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78-485: The majority of East German parents (85%) worked outside of the home which resulted in a significant need for childcare services throughout the country. Children up to the age of three attended state run crèches ( Kinderkrippen ) which were often situated next to kindergarten buildings. Throughout the history of East Germany, young women would serve or volunteer in crèches in order to have more of an influence in raising their children. Many crèches and polytechnic schools had

156-520: A geodesic is a generalization of the notion of a line to curved spaces . In Euclidean geometry a plane is a flat, two-dimensional surface that extends infinitely; the definitions for other types of geometries are generalizations of that. Planes are used in many areas of geometry. For instance, planes can be studied as a topological surface without reference to distances or angles; it can be studied as an affine space , where collinearity and ratios can be studied but not distances; it can be studied as

234-418: A parabola with the summation of an infinite series , and gave remarkably accurate approximations of pi . He also studied the spiral bearing his name and obtained formulas for the volumes of surfaces of revolution . Indian mathematicians also made many important contributions in geometry. The Shatapatha Brahmana (3rd century BC) contains rules for ritual geometric constructions that are similar to

312-425: A vector space and its dual space . Euclidean geometry is geometry in its classical sense. As it models the space of the physical world, it is used in many scientific areas, such as mechanics , astronomy , crystallography , and many technical fields, such as engineering , architecture , geodesy , aerodynamics , and navigation . The mandatory educational curriculum of the majority of nations includes

390-405: A common endpoint, called the vertex of the angle. The size of an angle is formalized as an angular measure . In Euclidean geometry , angles are used to study polygons and triangles , as well as forming an object of study in their own right. The study of the angles of a triangle or of angles in a unit circle forms the basis of trigonometry . In differential geometry and calculus ,

468-523: A decimal place value system with a dot for zero." Aryabhata 's Aryabhatiya (499) includes the computation of areas and volumes. Brahmagupta wrote his astronomical work Brāhmasphuṭasiddhānta in 628. Chapter 12, containing 66 Sanskrit verses, was divided into two sections: "basic operations" (including cube roots, fractions, ratio and proportion, and barter) and "practical mathematics" (including mixture, mathematical series, plane figures, stacking bricks, sawing of timber, and piling of grain). In

546-416: A level of sufficient fluency to have a small conversation with a local. There were only few opportunities for student exchanges. Appreciated by parents were the so-called head marks ( Kopfnoten ) which assessed behavior, industriousness, order, and cooperation. These were combined with a short teacher's essay about the student's character, success or progress, advice for future improvements - here and there from

624-440: A more rigorous foundation for geometry, treated congruence as an undefined term whose properties are defined by axioms . Congruence and similarity are generalized in transformation geometry , which studies the properties of geometric objects that are preserved by different kinds of transformations. Classical geometers paid special attention to constructing geometric objects that had been described in some other way. Classically,

702-438: A multi-level solution, a fully developed embedded course in geometry , the introduction to vectors, the handling and solving of simple equations etc. were taught from the beginning of the first grade. But nevertheless other subjects like arts , music and so forth were not neglected but emphasised to be important for an all-around, gapless general education . One lesson would last 45 minutes and students went to school six days

780-428: A multitude of forms, including the graphics of Leonardo da Vinci , M. C. Escher , and others. In the second half of the 19th century, the relationship between symmetry and geometry came under intense scrutiny. Felix Klein 's Erlangen program proclaimed that, in a very precise sense, symmetry, expressed via the notion of a transformation group , determines what geometry is . Symmetry in classical Euclidean geometry

858-451: A number of apparently different definitions, which are all equivalent in the most common cases. The theme of symmetry in geometry is nearly as old as the science of geometry itself. Symmetric shapes such as the circle , regular polygons and platonic solids held deep significance for many ancient philosophers and were investigated in detail before the time of Euclid. Symmetric patterns occur in nature and were artistically rendered in

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936-520: A number of specialists on site, such as their own doctors and dentists. A considerable number of crèches were built and developed during the post- Second World War construction period. The crèches were often situated within walking distance of residential blocks or on site at factories and collective farms . Crèches were able to support approximately 80% of young East German children with rates as high as 99% in several urban centres. It cost 27.50 East German marks per child per month for full day care at

1014-444: A physical system, which has a dimension equal to the system's degrees of freedom . For instance, the configuration of a screw can be described by five coordinates. In general topology , the concept of dimension has been extended from natural numbers , to infinite dimension ( Hilbert spaces , for example) and positive real numbers (in fractal geometry ). In algebraic geometry , the dimension of an algebraic variety has received

1092-528: A plane or 3-dimensional space. Mathematicians have found many explicit formulas for area and formulas for volume of various geometric objects. In calculus , area and volume can be defined in terms of integrals , such as the Riemann integral or the Lebesgue integral . Other geometrical measures include the curvature and compactness . The concept of length or distance can be generalized, leading to

1170-602: A purely algebraic context. Scheme theory allowed to solve many difficult problems not only in geometry, but also in number theory . Wiles' proof of Fermat's Last Theorem is a famous example of a long-standing problem of number theory whose solution uses scheme theory and its extensions such as stack theory . One of seven Millennium Prize problems , the Hodge conjecture , is a question in algebraic geometry. Algebraic geometry has applications in many areas, including cryptography and string theory . Complex geometry studies

1248-427: A size or measure to sets , where the measures follow rules similar to those of classical area and volume. Congruence and similarity are concepts that describe when two shapes have similar characteristics. In Euclidean geometry, similarity is used to describe objects that have the same shape, while congruence is used to describe objects that are the same in both size and shape. Hilbert , in his work on creating

1326-516: A socialistic point of view. From the seventh year onwards, students visited a factory, power station or farm one day per week for 4 hours, depending on their location. At some of these places, the student would work alongside regular employees. There were annual championships on various subjects with the winners receiving prizes. The Russian language and mathematics championships were seen and promoted as very prestigious and competitive as well as regular championships in sport, called Spartakiade (from

1404-407: A stable daily routine and were introduced to the idea of learning . The children stayed together in the same group with the same group educator during the three years. The groups were called the little group ( kleine Gruppe ) for the young children of the age of three, the middle group ( mittlere Gruppe ) for the children of the age of four and the big group ( große Gruppe ) for the older children of

1482-600: A technical sense a type of transformation geometry , in which transformations are homeomorphisms . This has often been expressed in the form of the saying 'topology is rubber-sheet geometry'. Subfields of topology include geometric topology , differential topology , algebraic topology and general topology . Algebraic geometry is fundamentally the study by means of algebraic methods of some geometrical shapes, called algebraic sets , and defined as common zeros of multivariate polynomials . Algebraic geometry became an autonomous subfield of geometry c.  1900 , with

1560-518: A theorem called Hilbert's Nullstellensatz that establishes a strong correspondence between algebraic sets and ideals of polynomial rings . This led to a parallel development of algebraic geometry, and its algebraic counterpart, called commutative algebra . From the late 1950s through the mid-1970s algebraic geometry had undergone major foundational development, with the introduction by Alexander Grothendieck of scheme theory , which allows using topological methods , including cohomology theories in

1638-494: A theory of ratios that avoided the problem of incommensurable magnitudes , which enabled subsequent geometers to make significant advances. Around 300 BC, geometry was revolutionized by Euclid, whose Elements , widely considered the most successful and influential textbook of all time, introduced mathematical rigor through the axiomatic method and is the earliest example of the format still used in mathematics today, that of definition, axiom, theorem, and proof. Although most of

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1716-448: A week. On Saturdays , there were approximately four or five lessons. The Ministry of Education determined a table of lessons ( Stundentafel ) which expressed the ideas of the curriculum by naming the subjects that were believed to be crucial for a modern general education together with the number of weekly lessons for every single subject. The table of lessons fragmented in two parts, the compulsory teaching ( obligatorischer Unterricht ) and

1794-411: Is diffeomorphic to Euclidean space. Manifolds are used extensively in physics, including in general relativity and string theory . Euclid defines a plane angle as the inclination to each other, in a plane, of two lines which meet each other, and do not lie straight with respect to each other. In modern terms, an angle is the figure formed by two rays , called the sides of the angle, sharing

1872-540: Is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic , one of the oldest branches of mathematics. A mathematician who works in the field of geometry is called a geometer . Until the 19th century, geometry was almost exclusively devoted to Euclidean geometry , which includes the notions of point , line , plane , distance , angle , surface , and curve , as fundamental concepts. Originally developed to model

1950-400: Is a part of some ambient flat Euclidean space). Topology is the field concerned with the properties of continuous mappings , and can be considered a generalization of Euclidean geometry. In practice, topology often means dealing with large-scale properties of spaces, such as connectedness and compactness . The field of topology, which saw massive development in the 20th century, is in

2028-413: Is a three-dimensional object bounded by a closed surface; for example, a ball is the volume bounded by a sphere. A manifold is a generalization of the concepts of curve and surface. In topology , a manifold is a topological space where every point has a neighborhood that is homeomorphic to Euclidean space. In differential geometry , a differentiable manifold is a space where each neighborhood

2106-409: Is defined. The earliest recorded beginnings of geometry can be traced to ancient Mesopotamia and Egypt in the 2nd millennium BC. Early geometry was a collection of empirically discovered principles concerning lengths, angles, areas, and volumes, which were developed to meet some practical need in surveying , construction , astronomy , and various crafts. The earliest known texts on geometry are

2184-437: Is not viewed as the set of the points through which it passes. However, there are modern geometries in which points are not primitive objects, or even without points. One of the oldest such geometries is Whitehead's point-free geometry , formulated by Alfred North Whitehead in 1919–1920. Euclid described a line as "breadthless length" which "lies equally with respect to the points on itself". In modern mathematics, given

2262-415: Is of importance to mathematical physics due to Albert Einstein 's general relativity postulation that the universe is curved . Differential geometry can either be intrinsic (meaning that the spaces it considers are smooth manifolds whose geometric structure is governed by a Riemannian metric , which determines how distances are measured near each point) or extrinsic (where the object under study

2340-482: Is represented by congruences and rigid motions, whereas in projective geometry an analogous role is played by collineations , geometric transformations that take straight lines into straight lines. However it was in the new geometries of Bolyai and Lobachevsky, Riemann, Clifford and Klein, and Sophus Lie that Klein's idea to 'define a geometry via its symmetry group ' found its inspiration. Both discrete and continuous symmetries play prominent roles in geometry,

2418-753: The Sulba Sutras . According to ( Hayashi 2005 , p. 363), the Śulba Sūtras contain "the earliest extant verbal expression of the Pythagorean Theorem in the world, although it had already been known to the Old Babylonians. They contain lists of Pythagorean triples , which are particular cases of Diophantine equations . In the Bakhshali manuscript , there are a handful of geometric problems (including problems about volumes of irregular solids). The Bakhshali manuscript also "employs

Education in East Germany - Misplaced Pages Continue

2496-690: The Egyptian Rhind Papyrus (2000–1800 BC) and Moscow Papyrus ( c.  1890 BC ), and the Babylonian clay tablets , such as Plimpton 322 (1900 BC). For example, the Moscow Papyrus gives a formula for calculating the volume of a truncated pyramid, or frustum . Later clay tablets (350–50 BC) demonstrate that Babylonian astronomers implemented trapezoid procedures for computing Jupiter's position and motion within time-velocity space. These geometric procedures anticipated

2574-523: The Lambert quadrilateral and Saccheri quadrilateral , were part of a line of research on the parallel postulate continued by later European geometers, including Vitello ( c.  1230  – c.  1314 ), Gersonides (1288–1344), Alfonso, John Wallis , and Giovanni Girolamo Saccheri , that by the 19th century led to the discovery of hyperbolic geometry . In the early 17th century, there were two important developments in geometry. The first

2652-518: The Oxford Calculators , including the mean speed theorem , by 14 centuries. South of Egypt the ancient Nubians established a system of geometry including early versions of sun clocks. In the 7th century BC, the Greek mathematician Thales of Miletus used geometry to solve problems such as calculating the height of pyramids and the distance of ships from the shore. He is credited with

2730-509: The Riemann surface , and Henri Poincaré , the founder of algebraic topology and the geometric theory of dynamical systems . As a consequence of these major changes in the conception of geometry, the concept of " space " became something rich and varied, and the natural background for theories as different as complex analysis and classical mechanics . The following are some of the most important concepts in geometry. Euclid took an abstract approach to geometry in his Elements , one of

2808-399: The complex plane using techniques of complex analysis ; and so on. A curve is a 1-dimensional object that may be straight (like a line) or not; curves in 2-dimensional space are called plane curves and those in 3-dimensional space are called space curves . In topology, a curve is defined by a function from an interval of the real numbers to another space. In differential geometry,

2886-493: The socialist life (visiting factories, traffic education, cultural life, introduction to professions deemed important), introduction to natural and scientific phenomena (weather, seasons, sky, stars, rocks etc.), music, sports, artistic and constructive handicrafts and esteeming pieces of art. There was no teaching of reading , writing or arithmetic , but the fundamental concepts were taught to develop intellectual and motor skills. For instance, introduction to set theory within

2964-631: The 19th century changed the way it had been studied previously. These were the discovery of non-Euclidean geometries by Nikolai Ivanovich Lobachevsky, János Bolyai and Carl Friedrich Gauss and of the formulation of symmetry as the central consideration in the Erlangen programme of Felix Klein (which generalized the Euclidean and non-Euclidean geometries). Two of the master geometers of the time were Bernhard Riemann (1826–1866), working primarily with tools from mathematical analysis , and introducing

3042-496: The 19th century several discoveries enlarged dramatically the scope of geometry. One of the oldest such discoveries is Carl Friedrich Gauss 's Theorema Egregium ("remarkable theorem") that asserts roughly that the Gaussian curvature of a surface is independent from any specific embedding in a Euclidean space . This implies that surfaces can be studied intrinsically , that is, as stand-alone spaces, and has been expanded into

3120-474: The 19th century, the discovery of non-Euclidean geometries by Nikolai Ivanovich Lobachevsky (1792–1856), János Bolyai (1802–1860), Carl Friedrich Gauss (1777–1855) and others led to a revival of interest in this discipline, and in the 20th century, David Hilbert (1862–1943) employed axiomatic reasoning in an attempt to provide a modern foundation of geometry. Points are generally considered fundamental objects for building geometry. They may be defined by

3198-743: The Soviet Union in the Eastern bloc. It was also available in English and French, but only as additional elective foreign languages (Universities required two foreign languages). The Russian lessons focused on the Cyrillic script, the writing, the reading and the grammar of the Russian language. To be able to have a substantial conversation was not an aim, but to be able to use professional and technical Russian literature. The speaking skills should reach

Education in East Germany - Misplaced Pages Continue

3276-891: The VHS, access to the ABF was restricted to workers and farmers with at least five years of working experience. This was usually organized by the HR department of the company where they worked. National service was 18 months for males between the age of 18 and 26. Often prospective students were intimidated into serving three years as NCO in order to begin university early. For popular subjects, such as information technology, or prestigious subjects, such as law or medicine , there were more applicants to university than places. Several criteria were taken into account: school exams, national service time, patriotism, ideology , religious affiliation (better to be atheist than religious), communist party membership, etc. In

3354-483: The age of five. Two times a day there were lesson-like pre-school activities ( Beschäftigungen ) which all children had to participate in. These activities were planned by the group educator and lasted 20 minutes in the little group, 25 minutes in the middle group and 30 minutes in the big group. The contents of the activities were regulated nationwide by a uniform teaching plan and included German language and speech, children's literature , mathematics , introduction to

3432-590: The angles between plane curves or space curves or surfaces can be calculated using the derivative . Length , area , and volume describe the size or extent of an object in one dimension, two dimension, and three dimensions respectively. In Euclidean geometry and analytic geometry , the length of a line segment can often be calculated by the Pythagorean theorem . Area and volume can be defined as fundamental quantities separate from length, or they can be described and calculated in terms of lengths in

3510-427: The compulsory lessons took place in the morning and the timetable for each class was organised in a way that there should not be any free periods while classes should end at the same time every day. Therefore, by allocating sufficient resources to the education system, East Germany employed a high number of teachers and educators, so the average number of students per class lessened from 26 in the fifties to 19 and less in

3588-412: The concept of angle and distance, finite geometry that omits continuity , and others. This enlargement of the scope of geometry led to a change of meaning of the word "space", which originally referred to the three-dimensional space of the physical world and its model provided by Euclidean geometry; presently a geometric space , or simply a space is a mathematical structure on which some geometry

3666-513: The contents of the Elements were already known, Euclid arranged them into a single, coherent logical framework. The Elements was known to all educated people in the West until the middle of the 20th century and its contents are still taught in geometry classes today. Archimedes ( c.  287–212 BC ) of Syracuse, Italy used the method of exhaustion to calculate the area under the arc of

3744-428: The crèches. Most crèches were open from 6 am to 6 pm. Unlike West Germany , East Germany accomplished large-scale education reform and introduced a dense network of high-standard education facilities, especially kindergartens. A unique characteristic of East German kindergartens was the strong educational background of these institutions. Children from age three to six learned to interact with other children, got used to

3822-454: The elective teaching ( fakultativer Unterricht ). Later a third component was introduced, the optionally compulsory teaching ( wahlweise obligatorischer Unterricht ). Together, with the introduction of the POS grading at schools was reorganized as well. School started early, usually between 7 am or 8 am. The POS was designed as a reliable all-day school ( verläßliche Tagesschule ), which means

3900-428: The field has been split in many subfields that depend on the underlying methods— differential geometry , algebraic geometry , computational geometry , algebraic topology , discrete geometry (also known as combinatorial geometry ), etc.—or on the properties of Euclidean spaces that are disregarded— projective geometry that consider only alignment of points but not distance and parallelism, affine geometry that omits

3978-520: The first use of deductive reasoning applied to geometry, by deriving four corollaries to Thales's theorem . Pythagoras established the Pythagorean School , which is credited with the first proof of the Pythagorean theorem , though the statement of the theorem has a long history. Eudoxus (408– c.  355 BC ) developed the method of exhaustion , which allowed the calculation of areas and volumes of curvilinear figures, as well as

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4056-526: The former in topology and geometric group theory , the latter in Lie theory and Riemannian geometry . A different type of symmetry is the principle of duality in projective geometry , among other fields. This meta-phenomenon can roughly be described as follows: in any theorem , exchange point with plane , join with meet , lies in with contains , and the result is an equally true theorem. A similar and closely related form of duality exists between

4134-544: The full-day care in kindergartens and there were enough places for 94% to 99% of East German children. The polytechnic secondary school , abbreviation POS for "Polytechnische Oberschule", was developed from 1957 to 1958 and established in 1959. The POS focused strongly on German language , mathematics , physics , chemistry , biology , astronomy , physical geography, sports , history , political education and of course, technology -related theoretical and practical work, including gardening , woodwork, metal work. Religion

4212-598: The idea of metrics . For instance, the Euclidean metric measures the distance between points in the Euclidean plane , while the hyperbolic metric measures the distance in the hyperbolic plane . Other important examples of metrics include the Lorentz metric of special relativity and the semi- Riemannian metrics of general relativity . In a different direction, the concepts of length, area and volume are extended by measure theory , which studies methods of assigning

4290-537: The idea of reducing geometrical problems such as duplicating the cube to problems in algebra. Thābit ibn Qurra (known as Thebit in Latin ) (836–901) dealt with arithmetic operations applied to ratios of geometrical quantities, and contributed to the development of analytic geometry . Omar Khayyam (1048–1131) found geometric solutions to cubic equations . The theorems of Ibn al-Haytham (Alhazen), Omar Khayyam and Nasir al-Din al-Tusi on quadrilaterals , including

4368-514: The juniors before retiring. Entrance to East German Universities was very limited. To attend University education in East Germany, one had to attend the erweiterte Oberschule . Access to these schools was restricted to the 2-3 best students per POS class. Entry to the EOS was after grade 8 for 4 years. At 18 years of age, every youth either had finished EOS or vocational training. A special form

4446-552: The latter section, he stated his famous theorem on the diagonals of a cyclic quadrilateral . Chapter 12 also included a formula for the area of a cyclic quadrilateral (a generalization of Heron's formula ), as well as a complete description of rational triangles ( i.e. triangles with rational sides and rational areas). In the Middle Ages , mathematics in medieval Islam contributed to the development of geometry, especially algebraic geometry . Al-Mahani (b. 853) conceived

4524-458: The lower classes. The beginning of the school year was September 1 unless that day was a Thursday , Friday or Saturday , then school started the following Monday . The school year always had 38 weeks of classes with 30 weeks covered by the nationwide unified curriculum. Since 1951, the learning of the Russian language as the foreign language was obligatory, because of the leading role of

4602-768: The mid-1980s, there was an important change: those who wanted to study informatics could have their national service halved to nine months. Childcare Too Many Requests If you report this error to the Wikimedia System Administrators, please include the details below. Request from 172.68.168.132 via cp1112 cp1112, Varnish XID 934298826 Upstream caches: cp1112 int Error: 429, Too Many Requests at Thu, 28 Nov 2024 08:02:35 GMT Geometry Geometry (from Ancient Greek γεωμετρία ( geōmetría )  'land measurement'; from γῆ ( gê )  'earth, land' and μέτρον ( métron )  'a measure')

4680-411: The most influential books ever written. Euclid introduced certain axioms , or postulates , expressing primary or self-evident properties of points, lines, and planes. He proceeded to rigorously deduce other properties by mathematical reasoning. The characteristic feature of Euclid's approach to geometry was its rigor, and it has come to be known as axiomatic or synthetic geometry. At the start of

4758-429: The multitude of geometries, the concept of a line is closely tied to the way the geometry is described. For instance, in analytic geometry , a line in the plane is often defined as the set of points whose coordinates satisfy a given linear equation , but in a more abstract setting, such as incidence geometry , a line may be an independent object, distinct from the set of points which lie on it. In differential geometry,

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4836-418: The numbers up to 10 , counting up to 20 , handling of quantities, crafts and motor skills exercises to prepare the hand to writing , the handling of pencils , scissors , fabrics and glue . Children were also encouraged to take an active role in the running of their kindergartens. Children often had to serve each other meals and help keep the kindergarten clean and tidy. There were no fees charged for

4914-441: The only instruments used in most geometric constructions are the compass and straightedge . Also, every construction had to be complete in a finite number of steps. However, some problems turned out to be difficult or impossible to solve by these means alone, and ingenious constructions using neusis , parabolas and other curves, or mechanical devices, were found. The geometrical concepts of rotation and orientation define part of

4992-514: The physical world, geometry has applications in almost all sciences, and also in art, architecture , and other activities that are related to graphics. Geometry also has applications in areas of mathematics that are apparently unrelated. For example, methods of algebraic geometry are fundamental in Wiles's proof of Fermat's Last Theorem , a problem that was stated in terms of elementary arithmetic , and remained unsolved for several centuries. During

5070-407: The placement of objects embedded in the plane or in space. Traditional geometry allowed dimensions 1 (a line or curve), 2 (a plane or surface), and 3 (our ambient world conceived of as three-dimensional space ). Furthermore, mathematicians and physicists have used higher dimensions for nearly two centuries. One example of a mathematical use for higher dimensions is the configuration space of

5148-482: The properties that they must have, as in Euclid's definition as "that which has no part", or in synthetic geometry . In modern mathematics, they are generally defined as elements of a set called space , which is itself axiomatically defined. With these modern definitions, every geometric shape is defined as a set of points; this is not the case in synthetic geometry, where a line is another fundamental object that

5226-554: The same definition is used, but the defining function is required to be differentiable. Algebraic geometry studies algebraic curves , which are defined as algebraic varieties of dimension one. A surface is a two-dimensional object, such as a sphere or paraboloid. In differential geometry and topology , surfaces are described by two-dimensional 'patches' (or neighborhoods ) that are assembled by diffeomorphisms or homeomorphisms , respectively. In algebraic geometry, surfaces are described by polynomial equations . A solid

5304-434: The seventies, the high number of compulsory lessons were evenly spread throughout the six schooldays of the week, there was de facto no loss of class time because of ill teachers or shortage of teachers, the compulsory teaching was finished around noon and the afternoon was free for a variety of optional activities like elective teaching, study groups, project groups, children's sports and organised afternoon care for students in

5382-428: The student was integrated into the team where they would work after the apprenticeship. The vocational training could take place in the students home town, but often occurred in another city. Students lived there in an Internat (boarding school). In most cases that was the first time in the young persons life they lived "independent" from their parents' home for one or two years. The students were allowed to visit home on

5460-589: The study of Euclidean concepts such as points , lines , planes , angles , triangles , congruence , similarity , solid figures , circles , and analytic geometry . Euclidean vectors are used for a myriad of applications in physics and engineering, such as position , displacement , deformation , velocity , acceleration , force , etc. Differential geometry uses techniques of calculus and linear algebra to study problems in geometry. It has applications in physics , econometrics , and bioinformatics , among others. In particular, differential geometry

5538-409: The theory of manifolds and Riemannian geometry . Later in the 19th century, it appeared that geometries without the parallel postulate ( non-Euclidean geometries ) can be developed without introducing any contradiction. The geometry that underlies general relativity is a famous application of non-Euclidean geometry. Since the late 19th century, the scope of geometry has been greatly expanded, and

5616-513: The weekends. With a completed apprenticeship the worker/farmer was qualified to have additional training for supervisor (Meister), or technological school (Ingenieurschule). To go to university, Abitur was necessary. Big companies often trained more apprentices than they could absorb. Because after apprenticeship people went to national service, moved town, went to universities, changed jobs to work closer to home, etc. Hence companies had all age levels from 16-year-olds to seniors, who then trained up

5694-659: The word Spartacus ). If a student did not enter into the EOS to take the Abitur (similar to the A-level in England) after the 8th year of POS, they applied for vocational training after the 9th year. The result of the 9th year was therefore often more important than the final exam. The contract was then signed during the tenth year, so that after the tenth year of polytechnical school, a student went on to 2 or 3 years (depending on subject) of vocational training. Vocational training

5772-438: The world to be of a very high standard. There were two ways to get into a university: either via EOS or via apprenticeship plus abitur. For those who found their calling later in life, there were Volkshochschulen (People's Colleges) for night classes and a special university preparation course in a boarding school lasting 1 year in an ABF ( Arbeiter- und Bauern Fakultät ; Workers and Farmers College). While everyone could visit

5850-444: Was not taught in school, but could be learned in the respective religious community outside of school. Instead of a comprehensive school with primary education followed by secondary education, the POS restructured the classic education process completely by establishing a systemic curriculum which expanded the concepts of secondary education into the lower classes. For instance in mathematics, handling of variables, math text problems with

5928-403: Was offered for every subject that was not taught at university, such as masonry, farming, accountancy, kindergarten teacher, nurse, mechanics, electricians, carpentry, butchery, etc. Vocational training was split in practical work and theoretical learning which focused both on the studied subject of career, and ended with a certificate and a formal title. In the latter part of the apprenticeship

6006-596: Was the creation of analytic geometry, or geometry with coordinates and equations , by René Descartes (1596–1650) and Pierre de Fermat (1601–1665). This was a necessary precursor to the development of calculus and a precise quantitative science of physics . The second geometric development of this period was the systematic study of projective geometry by Girard Desargues (1591–1661). Projective geometry studies properties of shapes which are unchanged under projections and sections , especially as they relate to artistic perspective . Two developments in geometry in

6084-439: Was vocational training with Abitur, which lasted three years after leaving the POS. East German universities were closely linked to both schools and to industry. The universities selected their own students from the applicants. As the school system was centralized, all school certificates were comparable. No university entry exam was necessary. Mostly focused on technical education, these universities were highly regarded all over

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