding chapters
also applies to nonEuclidean spaces. Thus, for example, we may have parallels, generalised triangles, etc, which have sides, heights, medians, etc, on any surface. Fig. 42 shows various generalised triangles ABC on various surfaces (S). 
   fig. 42 Obviously, sides a, b, c of generalised triangles ABC are the geodesic lines that connect vertices A, B, C to one another. GENERALISED SPACES Á. GENERALISED EUCLIDEAN SPACES 1.
GENERALISED EUCLIDEAN SPACES, TYPE G_{E,I}. Let¢s assume, fig. 43, that we have a Euclidean metric space of two dimensions, e.g. the plane (Å), which we consider a rectilineargeneratrix surface. 
   fig. 43 We now imagine a generalised triangle ABC whose vertices A, B, C are three random parallels A, B, C of the plane (Å). As we all know, the sides of generalised triangle ABC are a, b, c. As shown on fig. 43, from the sum S of generalised triangles A_{i}B_{i}C_{i} ,
((i = 1, 2, 3, …) which make up plane (Å), whichever the position of vertices A, B, C in all those generalised triangles, each one of their sides is always smaller than or equal to the sum of its two remaining sides. Thus, for example, in fig. 43 , for the generalised triangle ABC, we have 
   b = a + c c < a + b a < c + b   (A) 

 
 In this case, according to the above, it can be stated that the Euclidean space (Å) fig. 43 is a generalised twodimensional Euclidean space of type G_{E,I} , whose features are the parallel straight lines A_{i}B_{i}C_{i} . The above leads us to the following definition: DEFINITION: For a given partition Ä, an
ndimensional Euclidean space will be known as a Generalised ndimensional Euclidean space, of type G_{E,I} , when and only when each side of every generalised triangle A_{i}B_{i}C_{i} (i = 1, 2, 3,…) whose vertices A_{i ,}B_{i ,}C_{i} are features of partition Ä, is smaller or equal to the sum of its two remaining sides. The following figures show various Generalised
twodimensional Euclidean spaces of type G_{E,I}, with their corresponding partition Ä. 
   fig. 44 2. GENERALISED EUCLIDEAN SPACES, TYPE G_{E,II} Let¢s assume, fig. 45, that we have a twodimensional, metric Euclidean space, i.e. the plane (Å). 
   fig. 45 We divide the plane (Å) into parallel zones, e.g. of width d. We now take a random generalised triangle ABC whose vertices A, B, C are three of those parallel zones. As shown in fig. 45, from the sum S of generalised triangles A_{i}B_{i}C_{i},
(i = 1, 2, 3,…) which make up the plane (Å), whichever the position of their vertices A, B, C might be, only one of their sides is always larger that the sum of their two remaining sides. Thus, e.g. in figure 45, for the generalised triangle ABC, we have: 

  b > a + c c < b + a a < b + c   (B) 

  
In this case, in accordance with the above, it can be stated that the Euclidean space (E), fig. 45, is a generalised twodimensional Euclidean space of type G_{E,II}, whose features are the parallel zones A_{i ,}B_{i ,}C_{i} . The above leads us to the following definition: DEFINITION: For a given partition Ä, an ndimensional Euclidean space will be known as
a Generalised ndimensional Euclidean space, of type G_{E,II}, when and only when, for each generalised triangle A_{i}B_{i}C_{i}, (i= 1, 2, 3, …) whose vertices A_{i, }B_{i, }C_{i} are features of the partition Ä, only one of its sides is larger than the sum of its two remaining sides. The following figures show various Generalised twodimensional Euclidean spaces of type G_{E,II},
and their corresponding partition Ä (with zones of width d). 
   
   fig. 46 Â. GENERALISED RIEMANN SPACES 1. GENERALISED RIEMANN SPACES, TYPE G_{R,I}The same reasoning we used on twodimensional Generalised Euclidean spaces applies to twodimensional Generalised Riemann spaces. A simple example is the following: Let¢s take, fig. 47, the surface S of a sphere, which we divide into parallel circles. We now take a random generalised triangle ABC,
whose vertices A, B, C are three of the above circles. As shown in figure 47, in the sum P of generalised triangles A_{i}B_{i}C_{i}, (i = 1, 2, 3,…) that make up the surface S of the sphere, whichever the position of vertices A, B, C of each of these generalised triangles, each of their sides is always smaller than or equal to the sum of the two remaining sides.
Thus, e.g., in fig. 47 for the spherical generalised triangle ABC, we have: 
  
b = c + a c < a + b a < c + b   (C) 

   In this case, it can be said that the twodimensional Riemann space (i.e. the surface S of the sphere), fig. 47, is a generalised twodimensional
Riemann space of type G_{R,I} whose features are the parallel circles A_{i}B_{i}C_{i} . 
   fig. 47 The above leads us to the following definition: DEFINITION: For a given partition Ä, an ndimensional Riemann space will be known as a Generalised ndimensional Riemann space of type G_{R,I} , when and only when, , for each generalised triangle A_{i}B_{i}C_{i}, (i = 1, 2, 3,…) whose vertices A_{i ,}B_{i ,}C_{i} are features of the partition Ä, each one of its three sides is smaller than, or equal to the sum of its two remaining sides.
Figure 48 shows various generalised twodimensional Riemann spaces of type G_{R,I}, on a number of different surfaces. 
   fig. 49 We divide the surface (S) into parallel zones of width d. We now take a random generalised triangle ABC whose vertices A, B, C are three of the above parallel zones. As shown in fig. 49 from the sum S
of generalised triangles A_{i}B_{i}C_{i}, (i = 1, 2, 3,…) which make up the surface (S), whichever the position of their vertices A, B, C of all those triangles, only one of their sides is larger than the sum of their two remaining sides. Thus, for example, in fig. 49 for generalised triangle ABC we have: 
   b > a + c c < b + a a < b + c   (B) 

   In this case, and in accordance with the above, we can say that the Riemann space (S), fig. 49, is a generalised, twodimensional Riemann space, of type G_{R,II} whose features are the parallel zones A_{i ,}B_{i ,}C_{i} . Based on the above, we are led to the following definition:
DEFINITION: For a given partition Ä, an ndimensional Riemann space will be known as a Generalised ndimensional Riemann space of type G_{R,II} when and only when, for each generalised triangle A_{i}B_{i}C_{i}, (i = 1, 2, 3, …) whose vertices A_{i ,}B_{i ,}C_{i} , are features of the partition Ä, only one of its sides is larger than the sum of its remaining two sides. Fig. 50 shows various
Generalised twodimensional Riemann spaces of type G_{R,II}, on a number of different surfaces. 
   
   fig. 50 AN INTERESTING OBSERVATION It is clear that this area of Mathematics, i.e. Generalised Euclidean Spaces, type G_{E,I} and G_{E,II} and Generalised Riemann Spaces, type G_{R,I}, and G_{R,II}, can be the object of
indepth mathematical research, which may produce numerous interesting mathematical conclusions.
EPILOGUE Generalised Geometry, which we discussed at length in the preceding chapters, is a «New Geometry». We have looked at its basic principles and the reasoning behind this new field of Mathematics. As readers have surely realised, Generalised Geometry is a very wide field of research, within may lead to a
number of new Theorems, findings, definitions, properties, conclusions, etc. At this stage, it¢s still early days for Generalised Geometry. Only time can tell what contribution Generalised Geometry will make towards the development of Mathematics.
Copyright 2007: Christos A. Tsolkas Christos A. Tsolkas
June 2007

