Editing Real projective plane (section)

== Embedding into 4-dimensional space ==
The projective plane embeds into 4-dimensional Euclidean space.  The real projective plane '''P'''<sup>2</sup>('''R''') is the [[Quotient space (topology)|quotient]] of the two-sphere
: '''S'''<sup>2</sup> = {(''x'', ''y'', ''z'') ∈ '''R'''<sup>3</sup> : ''x''<sup>2</sup> + ''y''<sup>2</sup> + ''z''<sup>2</sup> = 1}
by the antipodal relation {{nowrap|(''x'', ''y'', ''z'') ~ (−''x'', −''y'', −''z'')}}.  Consider the function {{nowrap|'''R'''<sup>3</sup> → '''R'''<sup>4</sup>}} given by {{nowrap|(''x'', ''y'', ''z'') ↦ (''xy'', ''xz'', ''y''<sup>2</sup> − ''z''<sup>2</sup>, 2''yz'')}}.  This map restricts to a map whose domain is '''S'''<sup>2</sup> and, since each component is a homogeneous polynomial of even degree, it takes the same values in '''R'''<sup>4</sup> on each of any two antipodal points on '''S'''<sup>2</sup>.   This yields a map {{nowrap|'''P'''<sup>2</sup>('''R''') → '''R'''<sup>4</sup>}}. Moreover, this map is an embedding. Notice that this embedding admits a projection into '''R'''<sup>3</sup> which is the [[Roman surface]].