---
tag: C7WPR8W54K
kind: example
name: Boy's surface
msc: [57K20, 57R42]
instance_of: [6RW68JGBHC]
sources:
  - url: https://en.wikipedia.org/wiki/Boy%27s_surface
same_as:
  wikidata: Q896057
  wikipedia: Boy's surface
---

## Example

Boy's surface is an [[immersion|6RW68JGBHC]] 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|VGJJVBZ1MQ]], 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.

## Explanations

### L1 human

The projective plane cannot sit in ordinary space without crossing itself.
Boy's surface is a way to place it in space so that it crosses itself only along a curve, with no sharp points or creases.
It looks like a rounded shape with three lobes, whose walls pass through each other along three loops meeting at a single point.