Real projective plane
From Wikipedia, the free encyclopedia
 The fundamental polygon of the projective plane. |
 The Möbius strip with a single edge, can be closed into a projective plane by gluing opposite open edges together. |
 In comparison the Klein bottle is a mobius strip closed into a cylinder. |
In mathematics, the real projective plane is a non-orientable two-dimensional manifold, that is, a surface, that has basic applications to geometry, but which cannot be embedded in our usual three-dimensional space without intersecting itself. It has Euler characteristic 1, hence a demigenus (non-orientable genus, Euler genus) of 1.
A common construction of the real projective plane is as the space of lines in R3 passing through the origin. This is often taken as a geometric definition of the real projective plane. The plane is also often described topologically, in terms of a construction based on the Möbius strip: if one could glue the (single) edge of the Möbius strip to itself in the correct direction, one would obtain the projective plane. (This cannot be done in our three-dimensional space.) Equivalently, gluing a disk along the boundary of the Möbius strip gives the projective plane.
Since the Möbius strip, in turn, can be constructed from a square by gluing two of its sides together, the real projective plane can thus be represented as a unit square (that is, [0,1] × [0,1] ) with its sides identified by the following equivalence relations:
- (0, y) ~ (1, 1 − y) for 0 ≤ y ≤ 1
and
- (x, 0) ~ (1 − x, 1) for 0 ≤ x ≤ 1,
as in the diagram on the right.
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[edit] Construction
Consider a sphere, and let the great circles of the sphere be "lines", and let pairs of antipodal points be "points". It is easy to check that this system obeys the axioms required of a projective plane:
- any pair of distinct great circles meet at a pair of antipodal points; and
- any two distinct pairs of antipodal points lie on a single great circle.
This plane is the real projective plane.
If we identify each point on the sphere with its antipodal point, then we get a representation of the real projective plane in which the "points" of the projective plane really are points. This means that the projective plane is the quotient space of the sphere obtained by partitioning the sphere into equivalence classes under the equivalence relation ~, where x ~ y if y = −x. This quotient space of the sphere is homeomorphic with the collection of all lines passing through the origin in R3.
The resulting surface, a 2-dimensional compact non-orientable manifold, is a little hard to visualize, because it cannot be embedded in 3-dimensional Euclidean space without intersecting itself.
The quotient map from the sphere onto the real projective plane is in fact a two sheeted (i.e. two-to-one) covering map. It follows that the fundamental group of the real projective plane is the cyclic group of order 2, i.e. integers modulo 2. One can take the loop AB from the figure above to be the generator.
[edit] Immersing the real projective plane in three-space
The projective plane cannot be embedded (that is without intersection) in three-dimensional space. However, it can be immersed (local neighbourhoods do not have self-intersections). Boy's surface is an example of an immersion.
The Roman surface is a more degenerate map of the projective plane into 3-space, containing a cross-cap. The same goes for a sphere with a cross-cap.
The proof that the projective plane does not embed in three-dimensional Euclidean space goes like this: Assume that it does embed, it would bound a compact region in three-dimensional Euclidean space by the Generalized Jordan Curve Theorem. The outward-pointing unit normal vector field would then give an orientation of the boundary manifold, but the boundary manifold would be projective space, which is not orientable. This is a contradiction, and so our assumption that it does embed must have been false.
A polyhedral representation is the tetrahemihexahedron, which has the same general form as Steiner's Roman Surface, shown to the right.
Looking in the opposite direction, the hemi-cube, hemi-dodecahedron, and hemi-icosahedron, abstract regular polychora, can be constructed as a regular figure in the projective plane.
[edit] Homogeneous coordinates
The points in the plane can be represented by homogeneous coordinates. A point has homogeneous coordinates [x : y : z], where the coordinates [x : y : z] and [tx : ty : tz] are considered to represent the same point, for all nonzero values of t. The points with coordinates [x : y : 1] are the usual real plane, called the finite part of the projective plane, and points with coordinates [x : y : 0], called points at infinity or ideal points, constitute a line called the line at infinity. (The homogeneous coordinates [0:0:0] do not represent any point.)
The lines in the plane can also be represented by homogeneous coordinates. A projective line corresponding to the plane ax + by + c = 0 in R3 has the homogeneous coordinates (a : b : c). Thus, these coordinates have the equivalence relation (a : b : c) = (da : db : dc) for all nonzero values of d. Hence a different equation of the same plane dax+dby+dc=0 gives the same homogeneous coordinates. A point [x : y : z] lies on a line (a : b : c) if ax + by + c = 0. Therefore, lines with coordinates (a : b : c) where a, b are not both 0 correspond to the lines in the usual real plane, because they contain points that are not at infinity. The line with coordinates (0:0:1) is the line at infinity, since the only points on it are those with z = 0.
[edit] Embedding into 4-dimensional space
The projective plane embeds into 4-dimensional Euclidean space. Consider 
to be the quotient of the two-sphere 
by the antipodal relation 
. Consider the function 
given by 
. This map restricts to a map whose domain is S2 and, since it is a purely quadratic polynomial, it can be factorised to give a map 
. Moreover, this map is an embedding. Notice that this embedding admits a projection into R3 which is the Roman surface.
[edit] Higher non-orientable surfaces
By glueing together projective planes successively we get non-orientable surfaces of higher demigenus. The glueing process consists of cutting out a little disk from each surface and identifying (glueing) their boundary circles. Glueing two projective planes creates the Klein bottle.
The article on the fundamental polygon describes the higher non-orientable surfaces.
[edit] See also
[edit] References
- Coxeter, H.S.M. (1955), The Real Projective Plane, 2nd ed. Cambridge: At the University Press.
