GENERALISED GEOMETRY

Transcription

1 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: a. 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. b. 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. c. 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 well-known 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 well-known Euclidean and non-euclidean spaces. Subsequently, Euclidean and non-euclidean spaces, as we know them, are only a part of Generalised Euclidean and non-euclidean spaces of Generalised Geometry.

2 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 non-euclidean) 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, multi-segment, 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: 1. Parallel point sets. 2. Intersecting point sets. 3. Incompatible point sets. Definition Ι: Two point sets Α and Β are parallel when the definition of parallel point sets, as above, can be applied to them. Definition II: Two point sets Α and Β 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 Α and Β are incompatible, when they share no points and are not parallel. EXAMPLES Α. Parallel point sets are: 1. Two points A and B on the plane, fig. 1.

3 Fig The ends A and B of a rectilinear segment AB, fig. 2 Fig The centre Α of a circle, and its circumference Β, fig. 3 Fig Opposite sides AB and CD of a square ABCD, fig. 4 Fig Circumferences Α and Β of two concentric circles, fig. 5

4 Fig Surfaces Α and B of two concentric spheres, fig. 6 Fig Two parallel straight lines Α and Β on the plane, fig. 7 Fig The three vertices Α, Β, C of an equilateral triangle ABC are parallel to one another, fig. 8 Fig In the figures below, the point sets Α and Β are parallel to one another, fig. 9

5 Fig. 9 And numerous other parallel point sets Α and Β. Β. Intersecting point sets. 1. Two overlapping solid point sets Α and Β, fig. 10

6 Fig The circumferences Α and Β of two intersecting or tangential circles, fig. 11 Fig Two intersecting straight lines Α and Β, fig. 12 Fig The cover C of a closed rectilinear segment [AB] and point Α, fig. 13 Fig The cover C of a circle and its boundary (its circumference) Β, fig. 14 Fig The real axis ox, the closed interval Α = [1,10] the closed interval Β = [4,15], fig. 15

7 Fig. 15 And numerous other intersecting point sets. C. Incompatible point sets are: 1. A straight line Α on the plane, and a point Β, outside of it, fig. 16 Fig 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 On the plane, the circumference Α on an ellipse and a focal point Β, fig. 18 Fig The boundary (the circumference) Α of a circle, and its interior Β, fig. 19

8 fig On the real axis ox, the closed interval Α = [5,8] and the sum Β of all terms of the sequence a n =5n-1, fig. 20 fig Two identical triangles Α and Β, 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 Μ, which is outside a point set Α, you can either draw no parallel lines (Ν = 0) or you can only draw one (Ν = 1) or you can draw an infinite number of parallel lines ( =N ). Proof Let s take the sum S of all point sets. Then, according to the definition of parallel point sets, as above:

9 1. There are points sets to which we can never draw a parallel line from a point Μ. oyx(e.g. when point Μ 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 Μ. (e.g. when, on the plane, point Μ is outside of a straight line, (Euclides s well-known parallel postulate) and numerous other examples.) 3. There are points sets to which we can draw infinite parallel lines from a point Μ. (e.g. when, on the plane, point Μ is outside of a point Α, in which case (as we know), the infinite number of arcs of the circle whose centre is point Α and whose radius is R = (MA), are parallel to point Α.) Subsequently, we may conclude from the above, that there are point sets to which no parallels can be drawn, (Ν = 0), point sets to which one and only one parallel can be drawn (Ν = 1) and, finally, point sets to which infinite parallels can be drawn ( =N ). We will now prove that: When more than one parallel (Ν > 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 Α, (e.g. a semicurve ox) on the plane, to which more than one parallel (Ν > 1) can be drawn from a point Μ that is outside of it. xfig. 22 Given that more than one parallel (Ν > 1) can be drawn to point set Α, it follows that at least two parallels can be drawn to it (Ν xb= 2). Thus, one parallel is the semicurve 1 and the second parallel is the semicurve Bx2. xbbxin this case, as the two parallels 1 και 2 are different (i.e. they are not the same), then one parallel is a subset of the other parallel. xbwhich means there is a point set S = (B 2 B 1 ) that belongs to parallel 2 and not to xbparallel 1.

10 (,BB,BBut since the point set S consists of infinite points that belong to parallel xb2 it follows that infinite parallels can be drawn from point Μ to the point set Α, Bx,Bx,Bx,.i.e Subsequently, when more than one parallel (Ν > 1) can be drawn from point Μ, it follows that an infinite number of parallels can be drawn to point set Α. The above, then, proves the proposed theorem, i.e.: From a point Μ, which is outside a point set Α, you can either draw no parallel lines (Ν = 0) or you can only draw one (Ν = 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 (Ν > 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 Μ, which is on the inside of a convex rectilinear shape (e.g. inside a pentagon), you can never draw a parallel (Ν = 0) to its perimeter. THEOREM: On the plane, from a point Μ, which is outside (in the non-convex oyxpart) of an angle you can only draw one (Ν = 1) parallel Α towards it, fig. 23. fig. 23 NTHEOREM: On the plane, we can draw infinite parallels = xato Axa )rectilinear segment AB from a point M, which is outside of it, i.e. 1, 2, AxAx3, 4,, 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 well-known Parallel Postulate). This postulate does not apply in Generalised Geometry.

11 (fig. 24 fig. 25 And numerous other theorems. EXAMPLES x1. We have, in space, a straight line x and a point Μ outside of it, which is located at a Ndistance d from the straight line. In xthis case, we can draw infinite = from point Μ to the straight line. These infinite parallels (surfaces) xare parallels)subsets of the closed surface of the cylinder, whose axis is the straight line and whose radius is R = d. 2. From a point Μ, 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 Μ from their circumference (their boundary). 3. From a point Μ that is inside a cube, pyramid, tetrahedron, etc, you can never draw a parallel to its surface. 4. On the plane, from a point Μ, 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.

12 fig In a regular tetrahedron ABCD, each of its vertices, is parallel to the point set of the remaining three vertices, fig. 26 (a). fig. 26 (a) 6. Two topological surfaces Α and Β of a different genus n can never be parallel to one another, fig. 27. fig On the plane, two identical rectilinear shapes Α and Β can never be parallel to one another, e.g. two identical triangles Α and Β, fig. 28.

13 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 Α and Β. The object is to determine the maximum distance d max of a point Μ 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, multi-segment, connective, nonconnective, solid, non-solid, etc. GENERALISED TRIANGLE

14 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

15 Fig. 30 shows various generalised triangles ABC and their sides a, b, c.

16

17 fig. 30

18 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 a. The distance between point sets (vertices) Α and Β defines side c. b. The distance between vertices B and C defines side a. c. The distance between vertices C and Α defines side b. 2. Medians, fig. 32 fig. 31 a. The distance between vertex Α and the midpoint Μ 1 of side a defines the median μ a. b. The distance between vertex Β and the midpoint Μ 2 of side b defines the median μ b. c. The distance between vertex C and the midpoint Μ 3 of side c defines the median μ c. 3. Heights, fig. 33 fig. 32 a. The distance between vertex and the straight line of side a is height U a. b. The distance between vertex B and the straight line of side b is height U b. c. The distance between vertex C and the straight line of side c is height U c.

19 fig Angles, fig. 34 a. The extensions of sides b and c define the angle of vertex A. b. The extensions of sides c and a define the angle of vertex B. c. The extensions of sides a and b define the angle of vertex C. 5. Bisectrices, fig. 35 fig. 34 a. The rectilinear segment that lies upon the bicectrix of angle Aand is contained between vertex Α and the straight line of side a is the bicectrix δ a. b. The rectilinear segment that lies upon the bicectrix of angle Band is contained between vertex Β and the straight line of side b is the bicectrix δ b. c. The rectilinear segment that lies upon the bicectrix of angle Cand is contained between vertex C and the straight line of side ac is the bicectrix δ c.

20 fig Area, fig. 36 The space between vertices Α, Β, C and sides a, b, c is the area Ε. fig Incircle, fig. 37 The circle whose circumference p is tangent to all three straight lines defined by sides a, b, c is the incircle.

21 fig Circumscribed circle, fig. 37 p,ab,cthe circle whose circumference traverses the intersection points 111of,EE,Estraight lines 123, which are defined, respectively, by sides a, b, c is the circumscribed circle. 9. Escribed circle, fig. 37 pa. The circle whose circumference, is tangent to the straight line Ε 1, which is defined by side a, and the straight lines Ε 2 and Ε 3 are tangent to it on the inside, is the escribed circle of side a. b. 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: 1. 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. 2. 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. 3. In EEuclidean 0.Geometry, the area E of a triangle ABC is always a positive number, 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, Ε > 0, (e.g. three circle circumferences A, B, C which touch one another on the inside). 4. 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.

22 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 NON-REDUCIBLE 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 Α.

23 fig. 39 Definition: Any point set Α on the plane is known as non-reducible, 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 non-reducible point sets Α. 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.

24 fig. 41 Definition: The vertices of a generalised parallel shape C o, are the curved segments of the closed parallel C o, which correspond,ac,bto the vertices of the Euclidean rectilinear shape D o, e.g. CBAthe arches are the vertices of the parallel generalised triangle, etc. Parallel generalised shapes are of particular interest in CBAGeneralised Geometry. Thus, for example, in the generalised triangle, (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 Α of a sphere and its surface Β are parallel point sets, fig. 41 (a). Also, the surface Β of a «capsule» and its axis Β are parallel point sets, fig. 41 (a). The same applies to axis Α of a cylinder and its surface Β, fig. 41 (a).

25 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 in-depth mathematical research.

26 )EXAMPLES 1. In the coordinate system x,we have: :Ax2y242The circle + = :C15,0.The point (The object is to work out the sides and heights of the generalised triangle ABC. oyx2. On the real axis work out the sides and the medians of generalised triangle ABC, :Awhose,2vertices are the point sets: ( ],8[ ] 0,[ + ] 3 1B:C:3. In the coordinate xysystem xoy we have the ellipse: 2A: = with focal points Β and C. Work out the sides a, b, c and the area Ε of the generalised triangle ABC. x4. On the real axis we have: a110nα: The sum of terms of the sequence = 1a1Β: The sum of terms of the sequence, = n2a2n1c: The sum of terms of the sequence n= + Work out the sides of generalised triangle ABC. nn25. In the coordinate system xoy we have the equation: x26x50+ + = (1) Axxwhere: = 1+ 2Bxx1 2xC1= x= 2Work out the medians of generalised triangle ABC, where 1x, 2xare the roots of the above equation (1). 6. On the complex plane, a generalised triangle ABC has the vertices: Work out its medians. Α = 5 + 3i Β = 5 + 8i C = 5 10i

27 7. In the coordinate system coy we have the straight lines: xy1α: 2+ 4= xy1β: 3+ 3= xy1c: 10+ 2= Work out the sides a, b, c and the area Ε of the generalised triangle ABC. Β. GENERALISED GEOMETRY OF NON-EUCLIDEAN SPACES The reasoning we applied to Generalised Geometry of Euclidean spaces in the preceding chapters also applies to non-euclidean 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 Α. GENERALISED EUCLIDEAN SPACES

28 bca)1. GENERALISED EUCLIDEAN SPACES, TYPE,GIE. Let s assume, fig. 43, that we have a Euclidean metric space of two dimensions, e.g. the plane (Ε), which we consider a rectilinear-generatrix 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 (Ε). As we all know, the sides of generalised triangle ABC are a, b, c. BACAs shown on fig. 43, from the sum S of generalised triangles, (i = 1, 2, 3, ) which make up plane (Ε), 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(iiia= a< c< c+ b+ b+ A In this case, according to the above, it can be stated that the Euclidean space (Ε) fig. 43 is a generalised two-dimensional Euclidean space of type,gie, whose features BACare the parallel straight lines iii. The above leads us to the following definition: DEFINITION: For a given partition Δ, an n-dimensional Euclidean space will be known as a Generalised n-dimensional Euclidean space, of type,gie, when and BAConly when each side of every generalised triangle iii(i = 1, 2, 3, ) whose,ab,cvertices iiiare features of partition Δ, is smaller or equal to the sum of its two remaining sides. The following figures show various Generalised two-dimensional Euclidean spaces of type,gie, with their corresponding partition Δ.

29 fig. 44

30 2. GENERALISED EUCLIDEAN SPACES, TYPE,GIELet s assume, fig. 45, that we have a two-dimensional, metric Euclidean space, i.e. the plane (Ε). fig. 45 We divide the plane (Ε) 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 (Ε), 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: ba> b< b< cc+ a+ c+ ( )Ba In this case, in accordance with the above, it can be stated that the Euclidean space (E), fig. 45, is a generalised two-dimensional Euclidean space of type,gie, 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 Δ, an n-dimensional Euclidean space will be known as a Generalised n-dimensional Euclidean space, of type,gie, when and ABConly when, for each generalised triangle, (i= 1, 2, 3,...), whose vertices,ab,ciare features of the partition Δ, only one of its sides is larger than the iiiiisum of its two remaining sides. The following figures show various Generalised two-dimensional Euclidean spaces of type IE,G, and their corresponding partition Δ (with zones of width d).

31 fig. 46 Β. GENERALISED RIEMANN SPACES 1. GENERALISED RIEMANN SPACES, TYPE,GIRThe same reasoning we used on two-dimensional Generalised Euclidean spaces applies to two-dimensional 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. BACAs shown in figure 47, in the sum P of generalised triangles, (i = 1, 2, 3, ) that make up the surface S of the sphere, whichever the position of vertices A, B, C of iii

32 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: bca= + cab < + ( )cacb < + In this case, it can be said that the two-dimensional Riemann space (i.e. the surface S Gof the sphere), fig. 47, is a generalised two-dimensional,ab,criemann space of type,ir, whose features are the parallel circles iii. fig. 47 The above leads us to the following definition: DEFINITION: For a given partition Δ, an n-dimensional Riemann space will be known as a Generalised n-dimensional Riemann space of type,gir, when and ABConly when,, for each generalised triangle iii, (i = 1, 2, 3, ) whose vertices,ab,ciiiare features of the partition Δ, each one of its three sides is smaller than, or equal to the sum of its two remaining sides. Figure 48 shows various generalised two-dimensional Riemann spaces of type,gir, on a number of different surfaces.

33 fig GENERALISED RIEMANN SPACES, TYPE,GIRLet s assume, fig. 49, that we have a two-dimensional Riemann space, e.g. the surface (S) of a sphere. fig. 49 We divide the surface (S) into parallel zones of width d.

34 We now take a random generalised triangle ABC whose vertices A, B, C are three of the above parallel zones. BACAs shown in fig. 49 from the sum S of generalised triangles iii, (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: bac> + cba < + ( )Babc < + In this case, and in accordance with the above, we can say that the Riemann space (S), fig. 49, is a generalised, two-dimensional Riemann space, of type,girwhose features,ab,care the parallel zones iii. Based on the above, we are led to the following definition: DEFINITION: For a given partition Δ, an n-dimensional Riemann space will be known as a Generalised n-dimensional Riemann space of type,girwhen and ABConly when, for each generalised triangle iii, (i = 1, 2, 3, ) whose vertices,ab,ciii, are features of the partition Δ, only one of its sides is larger than the sum of its remaining two sides. Fig. 50 shows various Generalised two-dimensional Riemann spaces of type,gir, on a number of different surfaces.

35 fig. 50 AN INTERESTING OBSERVATION It is clear that this area of Mathematics, i.e. G1. Generalised Euclidean Spaces, type,ieand E,GI, and G2. Generalised Riemann Spaces, type,irand R,GI, can be the object of in-depth 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 tsolkas1@otenet.gr June 2007