In mathematics, the real projective plane is an example of a compact non-orientable two-dimensional manifold, that is, a one-sided surface. It cannot be embedded in our usual three-dimensional space without intersecting itself. It has basic applications to geometry, since the common construction of the real projective plane is as the space of lines in R3 passing through the origin.
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. Topologically, it has Euler characteristic 1, hence a demigenus (non-orientable genus, Euler genus) of 1.
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, × ) 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 leftmost diagram on the right.
Read more about Real Projective Plane: Examples, Homogeneous Coordinates, The Flat Projective Plane, Embedding Into 4-dimensional Space, Higher Non-orientable Surfaces
Famous quotes containing the words real and/or plane:
“Shakespeare carries us to such a lofty strain of intelligent activity, as to suggest a wealth which beggars his own; and we then feel that the splendid works which he has created, and which in other hours we extol as a sort of self-existent poetry, take no stronger hold of real nature than the shadow of a passing traveller on the rock. The inspiration which uttered itself in Hamlet and Lear could utter things as good from day to day, for ever.”
—Ralph Waldo Emerson (1803–1882)
“Have you ever been up in your plane at night, alone, somewhere, 20,000 feet above the ocean?... Did you ever hear music up there?... It’s the music a man’s spirit sings to his heart, when the earth’s far away and there isn’t any more fear. It’s the high, fine, beautiful sound of an earth-bound creature who grew wings and flew up high and looked straight into the face of the future. And caught, just for an instant, the unbelievable vision of a free man in a free world.”
—Dalton Trumbo (1905–1976)