The main goal of this thesis is to present the elementary proof for nonembeddability of real projective plane in the 3dimensional euclidean space and to study the embeddability of closed surfaces in general. A constructive real projective plane mark mandelkern abstract. The surface is called the real projective plane or just projective plane and it can be constructed in a variety of ways. We start with the short introduction of ideal points concept from projective geometry and present different geometrical presentations of real projective plane. Identifying both pairs of opposite sides in the same direction gives the torus t2. One of those is equivalent to tic tac toe on a torus.
Whereas a mobius strip is a surface with boundary, a klein bottle has no boundary for comparison, a sphere is an orientable surface with no boundary. Nonembeddability of real projective plane in r3 eprints. Triangulating the real projective plane 3 1 introduction the real projective plane p2 is in onetoone correspondence with the set of lines of the vector space r3. Anything that satisfies these rules is a projective plane, but when mathematicians refer to the projective plane, they generally mean a space more properly known as the real projective plane, or. Its discovery is attributed to the german mathematicians. If a circular disk is cut out of the real projective plane, what is left is a mobius strip. Like a normal loop, an ant crawling along it would never reach an end, but in a normal loop, an ant could only crawl along either the top or the bottom.
Looking at the constructions of real projective space and the mobius strip from squares though, i cant see. On dimensional grounds, sl2, c covers a neighborhood of the identity of so1,3. Projective geometry in a plane fundamental concepts undefined concepts. Homemorphism from projective plane to s1 and moebius strip. The recycling symbol of three folded arrow forms a mobius strip. We do not think about identification of the edge of mobius strip as well as the cruciform projective plane. We will calculate the euler characteristic of some surfaces to facilitate understanding. For the cylinder, since we identity awith a0, there are two vertices aa0 and bb0, four edges and two triangles. A quadrangle is a set of four points, no three of which are collinear. A simple ovoid in real 3space can be constructed by glueing together two suitable halves. If instead you make the belt extend all the way to the poles, youll have proved the stronger result that the real projective plane minus one point is homeomorphic to the mobius strip without boundary. Going in the other direction, if one glues a disk to a mobius strip by identifying their boundaries, the result is the projective plane. The topological result of attaching a mbius strip to a disk along its boundary is a real projective plane, which cannot be embedded in r3.
Any two lines l, m intersect in at least one point, denoted lm. We now consider several ways of viewing the real projective plane in a3. Mobius strip synonyms, mobius strip pronunciation, mobius strip translation, english dictionary definition of mobius strip. You can easily construct and experiment with a mobius strip using paper, scissors, tape, and a pencil. M on f given by the intersection with a plane through o parallel to c, will have no image on c. Im not necessarily looking for an explicit homeomorphism, just an intuitive argument of why this is the case. The classical theory of plane projective geometry is examined constructively, using both synthetic and analytic methods.
Why is the mobius strip so important in mathematics and. One may observe that in a real picture the horizon bisects the canvas, and projective plane. Figure genus number of sides can all sides be touched yesno number of dimensions needed to build disc square tube band. As a partial projective plane, a mobius strip will do.
Since for degrees higher than 4, it is practically impossible to compute manually control nets from. The mobius strip is the simplest example of something thats nonorientable, which basically means that if you lived on a mobius strip you wouldnt be able to have a consistent notion of clockwise that works everywhere, because if you went aroun. If we view the mobius strip as a fibre bundle over with fibres, then the fibrewise compactification is the klein bottle this is a fibre bundle of over. Another closely related manifold is the real projective plane.
Of course, we will also present a model of the mobius strip. The real numbers xand yare traditionally called the real and imaginary parts of zand are denoted by x rez. The second is formed by attaching two mobius bands along their common boundary to form a nonorientable surface called a klein bottle, named for its discoverer, felix klein. Mobius bands, real projective planes, and klein bottles. Deformation retraction of plane rp2 physics forums. Other related nonorientable objects include the mobius strip and the real projective plane. Identifying one pair of sides in the opposite direction gives the klein bottle. Feb 18, 2016 real projective plane and moebius strip jos leys. Let rp2 denote the real projective plane it can be obtained from glueing a mobius band and a disk whose boundary is the same as the boundary of the mobius band. A surface is nonorientable if there is no consistent notion of right handed versus left handed on it.
German mathematician johann benedict listing independently thought of the same idea in july 1858. And lines on f meeting on m will be mapped onto parallel lines on c. The orientable counterpart being handle attachment the mobius band, in contrast, has a twist. While the torus t generates the orientable closed surfaces, similarly, the real projective plane pgenerates the nonorientable closed surfaces. The klein bottle was first described in 1882 by the german mathematician felix klein. The simplest example is the mobius band, a twisted strip with one side, and one edge. The first, called a real projective plane, is obtained by attaching the boundary of a disc to the boundary of a mobius band. One way to think about it is to take the flat klein bottle. Sl2, c, and this action of sl2, c preserves the determinant of x because det a 1.
Apart from mc eschers painting mobius strip ii, which feature ants crawling around the surface of a mobius strip, the mobius strip has become a popular design for scarves. I know if one punches a hole off rp2 then the punched rp2 is homotopy equivalent to a mobius band which is in turn deformable to a circle, i also know that rp2 is not deformable to a. Give two proofs that the klein bottle k is the union of two mobius strips. An ovoid is a quadratic set and bears the same geometric properties as a sphere in a projective 3space. Why do these two procedures produce the same result. Projective plane p2 is obtained by identifying each pair of antipodal points p. Feb 16, 2012 according to this book im reading, if you cut out a closed disc in the projective plane, then the complement of the interior of this disc is topologically a mobius strip with boundary. The real projective plane minus a disc is a moebius strip.
What are the equivalence relations and equivalence classes for these identi cations. Or is the crosscap what results from the connected sum of a disk and a mobius band, i. The m obius strip 1 is a wellknown classical object in geometry and topology which attracts not only professional mathematicians, but many people. The opening ceremony is one of my favorite partsthe celebration of the host countrys history and culture, the athletes proudly marching in and representing their homeland.
For example, gluing two real projective planes together gives the klein bottle k. Difference between crosscap, mobius band, and real. The mobius strip is the simplest nonorientable surface. Unfortunately for listing, one of the most famous surfaces in mathematics was named for mobius, not listing. The whole of projective plane is represented by a hemisphere or disktype projective plane or crosscap. Meeks showed that if m is diffeomorphic to a real projective plane minus two points, then it does not admit a complete minimal immersion into e 3 with total curvature. It cannot be embedded in standard threedimensional space without intersecting itself. The topics include desarguess theorem, harmonic conjugates, projectivities, involutions, conics, pascals theorem, poles and polars. Fill in the following information about each of the figures in the open spaces below. Show that the projective plane p is the union of a mobius strip and a disk. Sep 06, 2019 a mobius strip can come in any shape and size. The mobius strip is named after mathematician and astronomer august ferdinand mobius. Any two points p, q lie on exactly one line, denoted pq. How to continuously deform a mobius strip into a crosscap homeomorphism duration.
It can be obtained by glueing a strip of paper with a twist i. Aug 31, 2017 anything that satisfies these rules is a projective plane, but when mathematicians refer to the projective plane, they generally mean a space more properly known as the real projective plane, or. And the best thing about a mobius strip is that we can actually build it. The mobius strip has the mathematical property of being unorientable. Cutting a hole in the projective plane physics forums. In mathematics, the real projective plane is an example of a compact nonorientable twodimensional manifold. Mobius strip definition of mobius strip by the free.
The cylinder is a double cover of the mobius strip. It has basic applications to geometry, since the common construction of the real projective plane is as the space of lines in r 3 passing through the origin. Since the determinant of x is identified with the quadratic form q, sl2, c acts by lorentz transformations. Real projective plane another important closed surface is the real projective plane, which can be formed by either gluing a disc to a m obius strip, or gluing the opposite points of the boundary circle of a disc together. According to this book im reading, if you cut out a closed disc in the projective plane, then the complement of the interior of this disc is topologically a mobius strip with boundary. The onepoint compactification of the mobius strip is the real projective plane.
Real projective plane 345 steiners roman surface when jakob steiner visited rome in 1844 he developed the concept of a surface that now carries his name see ap. However, there are three surfaces that are representations of the projective plane in r3 with selfintersections, namely the boy surface, crosscap, and roman surface. Stereographic projection of the real projective plane figure 2. The topics include desarguess theorem, harmonic conjugates, projectivities, involutions, conics. Its twodimensional, not threedimensional like actual sheets of paper. If an ant were to crawl along the surface of the mobius strip, it would walk along both the bottom and the top in an infinite loop.
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