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  "body": "\n## Example\n\nBoy's surface is an [[immersion|6RW68JGBHC]] of the real projective plane $\\R P^2$ in $\\R^3$.\nThat is, it is a smooth map $f \\colon \\R P^2 \\to \\R^3$ whose differential is injective at every point.\nSo the image has a tangent plane everywhere and no pinch points.\nThe map is not injective: its self-intersection is a closed curve made of three loops, all passing through one triple point.\nThe picture has a threefold rotational symmetry.\nWerner Boy found it in 1901.\n\n## Remarks\n\nThe 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.\nOther 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.\nAny 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.\n\n## Explanations\n\n### L1 human\n\nThe projective plane cannot sit in ordinary space without crossing itself.\nBoy'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.\nIt looks like a rounded shape with three lobes, whose walls pass through each other along three loops meeting at a single point.\n",
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