What is Geometry?
Geometry is the mathematics of shape and size. It is part of our everyday lives. Geometry helps us fit different shapes together.No one could design a car, airplane or sky_scraper without knowing a lot about geometry.Geometry has been used by man for many thousand of years. In Ancient Babylonia and Egypt , geometrical knowledge was applied to partical problems of land measurement and building. But in Ancient Greece geometry was studied as an independent subject, though much of the knowledge gained was useful to builders and land surveyors. In the third century the Greek mathematician Euclid organized the knowledge that he and previous geometers had gained into a deductive science. A deductive science is one in which all statements are proved, step by step, from a small number of basic statements. Some of Euclid!s basic statements were definitions like “a point is that which has no size”. Others were statements that Euclid thought were obvious and could not be proved from any simpler statement. He called these statements postulates. An example of a postulate is : ”only one line can be drawn through any point parallel to a given straight line.” from a small number of definitions and postulates Euclid was able to prove a huge number of geometrical facts. The facts that Euclid proved from his basic definitions and postulates he called theorems.
BRANCHES OF GEOMETRY: Geometry in fact consists of several geometries, or branches of geometry. The first geometrical concern of early mathematicians was to idealize the common geometrical figures and the space in which they reside. This study is called Euclidean geometry ,and it is commonly divided into plane and solid geometry_ that is the study of figures that lie on a flat surface, or plane , and the study of figures that extend into three dimensions, such as the cube, the sphere, and the tetrahedron. The study of figures on a sphere, although, technically a part of Euclidean geometry is in fact of sufficient extent and importance to be treated separately as sperical geometry . There must be some common properties in the true figure and in the figure perceived by the eye that enable the brain to recognize what this true figures is. The study of these common properties is the starting point for the subject know as projective geometry .
Babylonian and Egyptian Mathematics
The Babylonians lived in Mesopotamia, a fertile plain between the Tigris and Euphrates rivers.
They developed an abstract form of writing based on cuneiform (i.e. wedge-shaped) symbols. Their symbols were written on wet clay tablets which were baked in the hot sun and many thousands of these tablets have survived to this day. It was the use of a stylus on a clay medium that led to the use of cuneiform symbols since curved lines could not be drawn.
Perhaps the most amazing aspect of the Babylonian’s calculating skills was their construction of tables to aid calculation.
The Babylonians had an advanced number system, in some ways more advanced than our present system. It was a positional system with base 60 rather than the base 10 of our present system. Now 10 has only two proper divisors, 2 and 5. However 60 has 10 proper divisors so many more numbers have a finite form.
One major disadvantage of the Babylonian system however was their lack of a zero. This meant that numbers did not have a unique representation but required the context to make clear whether 1 meant 1, 61, 3601, etc.
In about 300 BC Euclid wrote The Elements, a book which was to become one of the most famous books ever written. Euclid stated five postulates on which he based all his theorems:
1. To draw a straight line from any point to any other.
2. To produce a finite straight line continuously in a straight line.
3. To describe a circle with any centre and distance.
4. That all right angles are equal to each other.
5. That, if a straight line falling on two straight lines make the interior angles on the same side less than two right angles, if produced indefinitely, meet on that side on which are the angles less than the two right angles.
It is clear that the fifth postulate is different from the other four. It did not satisfy Euclid and he tried to avoid its use as long as possible – in fact the first 28 propositions of The Elements are proved without using it. Another comment worth making at this point is that Euclid, and many that were to follow him, assumed that straight lines were infinite.
Proclus (410-485) wrote a commentary on The Elements where he comments on attempted proofs to deduce the fifth postulate from the other four, in particular he notes that Ptolemy had produced a false ‘proof’. Proclus then goes on to give a false proof of his own. However he did give the following postulate which is equivalent to the fifth postulate.
Playfair’s Axiom:- Given a line and a point not on the line, it is possible to draw exactly one line through the given point parallel to the line.
Although known from the time of Proclus, this became known as Playfair’s Axiom after John Playfair wrote
Born: about 325 BC
Died: about 265 BC in Alexandria, Egypt
Euclid of Alexandria is the most prominent mathematician of antiquity best known for his treatise on mathematics The Elements. The long lasting nature of The Elements must make Euclid the leading mathematics teacher of all time. However little is known of Euclid’s life except that he taught at Alexandria in Egypt. Proclus, the last major Greek philosopher, who lived around 450 AD wrote (see  or  or many other sources). Here Euclid a mathematician of docure origin, gathered into thirteen books most of the mathematicial knowledge of his time.Of these, seven books were devoted to geometry.
Born: 31 March 1596 in La Haye (now Descartes),Touraine, France
Died: 11 Feb 1650 in Stockholm, Sweden
René Descartes was a philosopher whose work, La géométrie, includes his application of algebra to geometry from which we now have Cartesian geometry.
Descartes was educated at the Jesuit college of La Flèche in Anjou. He entered the college at the age of eight years, just a few months after the opening of the college in January 1604. He studied there until 1612, studying classics, logic and traditional Aristotelian philosophy. He also learnt mathematics from the books of Clavius. While in the school his health was poor and he was granted permission to remain in bed until 11 o’clock in the morning, a custom he maintained until the year of his death.
School had made Descartes understand how little he knew, the only subject which was satisfactory in his eyes was mathematics. This idea became the foundation for his way of thinking, and was to form the basis for all his works.
Descartes spent a while in Paris, apparently keeping very much to himself, then he studied at the University of Poitiers. He received a law degree from Poitiers in 1616 then enlisted in the military school at Breda. In 1618 he started studying mathematics and mechanics under the Dutch scientist Isaac Beeckman, and began to seek a unified science of nature. After two years in Holland he travelled through Europe. Then in 1619 he joined the Bavarian army.
Nikolai Ivanovich Lobachevsky
Born: 1 Dec 1792 in Nizhny Novgorod (was Gorky from 1932-1990), Russia
Died: 24 Feb 1856 in Kazan, Russia
Nikolai Ivanovich Lobachevskii’s father Ivan Maksimovich Lobachevskii, worked as a clerk in an office which was involved in land surveying while Nikolai Ivanovich’s mother was Praskovia Alexandrovna Lobachevskaya. Nikolai Ivanovich was one of three sons in this poor family. When Nikolai Ivanovich was seven years of age his father died and, in 1800, his mother moved with her three sons to the city of Kazan in western Russia on the edge of Siberia. There the boys attended Kazan Gymnasium, financed by government scholarships, with Nikolai Ivanovich entering the school in 1802.
In 1807 Lobachevskii graduated from the Gymnasium and entered Kazan University as a free student. Kazan State University had been founded in 1804, the result of one of the many reforms of the emperor Alexander I, and it opened in the following year, only two years before Lobachevskii began his undergraduate career. His original intention was to study medicine but he changed to study a broad scientific course involving mathematics and physics.
Some History of Geometry
In the beginning there was geometry, and it was codified by Euclid about 300 BC. Some 1500 years later, the contemporary thought was that there could be only this one, true geometry. This culminated in Kant’s argument in his Critique of Pure Reason in 1781 that Euclidean geometry was an a priori synthetic truth. Others appealed to experience or relied on innate truths, but almost all philosophers and mathematicians were agreed.
Unfortunately for their certitude, however, Lambert had already proved the existence of a noneuclidean geometry in 1766, but this was not generally known until somewhat later  and was not recognized until much later. In 1799, Gauss was already doubting the privileged rôle of Euclidean geometry, and by 1817 his doubt had become assured . Meanwhile, Schweikart had found a noneuclidean geometry by 1816 , but did not take the final step of observing that it could not be determined if the universe were Euclidean or not. This remained for Bolyai , beginning about 1820, and independently Lobachevsky , about 1826.
Thus by 1840 it had become clear to the experts that there were at least two geometries. In 1854, Riemann gave his inaugural lecture at Göttingen on the topic of Gauss’s choice: the foundations of geometry . He presented the audience with an infinity of new geometries, those now called metric geometries.
In this lecture, Riemann also gave what turned out to be the notion of space most suitable for geometry. This was an “n-fold extended quantity,” now called a manifold of dimension n. In dimension 3, a manifold can be envisaged as various blobs of ordinary space glued together in a precisely specified way. Clearly, there are a lot of these spaces. On each such space, one then specifies a geometry by means of an auxiliaryobject called a metric.
Some harbored doubts about the “truth” of noneuclidean geometries for several years, but by 1872 when Klein gave his inaugural lecture at Erlangen  they were mostly assuaged. He enlarged the world of geometries yet again in another major way, declaring that a geometry is the study of those properties which are preserved by a group of transformations, in any space, whether metric or not.
As one might guess, Riemann’s and Klein’s notions of a geometry do not coincide. Both are very extensive theories, each including vast arrays of examples of more or less practical applicability. The part in common has turned out to be the best place to test our further understanding of geometrical concepts and notions. The particular subset of the common part where they most closely mesh consists of those groups which are simultaneously manifolds and in which the group somehow describes its own geometry via a metric. Thus we combine Riemann’s and Klein’s geometries on the same space.
Now each group has an operation by which one may compose two elements and obtain a third. If the order in which two elements are composed does not affect the outcome, the group is called commutative. Thus in a commutative group, which element is to the right and which is to the left in a composition does not matter. But the operations of these geometric groups are not commutative in general, so one must make a choice of whether one will write down the left- or right-handed version of the theory. It has become traditional to write down the left-handed version and leave it to the reader as an exercise to work out the right-handed version, if desired. Because the geometry is preserved by or remains invariant under the group operation, one then speaks of left-invariant geometries on groups which are simultaneously manifolds. The Norwegian mathematician S. Lie was the first to study such groups extensively, so they are called Lie groups. Thus we arrive at left-invariant metrics on Lie groups as the tightest combination of the geometries of Riemann and Klein.
Traditionally, only some of these metric geometries were studied: the so-called definite ones. These are the most natural generalizations of Euclidean geometry. A summary of the state of knowledge in 1976 may be found in . As early as 1905, however, some applications in physics required the use of the much larger class of indefinite metric geometries. Unfortunately, very little is yet known about indefinite left-invariant metrics on Lie groups, and only one particular type has been studied in any generality [1, 14]. In dimension 3, we have determined all possible left-invariant metric geometries on Lie groups . In this dimension there are only two types; but in each higher dimension, there are many types of indefinite metrics while there is essentially only one type of definite metric.
One class of Lie groups that is most tractable consists of the nilpotent Lie groups. In a certain technical sense, they are the ones that are the closest to being commutative. The left-invariant metric geometries on commutative Lie groups are Euclidean geometry and its nearest indefinite relatives. Thus the left-invariant metric geometries on nilpotent Lie groups are those that are closest to being familiar while still exhibiting distinctive new features. This makes them an ideal place to enhance our limited understanding of indefinite metric geometric concepts andnotions.
Among the nilpotent Lie groups, the very nearest to the commutative groups are those called 2-step. In recent years, some of the most exciting new results in definite metric geometries have been obtained with 2-step nilpotent Lie groups; e.g., . The definite left-invariant metric geometry of these was recently studied in some detail , and this is a focus of continuing research; e.g., .
We are now studying indefinite left-invariant metric geometries on 2-step nilpotent Lie groups. This is another step in a long-term program to increase significantly our knowledge of the general features of indefinite metric geometries. Other recent parts of this program include [2, 4, 6]. Indefinite left-invariant metric geometries on 2-step nilpotent Lie groups exhibit several new phenomena that do not occur in the definite case. Some of these provide links to other areas of current mathematical research, such as splitting of foliations or decoupling of systems of differential equations, thereby enriching much more than just geometric study.
Historical Remarks on Finsler Geometry
The fundamental idea of a Finsler space may be traced back to the famous lecture of Riemann:”Uber die Hypothesen, welche der Geometrie zugrnde liegen.” In this memoir of 1854 Riemann discusses various possibilities by means of which an n-dimensional manifold may be endowed with a metric, and pays particular attention to a metric defined by the positive square root of a positive definite quadratic differentialform. Thus the foundations of Riemannian geometry are laid; nevertheless, it is also suggested that the positive fourth root of a fourth order differential form might serve as a metric function. These functions have three properties in common: they are positive, homogeneous of the first degree in the differentials, and are also convex in the latter. It would seem natural, therefore, to introduce a further generalisation to the effect that the distance ds between two neighbouring points represented by the coordinates x and x +dx be defined by some functions F(x, dx):
where this function satisfies these three properties.
It is remarkable that the first systematic study of manifolds endowed with such a metric was delayed by more than 60 years. It was an investigation of this kind which formed the subject matter of the thesis of Finsler in 1918, after whom such spaces were eventually named. It would appear that this new impulse was derived almost directly from the calculus of variations, with particular reference to the new geometrical background which was introduced by Caratheodory in connection with problems in parametric form. The kernel of these methods is the so-called indicatrix, while the property of convexity is of fundamental importance with regard to the necessary conditions for a minimum in the calculus of variations. In fact, the remarkable affinity between some aspects of differential geometry and the calculus of variations had been noticed some years prior to the publication of Finsler’s thesis, in particular by Bliss, Landsberg and Blaschke. Both Bliss and Landsberg introduced (distinct) definitions of angle in terms of invariants of a parametric problem in the calculus of variations, while an analytic study of such invariants had been made by E. Noether and A. Underhill. Yet the geometrical theories of Bliss and Landsberg were developed against an Euclidean background and cannot, therefore, be regarded as fulfilling the true objectives of the generalisation of Riemann’s proposal. Clearly, Finsler’s thesis must be regared as the first step in this direction.
A few years later, however, the general development took a curious turn away from the basic aspects and methods of the theory as developed by Finsler. The latter did not make use of the tensor calculus, being guided in principle by the notions of the calculus of variations; and in 1925 the methods of the tensor calculus were applied to the theory independently but almost simultaneously by Synge, Taylor and Berwald. It was found that the second derivatives of the half of the squre of F(x,dx) with respect to the differentials, dx, served admirably as components of a metric tensor in analogy with Riemannian geometry, and from the differential equations of the geodesics connection coefficients could be derived by means of which a generalisation of Levi-Civita’s parallel displacement could be defined. While the corresponding covariant derivatives as introduced by Synge and Taylor coincide, the theory of Berwald shows a marked difference, in the sense that in his geometry the lemma of Ricci (which in Riemannian geometry implies the vanishing of the covariant derivative of the metric tensor) is no longer valid. Nevertheless, Berwald continued to develop his theory with particular reference to the theory of curvature as well as to two-dimensional spaces. The significance of his work was enhanced by the advent of the general geometry of paths (a generalisation of the so-called Non-Riemannian geometry) due to Douglas and Knebelman, for the initial approach of Berwald was such as to establish a close affinity between these branches of metric and non-metric differential geometry.
Again, the theory took a new and unexpected turn in 1934 when E. Cartan published his tract on Finsler spaces. He showed that it was indeed possible to define connection coefficients and hence a covariant derivative such that the preservation of Ricci’s lemma was ensured. On this basis Cartan developed a theory of curvature, and practically all subsequent investigations concerning the geometry of Finsler spaces were dominated by this approach. Several mathematicians expressed the opinion that the theory had thus attained its final form. To a certain extent this was correct, but not altogether so, as we shall now indicate.
The above-mentioned theories make use of a certain device which basically involves the consideration of a space whose elements are not the points of the underlying manifold, but the line-elements of the later, which form a (2n-1)-dimensional variety. This facilitates the introduction of what Cartan calls the “Euclidean connection”, which , by means of certain postulates, may be derived uniquely from the fundamental metric function F(x, dx). The method also depends on the introduction of a so-called “element of support”, namely, that at each point a previously assigned direction must be given, which then serves asdirectional argument in all functions depending on direction as well as position. Thus, for instance, the length of a vector and the vector obtained from it by an infinitesimal parallel displacement depend on the arbitrary choice of the element of support. It is this device which led to the development of Finsler geometry in terms of direct generalisations of the methods of Riemannian geometry.
It was felt, however, that the introduction of the element of support was undesirable from a geometrical point of view, while the natural link with the calculus of variations was seriously weakened. This view was expressed independently by several authors, in particular by Vagner, Busemann and the present writer. It was emphasised that the natural local metric of a Finsler space is a Minkowskian one, and that the arbitrary imposition of a Euclidean metric would to some extent obscure some of the most interesting characteristics of the Finsler space. Thus at the beginning of the present decade further theories were put forward. The rejection of the use of the element of support, however desirable from a geometrical point of view, led to new difficulties: for instance, the natural orthogonality between two vectors is not in general symmetric, while the analytical difficulties are certainly enhanced, particularly since Ricci’s lemma cannot be generalised as before. Fortunately, from the point of view of differential invariants, there exist marked similarities between all these theories, which is a perfectly natural phenomenon and could have been expected. It is in the application and in the interpretation of these invariants that the two points of view appear to be irreconcilable.
Geometry Before the Greeks
The beginnings of geometry are shrouded in the mists of pre-history. Eves calls this stage “subconscious geometry.” Later, humans came to recognize certain principles, such as the fact that the circumference and diameter of circles are always in the same ratio. Eves calls this stage “scientific geometry.”
Geometry as a science may have begun in Egypt, where the rulers had need to measure the areas of fields in order to assess taxes on them. (The word “geometry” means “earth measurement.”) The Moscow papyrus (19th cen. BCE) and the Rhind papyrus (17th cen. BCE) make it clear that the Egyptians had a significant amount of geometrical knowledge at least 4000 years ago, and perhaps much earlier than that. (The great Pyramid of Gizeh was built nearly 5000 years ago.) Many of the methods used to calculate areas and volumes were only approximately correct. One of the most remarkable results in the Moscow papyrus is a correct procedure for calculating the volume of a truncated pyramid (a pyramid with the top sliced off.)
wherever it may have appeared first, it is clear that some significant geometry was developed–probably independently–in Egypt, Mesopotamia (Babylonia), China, India, and perhaps in other places and cultures as well. wherever it developed it seems likely that this development came about to meet the practical needs of surveying, engineering, and agriculture. Geometry remained empirical and utilitarian until the Greeks made it into a science which could be, and was, studied independently of its practical applications.
Geometry in Greece
The Greeks of the Classical Period (600-300 BCE) not only increased the quantity of geometry, they changed the very nature of the subject,and of mathematics in general. Some Greek mathematicians traveled to Egypt and Babylonia and learned what was known of geometry in those places. They “transformed the subject into something vastly different from the set of empirical conclusions worked out by their predecessors. The Greeks insisted that geometric fact must be established, not by empirical procedures, but by deductive reasoning; geometrical truth was to be obtained in the study room rather than in the laboratory.” They created what we may call “demonstrative geometry,” whose truths are supported by (deductive) proofs rather than only by inductive evidence.
This work began with a philosopher/mathematician named Thales in the first part of the 6th century BCE. Thales, who had traveled in Egypt, “is the first known individual with whom the use of deductive methods in geometry is associated.” This axiomatic-deductive method is the cornerstone of modern mathematics, which thus may be truly said to have begun with the classical Greeks. (It should be noted that other important aspects of modern western civilization also find their roots in classical Greece.)
The next important name is that of Pythagoras, who may have studied under Thales. He founded “the celebrated Pythagorean school, a brotherhood knit together with secret and cabalistic rites and observances and committed to the study of philosophy, mathematics, and natural science.” The Pythagorean belief that everything is explainable by numbers may be regarded as one of the origins of the quantitative emphasis in modern science. Pythagoras may have been the first to provide a proof of the theorem named for him, but the result had been understood by many peoples for many centuries. Demonstrative geometry was considerably advanced by Pythagoras and his followers.
Several attempts were made to unite all of the truths of mathematics into a single chain which could be deduced from explicitly-stated “self-evident” assumptions (or axioms.) “and then, about 300 BCE, Euclid produced his epoch-making effort, the Elements, a single deductive chain of 465 propositions neatly and beautifully comprising plane and solid geometry, number theory, and Greek geometrical algebra. from its very first appearance this work was accorded the highest respect, and it so quickly and so completely superseded all previous efforts of the same nature that now no trace remains of the earlier efforts. This single work on the future development of geometry has been enormous and is difficult to overstate.” Euclid’s Elements remained the standard textbook in geometry until the early years of the 20th century.
After Euclid the most exceptional of the Greek mathematicians were Archimedes and Apollonius. Archimedes (287-212 BCE), universally acclaimed as the greatest mathematician of antiquity and among the greatest of all time, wrote extensively on geometry. He proved that the value of pi (the ration of the circumference of a circle to its diameter) must lie between 3-10/71 and 3-1/7. He discovered and established the relationship between pi and the area of a circle. He found and proved formulas for the volume and surface area of a sphere (using methods which anticipated the development of integral calculus.)
Apollonius (262-190 BCE) wrote a number of works, most of which have been lost. We do have his monumental work on parabolas, hyperbolas, and ellipses, The Conic Sections, which has been called one of the greatest scientific works of antiquity.
Arabic and Hindu Contributions
The five or six centuries following the fall of Rome (in the 5th century CE) is often referred to as the Dark Ages of Europe. Leadership in world of mathematics during this period passed to the Arabs and the Hindus. It has been said that the greatest mathematical contribution of the Arabs was the preservation and transmission of Greek achievements to the modern period. But it is important to note that the Arabs and Hindus made many important contributions of their own. The most important of these are in the areas of numeration, computation, algebra, and trigonometry,and are thus not included in these notes on the history of geometry.
Noteworthy geometrical achievements include the work of the Hindu Brahmagupta (fl. 630 CE) on cyclic quadrilaterals, the Arab Abu’l-Wefa on constructions, the Arab poet and mathematician Omar Khayyam on geometric solutions of cubic equations, and the work of the Arab Nasir Eddin on Euclid’s parallel postulate.
The Modern Period
The modern period has seen much activity in geometry, including the creation of a range of new kinds of geometry. These notes speak briefly of Projective Geometry, Coordinate Geometry, Differential Geometry, Non-Euclidean Geometry, and Topology. The first three of these arose in the 17th century. Projective geometry results from the application of the concept of a “projection” to geometry. Coordinate geometry results fromthe application of algebra to geometry. Differential geometry results from the application of calculus to geometry. The previously existing geometry is sometimes called “synthetic geometry” to distinguish it from these newer branches of the subject.
Projective geometry grew out of the efforts of Italian artists in the 15th century who created a geometrical theory called “mathematical perspective” to aid them in the production of realistic pictures. Later this theory was taken into mathematics and considerably expanded by several Frenchmen, led by Gerard Desargues and Blais Pascal. Descriptive geometry, which is related to projective geometry, was created andextended by Frenchmen Gaspard Monge, Jean Victor Poncelet, and others in the 18th and 19th centuries.
Analytic geometry, also called coordinate geometry or Cartesian geometry, was one of the great mathematical achievements of the 17th century–a century which also saw the creation of calculus and foundations laid in the theory of probability. Created independently by Frenchmen Rene Descartes and Pierre de Fermat, coordinate geometry is the fruitful marriage of algebra and geometry. It made it possible to use algebra as a tool for solving geometric problems (and vice versa.)
Differential geometry is that geometry which uses calculus as a tool to investigate the properties of curves and surfaces. The most prominent names in the history of differential geometry are Gaspard Monge (1746-1818), Carl Gauss (1777-1855), and Bernard Riemann (1826-1866).
Non-Euclidean Geometry was created by Gauss, Nikolai Lobachevsky (1793-1856), and Janos Bolyai (1802-1860). Bernard Riemann extended their researches. Non-Euclidean geometry is obtained by replacing Euclid’s parallel postulate by one of its contradictory forms, anddeducing theorems from this new set of axioms. “It took unusual imagination to entertain the possibility of a geometry different from Euclid’s, for the human mind had for two millennia been bound by the prejudice of tradition to the firm belief that Euclid’s system was most certainly the only way geometrically to describe physical space, and that any contrary geometric system simply could not be consistent.” One result beyond the world of mathematics was the doubt which this result cast upon human ability to know the truth about anything.
Topology, sometimes described playfully as “rubber-sheet geometry,” is the study of those properties of objects which are not altered by stretching or bending (more precisely, by “continuous deformation.”) It began in the 19th century as a branch of geometry. It has been a major area of mathematical research in the 20th century and has come to be regarded as a fourth division of mathematics, along with algebra, geometry, and analysis. Among the many important contributors to topology are Riemann and Henri Poincare (1854-1912).
Other recent developments in geometry include “The Erlnager Programm” of Felix Kline, the abstract spaces of Maurice Frechet, and the “Grundlagen” of David Hilbert. See Eves article for more information on these.