Boy's surface

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An animation of Boy's surface

In geometry, Boy's surface is an immersion of the real projective plane in 3-dimensional space found by Werner Boy in 1901 (he discovered it on assignment from Hilbert to prove that the projective plane could not be immersed in 3-space). Unlike the Roman surface and the cross-cap, it has no singularities (pinch points), but it does self-intersect.

To make a Boy's surface:

Start with a sphere. Remove a cap.
Attach one end of each of three strips to alternate sixths of the edge left by removing the cap.
Bend each strip and attach the other end of each strip to the sixth opposite the first end, so that the inside of the sphere at one end is connected to the outside at the other. Make the strips skirt the middle rather than go through it.
Join the loose edges of the strips. The joins intersect the strips.

Boy's surface is discussed (and illustrated) in Jean-Pierre Petit's Le Topologicon.

Boy's surface was first parametrized correctly by Bernard Morin in 1978. See below for another parametrization, discovered by R. Bryant.

1 Parametrization of Boy's surface
2 Property of R. Bryant's parametrization
- 2.1 Relating the Boy's surface to the real projective plane
3 Symmetry of the Boy's surface
4 Model at Oberwolfach
5 References

[edit] Parametrization of Boy's surface

Boy's surface can be parametrized in several ways. One parametrization, discovered by R. Bryant, is the following: given a complex number z whose magnitude is less than or equal to one, let

$g_1 = -{3 \over 2} \mathrm{Im} \left( {z (1 - z^4) \over z^6 + \sqrt{5} z^3 - 1} \right),$

$g_2 = -{3 \over 2} \mathrm{Re} \left( {z (1 + z^4) \over z^6 + \sqrt{5} z^3 - 1} \right),$

$g_3 = \mathrm{Im} \left( {1 + z^6 \over z^6 + \sqrt{5} z^3 - 1} \right) - {1 \over 2},$

$g = g_1^2 + g_2^2 + g_3^2,$

so that

$X = {g_1 \over g},$

$Y = {g_2 \over g},$

$Z = {g_3 \over g},$

where X, Y, and Z are the desired Cartesian coordinates of a point on the Boy's surface.

[edit] Property of R. Bryant's parametrization

If z is replaced by the negative reciprocal of its complex conjugate, $- {1 \over z^\star},$ then the functions g₁, g₂, and g₃ of z are left unchanged. (proof)

[edit] Relating the Boy's surface to the real projective plane

Let $P (z) = (X (z), Y (z), Z (z))$ denote a point on Boy's surface, where $\| z \| \le 1.$ Then

$P(z) = P\left( -{1 \over z^\star} \right)$

but only if $\| z \| = \sqrt{z z^\star} = 1.$ What if $\| z \| < 1 ?$ Then

$\left\| - {1 \over z^\star} \right\| > 1$

because

$- {1 \over z^\star} = {- z \over z^\star z} = {-z \over \| z \|^2}$

whose magnitude is

${\| z \| \over \| z \|^2} = {1 \over \| z \|},$

but $\| z \| < 1,$ so that

${1 \over \| z \|} > 1.$

Since P(z) belongs to the Boy's surface only when $\|z\| \le 1,$ this means that $P\left( - {1 \over z^\star} \right)$ belongs to Boy's surface only if $\| z \| = 1.$ Thus $P (z) = P ( - z)$ if $\| z \| = 1,$ but all other points on the Boy's surface are unique. The Boy's surface has been parametrized by a unit disk such that pairs of diametrically opposite points on the perimeter of the disk are equivalent. Therefore the Boy's surface is homeomorphic to the real projective plane, RP².

[edit] Symmetry of the Boy's surface

Boy's surface has 3-fold symmetry. This means that it has an axis of discrete rotational symmetry: any 120° turn about this axis will leave the surface looking exactly the same. The Boy's surface can be cut into three mutually congruent pieces. (proof)

[edit] Model at Oberwolfach

Model of a Boy's surface in Oberwolfach

The Mathematisches Forschungsinstitut Oberwolfach has a large model of a Boy's surface outside the entrance, constructed and donated by Mercedes-Benz in 1991. It consists of steel strips which represent the image of a polar coordinate grid under the above parametrization by Bryant. The meridians (rays) become ordinary Möbius strips, i.e. twisted by 180 degrees. All but one of the strips corresponding to circles of latitude (radial circles around the origin) are untwisted, while the one corresponding to the boundary of the unit circle is a Möbius strip twisted by three times 180 degrees - as is the emblem of the institute.^[1]