, 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 David Hilbert
to prove that the projective plane ''could not'' be immersed in 3-space
Boy's surface was first parametrized
explicitly by Bernard Morin
Another parametrization was discovered by Rob Kusner and Robert Bryant
Boy's surface is one of the two possible immersions of the real projective plane which have only a single triple point.
Unlike the Roman surface
and the cross-cap
, it has no other singularities
s (that is, it has no pinch-points
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.
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
Model at Oberwolfach
Model of a Boy's surface in Oberwolfach
The Mathematical Research Institute of Oberwolfach
has a large model of a Boy's surface outside the entrance, constructed and donated by Mercedes-Benz
in January 1991. This model has 3-fold rotational symmetry
and minimizes the Willmore energy
of the surface. It consists of steel strips which represent the image of a polar coordinate grid
under a parameterization given by Robert Bryant and Rob Kusner. The meridians (rays) become ordinary Möbius strip
s, 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 .
Boy's surface can be used in sphere eversion
, as a half-way model
. A half-way model is an immersion of the sphere with the property that a rotation interchanges inside and outside, and so can be employed to evert (turn inside-out) a sphere. Boy's (the case p = 3) and Morin's
(the case p = 2) surfaces begin a sequence of half-way models with higher symmetry first proposed by George Francis, indexed by the even integers 2p (for p odd, these immersions can be factored through a projective plane). Kusner's parametrization yields all these.
Parametrization of Boy's surface
Boy's surface can be parametrized in several ways. One parametrization, discovered by Rob Kusner and Robert Bryant
is the following: given a complex number ''w'' whose magnitude
is less than or equal to one (
where ''x'', ''y'', and ''z'' are the desired Cartesian coordinates
of a point on the Boy's surface.
If one performs an inversion of this parametrization centered on the triple point, one obtains a complete minimal surface
with three ends
(that's how this parametrization was discovered naturally). This implies that the Bryant–Kusner parametrization of Boy's surfaces is "optimal" in the sense that it is the "least bent" immersion of a projective plane
Property of Bryant–Kusner parametrization
If ''w'' is replaced by the negative reciprocal of its complex conjugate
then the functions ''g''1
, and ''g''3
of ''w'' are left unchanged.
By replacing in terms of its real and imaginary parts , and expanding resulting parameterization, one may obtain a parameterization of Boy's surface in terms of rational function
s of and . This shows that Boy's surface is not only an algebraic surface
, but even a rational surface
. The remark of the preceding paragraph shows that the generic fiber
of this parameterization consists of two points (that is that almost every point of Boy's surface may be obtained by two parameters values).
Relating the Boy's surface to the real projective plane
be the Bryant–Kusner parametrization of Boy's surface. Then
This explains the condition
on the parameter: if
However, things are slightly more complicated for
In this case, one has
This means that, if
the point of the Boy's surface is obtained from two parameter values:
In other words, the Boy's surface has been parametrized by a disk such that pairs of diametrically opposite points on the perimeter
of the disk are equivalent. This shows that the Boy's surface is the image of the real projective plane
by a smooth map
. That is, the parametrization of the Boy's surface is an immersion
of the real projective plane into the Euclidean space
* This describes a piecewise linear model of Boy's surface.
** Article on the cover illustration that accompanies the Rob Kirby article.
* Sanderson, B''Boy's will be Boy's''
(undated, 2006 or earlier).
at MathCurve; contains various visualizations, various equations, useful links and references
– applet from ''Plus Magazine''.
including thoriginal article
of a topologist in thOberwolfach Boy's surface
A paper model of Boy's surface
– pattern and instructions
in Constructive Solid Geometry
together with assembling instructions
visualization video from the Mathematical Institute of the Serbian Academy of the Arts and Sciences