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.
Read more about this topic: Real Projective Plane
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“It is the space inside that gives the drum its sound.”
—Hawaiian saying no. 1189, ‘lelo No’Eau, collected, translated, and annotated by Mary Kawena Pukui, Bishop Museum Press, Hawaii (1983)