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 the, relation to, relation, real and/or plane:
“It would be disingenuous, however, not to point out that some things are considered as morally certain, that is, as having sufficient certainty for application to ordinary life, even though they may be uncertain in relation to the absolute power of God.”
—René Descartes (1596–1650)
“In relation to God, we are like a thief who has burgled the house of a kindly householder and been allowed to keep some of the gold. From the point of view of the lawful owner this gold is a gift; From the point of view of the burglar it is a theft. He must go and give it back. It is the same with our existence. We have stolen a little of God’s being to make it ours. God has made us a gift of it. But we have stolen it. We must return it.”
—Simone Weil (1909–1943)
“We must get back into relation, vivid and nourishing relation to the cosmos and the universe. The way is through daily ritual, and is an affair of the individual and the household, a ritual of dawn and noon and sunset, the ritual of the kindling fire and pouring water, the ritual of the first breath, and the last.”
—D.H. (David Herbert)
“Great ambition, the desire of real superiority, of leading and directing, seems to be altogether peculiar to man, and speech is the great instrument of ambition.”
—Adam Smith (1723–1790)
“In time the scouring of wind and rain will wear down the ranges and plane off the region until it has the drab monotony of the older deserts. In the meantime—a two-million-year meantime—travelers may enjoy the cruel beauties of a desert in its youth,....”
—For the State of California, U.S. public relief program (1935-1943)