Self-intersecting compact surface, an immersion of the real projective plane
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
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
in 1978.
[1]
Another parametrization was discovered by Rob Kusner and
Robert Bryant
.
[2]
Boy's surface is one of the two possible immersions of the real projective plane which have only a single triple point.
[3]
Unlike the
Roman surface
and the
cross-cap
, it has no other
singularities
than
self-intersections
(that is, it has no
pinch-points
).
Parametrization
[
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]
A view of the Kusner?Bryant parametrization of the Boy's surface
Boy's surface can be parametrized in several ways. One parametrization, discovered by Rob Kusner and
Robert Bryant
,
[4]
is the following: given a complex number
w
whose
magnitude
is less than or equal to one (
), let
![{\displaystyle {\begin{aligned}g_{1}&=-{3 \over 2}\operatorname {Im} \left[{w\left(1-w^{4}\right) \over w^{6}+{\sqrt {5}}w^{3}-1}\right]\\[4pt]g_{2}&=-{3 \over 2}\operatorname {Re} \left[{w\left(1+w^{4}\right) \over w^{6}+{\sqrt {5}}w^{3}-1}\right]\\[4pt]g_{3}&=\operatorname {Im} \left[{1+w^{6} \over w^{6}+{\sqrt {5}}w^{3}-1}\right]-{1 \over 2}\\\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/24e2d30d1e2883a59c352c8e82321c4d4bca4bff)
and then set
![{\displaystyle {\begin{pmatrix}x\\y\\z\end{pmatrix}}={\frac {1}{g_{1}^{2}+g_{2}^{2}+g_{3}^{2}}}{\begin{pmatrix}g_{1}\\g_{2}\\g_{3}\end{pmatrix}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d92295287a38b329dca7776d1d3b4579f66e9ea0)
we then obtain the
Cartesian coordinates
x
,
y
, and
z
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
into
three-space
.
Property of Bryant?Kusner parametrization
[
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]
If
w
is replaced by the negative reciprocal of its
complex conjugate
,
then the functions
g
1
,
g
2
, and
g
3
of
w
are left unchanged.
By replacing
w
in terms of its real and imaginary parts
w
=
s
+
it
, and expanding resulting parameterization, one may obtain a parameterization of Boy's surface in terms of
rational functions
of
s
and
t
. 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).
Relation to the real projective plane
[
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]
Let
be the Bryant?Kusner parametrization of Boy's surface. Then
![{\displaystyle P(w)=P\left(-{1 \over w^{\star }}\right).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0e782c98c2f3a0397573d9af4de87f767f6c1f39)
This explains the condition
on the parameter: if
then
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
, RP
2
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
.
Symmetries
[
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]
STL 3D model
of 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.
Applications
[
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]
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.
Model at Oberwolfach
[
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]
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
Mobius 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 Mobius strip twisted by three times 180 degrees — as is the emblem of the institute (
Mathematisches Forschungsinstitut Oberwolfach 2011
).
Model made for Clifford Stoll
[
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]
A model was made in glass by glassblower Lucas Clarke, with the cooperation of
Adam Savage
, for presentation to
Clifford Stoll
, It was featured on Adam Savage's
YouTube
channel,
Tested
. All three appeared in the video discussing it.
[5]
References
[
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]
Citations
[
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]
Sources
[
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]
- Kirby, Rob
(November 2007),
"What is Boy's surface?"
(PDF)
,
Notices of the AMS
,
54
(10): 1306?1307
This describes a piecewise linear model of Boy's surface.
- Casselman, Bill (November 2007),
"Collapsing Boy's Umbrellas"
(PDF)
,
Notices of the AMS
,
54
(10): 1356
Article on the cover illustration that accompanies the Rob Kirby article.
- Mathematisches Forschungsinstitut Oberwolfach (2011),
The Boy surface at Oberwolfach
(PDF)
.
- Sanderson, B.
Boy's will be Boy's
, (undated, 2006 or earlier).
- Weisstein, Eric W.
"Boy's Surface"
.
MathWorld
.
External links
[
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]
Compact topological surfaces and their immersions in 3D
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Without boundary
| Orientable
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- Sphere
(genus 0)
- Torus
(genus 1)
- Number 8 (genus 2)
- Pretzel (genus 3) ...
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Non-orientable
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With boundary
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Related
notions
| Properties
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Characteristics
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Operations
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