# Non-Euclidean Geometry and Nonorientable Surfaces

## Timothy Faulkner

The diagram illustrating that two perpendicular planes can meet in just one point in four space is on page 144, not page 148.

I would like to see the example of the real projective plane illustrated in four-space a little more fully with the twisted icosahedron. When I learned projective geometry, the shape of projective space was only conveyed with the three-dimensional analogy of the hemisphere with antipodal points of the hemisphere twisted across the disk--kind of like a leather change purse. . . I would like to see that ³point from which we could see all five edges of the pentagon.²

This chapter seems awfully mean to Kant--lots of people are resistant to nonorientability, and I think that if ³some future interstellar explorer [discovered] an orientation-reversing path in our own three-dimensional space,² it would dismay a great many more people than just ³the future followers of Kant.²

Prof. Banchoff's response