Real Projective Plane - Embedding Into 4-dimensional Space

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 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 which is the Roman surface.