Manifolds and cell complexes Surfaces Example

Boy's surface

Example

Boy's surface is an immersion of the real projective plane $\R P^2$ in $\R^3$ . That is, it is a smooth map $f \colon \R P^2 \to \R^3$ whose differential is injective at every point. So the image has a tangent plane everywhere and no pinch points. The map is not injective: its self-intersection is a closed curve made of three loops, all passing through one triple point. The picture has a threefold rotational symmetry. Werner Boy found it in 1901.

Remarks

The real projective plane is a closed non-orientable surface, so it has no embedding in $\R^3$ ; Boy's surface shows that it does have an immersion. Other classical models of $\R P^2$ in $\R^3$ , such as the cross-cap and the Roman surface, have singular points where the differential is not injective, so they are not immersions. Any generic immersion of a closed surface with odd Euler characteristic in $\R^3$ has an odd number of triple points, and $\chi(\R P^2) = 1$ , so one triple point is the least possible.

Tag	`C7WPR8W54K`
URI	`m2://C7WPR8W54K`
Kind	example
Writing	W0 stub
MSC	57K20, 57R42
Instance of	Immersion
Same as	Wikidata Q896057 Wikipedia Boy's surface
Changed	2 November 2026

Boy's surface

Example

Remarks

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