A sphere with a single cross-cap. This term is more appropriate in purely topological applications than the more common term real projective plane, which implies the presence of an affine structure (Francis and Weeks 1999). The double cross surface is the Klein bottle and the triple cross surface is called Dyck's surface (Francis and Collins 1993, Francis and Weeks 1999).

# Cross Surface

## See also

Cross-Cap, Real Projective Plane## Explore with Wolfram|Alpha

## References

Francis, G. and Collins, B. "On Knot-Spanning Surfaces: An Illustrated Essay on Topological Art." Ch. 11 in*The Visual Mind: Art and Mathematics*(Ed. M. Emmer). Cambridge, MA: MIT Press, 1993.Francis, G. K. and Weeks, J. R. "Conway's ZIP Proof."

*Amer. Math. Monthly*

**106**, 393-399, 1999.

## Referenced on Wolfram|Alpha

Cross Surface## Cite this as:

Weisstein, Eric W. "Cross Surface." From
*MathWorld*--A Wolfram Web Resource. https://mathworld.wolfram.com/CrossSurface.html