Relation To The Real Projective Plane
The sphere, before being transformed, is not homeomorphic to the real projective plane, RP2. But the sphere centered at the origin has this property, that if point (x,y,z) belongs to the sphere, then so does the antipodal point (-x,-y,-z) and these two points are different: they lie on opposite sides of the center of the sphere.
The transformation T converts both of these antipodal points into the same point,
If this were true for only one or small subset of points of the sphere, then these points would just be double points. But since it is true of all points, then it is possible to consider the Roman surface to be homeomorphic to a "sphere modulo antipodes". However, these are not the only identifications that occur under this map. Consequently, the Roman surface is a quotient of the real projective plane RP2 = S2 / (x~-x). Furthermore, this quotient has the special property that it is locally injective, making this an immersion of RP2 into 3-space. It was previously stated that the Roman surface is homeomorphic to RP2, but this was in error.
Read more about this topic: Roman Surface
Famous quotes containing the words relation to, relation, real and/or plane:
“The foregoing generations beheld God and nature face to face; we, through their eyes. Why should not we also enjoy an original relation to the universe? Why should not we have a poetry and philosophy of insight and not of tradition, and a religion by revelation to us, and not the history of theirs?”
—Ralph Waldo Emerson (1803–1882)
“There is undoubtedly something religious about it: everyone believes that they are special, that they are chosen, that they have a special relation with fate. Here is the test: you turn over card after card to see in which way that is true. If you can defy the odds, you may be saved. And when you are cleaned out, the last penny gone, you are enlightened at last, free perhaps, exhilarated like an ascetic by the falling away of the material world.”
—Andrei Codrescu (b. 1947)
“I have no connections here; only gusty collisions,
rootless seedlings forced into bloom, that collapse.
...
I am the Visiting Poet: a real unicorn,
a wind-up plush dodo, a wax museum of the Movement.
People want to push the buttons and see me glow.”
—Marge Piercy (b. 1936)
“We’ve got to figure these things a little bit different than most people. Y’know, there’s something about going out in a plane that beats any other way.... A guy that washes out at the controls of his own ship, well, he goes down doing the thing that he loved the best. It seems to me that that’s a very special way to die.”
—Dalton Trumbo (1905–1976)