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

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 two geodesics can either intersect or be parallel, in general and in hyperbolic space in particular there are three possibilities.

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 in Athens, his wife Pythias died and Aristotle became involved with Herpyllis of Stageira, who bore him a son whom he named after his father, Nicomachus.

While amber is not actually named, it is called the concreti maris purgamentum, " the leavings of the frozen sea " after the spring melt.

While there is an official locality named Akibahara, which is also 秋葉原 in kanji, nearby ( as part of Taitō-ku ), the area known to most people as Akihabara ( including the railway station of the same name ) also include Soto-Kanda, a part of Chiyoda-ku.

While he was still on the Massachusetts Supreme Judicial Court, and before being named to the U. S. Supreme Court, Justice Oliver Wendell Holmes, Jr. published a short volume called The Common Law, which remains a classic in the field.

While the Browns ' on-field play in 1956 was uninspiring, off-the-field drama developed after a Cleveland-based inventor named George Sarles let Brown test a helmet with a radio transmitter inside.

While not musically related, instruments of the Zink family ( which includes serpents ) are named " cornetto-" with a tonal or pitch related Latin word following the hyphen to describe the particular variant.

While at Iwerks, he met a cel painter named Dorothy Webster, who would later become his first wife.

While the Darwin Awards are named after Charles Darwin, due to his theory of natural selection, there is no evidence that the stories depicted actually represent the removal of " judgement impairment genes " from the gene pool.

While Snake is forced to fight with a prisoner named Slag ( Ox Baker ), Brain and Maggie trick the Duke's men into letting them see the President.

While Cabrera, Willis, and several others posted very good first-half numbers, Lowell was one of the least productive regular major-league starters, and Leiter went 3 – 7 with an ERA of 6. 64 before being traded to the New York Yankees on July 15 for a player to be named later.

While Brooks's musical style placed him squarely within the boundaries of country music, he was strongly influenced by the 1970s singer-songwriter movement, especially the works of James Taylor ( whom he idolized and named his first child after ) and Dan Fogelberg.

While the organization was founded under a different name in 1970 and was officially named Greenpeace in 1972, the organization itself dates its birth to the first protest of 1971.

While some of the formerly named streets ( including Factory, Herring and Amity Streets ) are now numbered, even they do not always conform to the usual grid pattern when they enter the neighborhood.

While the Sidhe retreats, one elf ( revealed in the sequel to be named Corvus ), sets off on a quest of vengeance against the weakest of the three Serpent Riders, D ' Sparil.

While visiting New York City in the late ' 30s, Pei met a Wellesley College student named Eileen Loo.

While Bobby sits in his cell at the sheriff station, the outer wall is blown open, and a man named Scott Summers ( Cyclops ) walks in and offers to take Bobby with him.

While in prison, he formed a gospel quartet with fellow cell mates Johnny Terry, " Hucklebuck " Davis and a person named " Shag ", and made homemade instruments-a comb and paper, a washtub bass, a drum kit made from lard tubs and for Brown, what he called " a sort of mandolin out of a wooden box.

While both serving on the Tsaritsyn Front in 1918, where Ekaterina was helping orphans, they adopted a four year old orphan boy who they named Petya.

While the possible colors are virtually limitless, breeders have identified and named a number of specific categories.

While all the other outcasts lay idly lamenting, one of them, named Moses, advised them not to look for help to gods or men, since both had deserted them, but to trust rather in themselves, and accept as divine the guidance of the first being, by whose aid they should get out of their present plight.

While recuperating at Youpon Dunes, a home owned by Elizabeth Patterson, in Myrtle Beach, he worked on the charter for a group of Sufis, which he named Sufism Reoriented.

While Paganini was still a teenager in Livorno, a wealthy businessman named Livron lent him a violin, made by the master luthier Giuseppe Guarneri, for a concert.

While the infectious agent was named a prion, the specific protein that the prion was composed of is also known as the Prion Protein ( PrP ), though this protein may occur both in infectious and non-infectious forms.

While most other necklaces are simply referred to by their physical measurement, pearl necklaces are named by how low they hang when worn around the neck.

While living like this he founded, in 1244, the order subsequently named after him, the Celestines.