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geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is ...
, Boy's surface is an immersion of the
real projective plane In mathematics, the real projective plane is an example of a compact non- orientable two-dimensional manifold; in other words, a one-sided surface. It cannot be embedded in standard three-dimensional space without intersecting itself. It has b ...
in 3-dimensional space found by
Werner Boy Werner Boy (; 4 May 1879 − 6 September 1914) was a German mathematician. He was the discoverer and eponym of Boy's surface—a three-dimensional projection of the real projective plane without singularities, the first of its kind. He discov ...
in 1901. He discovered it on assignment from
David Hilbert David Hilbert (; ; 23 January 1862 – 14 February 1943) was a German mathematician, one of the most influential mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas in many a ...
to prove that the projective plane ''could not'' be immersed in
3-space Three-dimensional space (also: 3D space, 3-space or, rarely, tri-dimensional space) is a geometric setting in which three values (called ''parameters'') are required to determine the position of an element (i.e., point). This is the informal ...
. Boy's surface was first parametrized explicitly by
Bernard Morin Bernard Morin (; 3 March 1931 in Shanghai, China – 12 March 2018) was a French mathematician, specifically a topologist. Early life and education Morin lost his sight at the age of six due to glaucoma, but his blindness did not prevent him f ...
in 1978. 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 In mathematics, the Roman surface or Steiner surface is a self-intersecting mapping of the real projective plane into three-dimensional space, with an unusually high degree of symmetry. This mapping is not an immersion of the projective pla ...
and the cross-cap, it has no other singularities than
self-intersection In mathematics, intersection theory is one of the main branches of algebraic geometry, where it gives information about the intersection of two subvarieties of a given variety. The theory for varieties is older, with roots in Bézout's theorem ...
s (that is, it has no pinch-points).


Symmetry of the Boy's surface

Boy's surface has 3-fold
symmetry Symmetry (from grc, συμμετρία "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more precise definit ...
. 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.


Model at Oberwolfach

The
Mathematical Research Institute of Oberwolfach The Oberwolfach Research Institute for Mathematics (german: Mathematisches Forschungsinstitut Oberwolfach) is a center for mathematical research in Oberwolfach, Germany. It was founded by mathematician Wilhelm Süss in 1944. It organizes week ...
has a large model of a Boy's surface outside the entrance, constructed and donated by
Mercedes-Benz Mercedes-Benz (), commonly referred to as Mercedes and sometimes as Benz, is a German luxury and commercial vehicle automotive brand established in 1926. Mercedes-Benz AG (a Mercedes-Benz Group subsidiary established in 2019) is headquartere ...
in January 1991. This model has 3-fold rotational symmetry and minimizes the
Willmore energy In differential geometry, the Willmore energy is a quantitative measure of how much a given surface deviates from a round sphere. Mathematically, the Willmore energy of a smooth closed surface embedded in three-dimensional Euclidean space is def ...
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 In mathematics, a Möbius strip, Möbius band, or Möbius loop is a surface that can be formed by attaching the ends of a strip of paper together with a half-twist. As a mathematical object, it was discovered by Johann Benedict Listing and Aug ...
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 .


Applications

Boy's surface can be used in
sphere eversion In differential topology, sphere eversion is the process of turning a sphere inside out in a three-dimensional space (the word '' eversion'' means "turning inside out"). Remarkably, it is possible to smoothly and continuously turn a sphere in ...
, as a
half-way model In geometry, minimax eversions are a class of sphere eversions, constructed by using half-way models. It is a variational method, and consists of special homotopies (they are shortest paths with respect to Willmore energy); contrast with Thurston' ...
. 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 Magnitude may refer to: Mathematics *Euclidean vector, a quantity defined by both its magnitude and its direction *Magnitude (mathematics), the relative size of an object *Norm (mathematics), a term for the size or length of a vector *Order of ...
is less than or equal to one ( \, w \, \le 1), let :\begin g_1 &= - \operatorname \left \right\ pt g_2 &= - \operatorname \left \right\ pt g_3 &= \operatorname \left \right- \\ \end and then set :\beginx\\ y\\ z\end = \frac \beging_1\\ g_2\\ g_3\end we then obtain the
Cartesian coordinates A Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in ...
''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 In mathematics, a minimal surface is a surface that locally minimizes its area. This is equivalent to having zero mean curvature (see definitions below). The term "minimal surface" is used because these surfaces originally arose as surfaces tha ...
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 In mathematics, a projective plane is a geometric structure that extends the concept of a plane. In the ordinary Euclidean plane, two lines typically intersect in a single point, but there are some pairs of lines (namely, parallel lines) that ...
into
three-space Three-dimensional space (also: 3D space, 3-space or, rarely, tri-dimensional space) is a geometric setting in which three values (called ''parameters'') are required to determine the position of an element (i.e., point). This is the informa ...
.


Property of Bryant–Kusner parametrization

If ''w'' is replaced by the negative reciprocal of its
complex conjugate In mathematics, the complex conjugate of a complex number is the number with an equal real part and an imaginary part equal in magnitude but opposite in sign. That is, (if a and b are real, then) the complex conjugate of a + bi is equal to a - ...
, -, then the functions ''g''1, ''g''2, 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 In mathematics, a rational function is any function that can be defined by a rational fraction, which is an algebraic fraction such that both the numerator and the denominator are polynomials. The coefficients of the polynomials need not be ...
s of and . This shows that Boy's surface is not only an
algebraic surface In mathematics, an algebraic surface is an algebraic variety of dimension two. In the case of geometry over the field of complex numbers, an algebraic surface has complex dimension two (as a complex manifold, when it is non-singular) and so of di ...
, but even a
rational surface In algebraic geometry, a branch of mathematics, a rational surface is a surface birationally equivalent to the projective plane, or in other words a rational variety of dimension two. Rational surfaces are the simplest of the 10 or so classes of s ...
. The remark of the preceding paragraph shows that the
generic fiber In algebraic geometry, a generic point ''P'' of an algebraic variety ''X'' is, roughly speaking, a point at which all generic properties are true, a generic property being a property which is true for almost every point. In classical algebraic g ...
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

Let P(w) = (x(w), y(w), z(w)) be the Bryant–Kusner parametrization of Boy's surface. Then : P(w) = P\left(- \right). This explains the condition \left\, w \right\, \le 1 on the parameter: if \left\, w \right\, < 1, then \left\, - \right\, > 1 . However, things are slightly more complicated for \left\, w \right\, = 1. In this case, one has - = -w . This means that, if \left \, w \right\, = 1, the point of the Boy's surface is obtained from two parameter values: P(w) = P(-w). In other words, the Boy's surface has been parametrized by a disk such that pairs of diametrically opposite points on the
perimeter A perimeter is a closed path that encompasses, surrounds, or outlines either a two dimensional shape or a one-dimensional length. The perimeter of a circle or an ellipse is called its circumference. Calculating the perimeter has several pract ...
of the disk are equivalent. This shows that the Boy's surface is the image of the
real projective plane In mathematics, the real projective plane is an example of a compact non- orientable two-dimensional manifold; in other words, a one-sided surface. It cannot be embedded in standard three-dimensional space without intersecting itself. It has b ...
, RP2 by a
smooth map In mathematical analysis, the smoothness of a function is a property measured by the number of continuous derivatives it has over some domain, called ''differentiability class''. At the very minimum, a function could be considered smooth if ...
. That is, the parametrization of the Boy's surface is an immersion of the real projective plane into the
Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean s ...
.


References


Citations


Sources

* 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). *


External links


Boy's surface
at MathCurve; contains various visualizations, various equations, useful links and references

– applet from ''Plus Magazine''.

including th
original article
and a
embedding
of a topologist in th
Oberwolfach Boy's surface



A paper model of Boy's surface
– pattern and instructions



in
Constructive Solid Geometry Constructive solid geometry (CSG; formerly called computational binary solid geometry) is a technique used in solid modeling. Constructive solid geometry allows a modeler to create a complex surface or object by using Boolean operators to combin ...
together with assembling instructions
''Boy's surface''
visualization video from the Mathematical Institute of the Serbian Academy of the Arts and Sciences {{DEFAULTSORT:Boy's Surface Surfaces Geometric topology