While the Möbius strip can be embedded in three-dimensional Euclidean space R < sup > 3 </ sup >, the Klein bottle cannot.

While Euclidean geometry, named after the Greek mathematician Euclid, includes some of the oldest known mathematics, non-Euclidean geometries were not widely accepted as legitimate until the 19th century.

While Lobachevsky created a non-Euclidean geometry by negating the parallel postulate, Bolyai worked out a geometry where both the Euclidean and the hyperbolic geometry are possible depending on a parameter k. Bolyai ends his work by mentioning that it is not possible to decide through mathematical reasoning alone if the geometry of the physical universe is Euclidean or non-Euclidean ; this is a task for the physical sciences.

While seldom used explicitly, the geometric view of the complex numbers is implicitly based on its structure of a Euclidean vector space of dimension 2, where the inner product of complex numbers and is given by ; then for a complex number its absolute value || coincides with its Euclidean norm, and its argument with the angle turning from 1 to.

While a Euclidean space has only spacelike dimensions, a Minkowski space also has one timelike dimension.

While each side would have length √ 2r using a Euclidean metric, where r is the circle's radius, its length in taxicab geometry is 2r.

While in Euclidean geometry a geodesic intersects its parallels in both directions in the limit to infinity, in hyperbolic geometry both directions have their own line of parallelism.

While Grothendieck topologies are most often used to define cohomology theories, they have found other applications as well, such as to John Tate's theory of rigid analytic geometry.

While the supergravity description assumes a continuous space-time, Matrix theory predicts that, at short distances, non-commutative geometry takes over, somewhat similar to the way the continuum of water breaks down at short distances in favor of the graininess of molecules.

While all motion in a mechanical system occurs in three dimensional space, planar motion can be analyzed using plane geometry, if all point trajectories are parallel to a plane.

While this definition is the most abstract, it is also the one most easily transferred to other settings, for instance to the varieties considered in algebraic geometry.

While he is best known for the Kolmogorov – Arnold – Moser theorem regarding the stability of integrable Hamiltonian systems, he made important contributions in several areas including dynamical systems theory, catastrophe theory, topology, algebraic geometry, classical mechanics and singularity theory, including posing the ADE classification problem, since his first main result — the partial solution of Hilbert's thirteenth problem in 1957 at the age of 19.

While the ancient Egyptians empirically discovered some truths of geometry, the great achievement of the ancient Greeks was to replace empirical methods by demonstrative science.

While earlier Greek proofs were largely geometric demonstrations, the development of arithmetic and algebra by Islamic mathematicians allowed more general proofs that no longer depended on geometry.

While his original writings demonstrate that he had a sincere passion for the seven liberal arts ( grammar, rhetoric, logic, mathematics, geometry, music, and astronomy ), his work in Quaestiones naturales illustrated a more encompassing dedication to subjects such as physics, the natural sciences, and possibly even metaphysics.

While the local geometry does not determine the global geometry completely, it does limit the possibilities, particularly a geometry of a constant curvature.

While z-buffering allows the geometry to be unsorted, sorting polygons by increasing depth ( thus using a reverse painter's algorithm ) allows each screen pixel to be rendered fewer times.

While the ideas were available earlier, projective geometry was mainly a development of the nineteenth century.

While it is impossible to construct a perfect regular 56-sided polygon using simple ' square and circle ' geometry, a close approximation has recently been discovered which it is claimed might have been used at Stonehenge.

While the former picture looks across the bridge with the ironworks diagonally crossing the picture to the right, with a scene of partially interacting chickens on the bridge to the left, the latter depicts the iron structure of the bridge face-on in a strong close-up of its industrial geometry, with three male figures to the left side of the painting all looking in different directions ( the Pont de l ' Europe is a massive bridge spanning the railyard of the newly-expanded station, which at that time had an iron-work trellis ).

While the Quillen algebraic K-theory has provided deep insight into various aspects of algebraic geometry and topology, the K-groups have proved particularly difficult to compute except in a few isolated but interesting cases.

While all these things made the game appear to be 3D, it wouldn't be until later first-person shooters, like Quake, that the engine actually stored the world geometry as true 3D information.

While algebraic geometry studies algebraic varieties, analytic geometry deals with complex manifolds and the more general analytic spaces defined locally by the vanishing of analytic functions of several complex variables.

While several yards from it, still concealed by the shrubbery, she'd seen two men on her left at the pool's edge.

While this was being discussed, we saw the militia to the west of us fanning out and breaking into little clusters of two and three men as they approached the road.

While prowling around these buildings, two of the posse recognized the voice of Gonzales speaking to the people inside.

While Communists were undermining United Nations efforts to rescue the Congo from chaos, two other Communist offensives stirred the Eisenhower Administration into emergency conferences and serious decisions.

While the two are heading a caravan from Salalah, Oman, they cross the border into Yemen and find the unexplored desert area that the Necronomicon calls " Roba el Ehaliyeh " or " Roba el Khaliyeh " — presumably a reference to the Empty Quarter or " Rub al Khali ".

While there are ancient relations between the Indian Vedas and the Iranian Avesta, the two main families of the Indo-Iranian philosophical traditions were characterized by fundamental differences in their implications for the human being's position in society and their view on the role of man in the universe.

While Andronikos was on one of his incursions, his castle was surprised by the governor of Trebizond, and Theodora and her two children were captured and sent to Constantinople.

While encysted it may undergo a peculiar process of autogamy or self-fertilization, where it goes through meiosis and divides to form two gametes, which then fuse together again.

While the complete title of the book is the Acts of the Apostles, really the book focuses on only two men: The Apostle Peter ( chs.

While there are some similarities between the two movements, Theodism derived its origins primarily as a reaction to Wicca.

While Aristotle likewise identifies the first two characteristics, St. Thomas conceives of the third as an appropriation from principles developed by neo-Platonic and Augustinian thinkers.

While the two herbicides that make up Agent Orange, 2, 4-D and 2, 4, 5-T, remain toxic over a short period — a scale of days or weeks — they quickly degrade.

While 700 Ethiopians are killed and many more injured, only two die from the British / Indian troops.

While in the view of most American academics the two were as diametrically opposed as good and evil, Sakharov believed that in this " tragic confrontation of two outstanding people ," both deserved respect, because " each of them was certain he had right on his side and was morally obligated to go to the end in the name of truth.

While there, Desiree notices Fredrik ; the two were lovers years before.

While some women's bicycles continue to use this frame style, there is also a variation, the mixte, which splits the top tube laterally into two thinner top tubes that bypass the seat tube on each side and connect to the rear fork ends.

While most casts have five toes — like all known apes — some casts of alleged Bigfoot tracks have had numbers ranging from two to six.

While traditional teaching maintains that any bishop with apostolic succession can validly perform the ordination of another bishop, some churches require two or three bishops participate, either to ensure sacramental validity or to conform with church law.

While their coordinate distance ( comoving distance ) remains constant, the physical distance between two such comoving points expands proportionally with the scale factor of the Universe.

While Brown is considered a small research university with 682 full-time faculty and 1, 919 graduate students, five of its professors and two of its alumni have been honored as Nobel Laureates.

While the British military historian Sir John Keegan suggested an ideal definition of battle as " something which happens between two armies leading to the moral then physical disintegration of one or the other of them ", the origins and outcomes of battles can rarely be summarized so neatly.

While the town's name is generally seen as a diminutive form of Barcelona in Spain, Albert Dauzat and Charles Rostaing point out an earlier attestation of the name Barcilona in Barcelonnette in around 1200, and suggest that it is derived instead from two earlier stems signifying a mountain, * bar and * cin ( the latter of which is also seen in the name of Mont Cenis ).

While some layer two network devices are able to divide the collision domains, broadcast domains are only divided by layer 3 network devices such as routers or layer 3 switches.

While the battle raged in the bay, the two straggling British ships made strenuous efforts to join the engagement, focusing on the flashes of gunfire in the darkness.