Boy's Surface

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Boy's Surface

While trying to prove that an immersion (a special representation) of the projective plane did not exist, German mathematician Werner Boy discovered Boy’s Surface in 1901. Boy’s Surface is an immersion of the projective plane in three-dimensional space. This object is a single-sided surface with no edges.

Boy's Surface

Field: Geometry
Created By: Paul Nylander

1 Basic Description
2 Teaching Materials
3 References

Basic Description

Boy's surface is a non-orientable surface that is one possible parametrization of the surface obtained by sewing a Möbius strip to the edge of a disk.

Boy's surface is one of the shapes that is well known in topology, a branch of mathematics (you can think of as an abstract and more advanced version of geometry). Topology is the study of properties that remain constant regardless how distorted the object is.

In geometry, we have many shapes with specific names, dimensions and properties. For instance, everyone knows a square. It has 2 dimensions, a width, a base, and other cool math properties. In topology, we have manifolds. A manifold is a broad definition of a shape. Manifolds are thought of as surfaces without any boundaries or edges.

Manifolds can be categorized by their dimensions. A one-dimensional manifold is just a one-dimensional shape or surface. This means each section of a one-dimensional manifold looks like a line. A two-dimensional manifold is just a two dimensional shape or surface. This means each section of a two dimensional manifold looks like a plane. In fact, a surface is a two dimensional manifold. Manifolds are the first step in understanding what type of surface Boy's Surface actually is.

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An example of a manifold could be a tossed blanket. The tossed blanket is a shape. It has dimensions just like the typical square, even if we do not really think about it in that way. We don't have a particular name for this shape, but it is a shape nonetheless with many properties.

Beside the dimensions, a manifold can be labeled as an orientable or non-orientable surface as well. A manifold is orientable if vectors, placed at every location, point toward the same direction. The concept of orientability and non-orientability is essential in understanding Boy's Surface. It will be discussed later.

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Think of a sphere. A horizontal cross section of a sphere is a circle. If a vector outlines the cross section of the sphere such that its tail is on the edge of the circle, the vector will always point into the center point of the sphere. So, after traveling the path the vector is in the same orientation as it was initially.

A non-orientable manifold on the other hand has a path where the vector ends up as the mirror-image of the initial orientation. Any surface with a reversing path is non-orientable. A good example is the Möbius strip.

A Möbius strip is shown below. The arrow is shown on what is essentially the "top" side of the strip, but if you continue in the direction of the arrow until you come back to your starting point, you will find yourself "below" where you started. This is the definition of non-orientable. However, from our view, it is easy to realize that when we travel in the direction of the arrow we aren't always on the top or the bottom of the strip. However, with a more local view the orientation isn't noticeable. To us, locally, the planet seems flat, but when we "zoom out" we get a more worldly view and see that planet earth is actually more spherical.

A Mobius Strip has an edge. Because, the Mobius Strip has an edge there is no self intersection within the shape, meaning sections of the strip pass through each other. However, it was mentioned that Boy's Surface has no edges yet is made from something with edges. As mentioned, sewing a Mobius Strip and a disk can construct Boy's Surface. This closes up the edge such that Mobius Strip fails to have boundaries.

add more Boy's surface is a non-orientable manifold. If a creature that lived within Boy's Surface went on a stroll around his "block", he would realize that he's facing the opposite way than what he originally started from.

Immersion, The Real Projective Plane, and Embedding

The Real Projective Space is a modified Euclidean space (the typical x-y-z space) where every line in the projective space forms into a circle by meeting another point in the space. This is true for all line, even parallel lines. An example, would be on the road. Solid white lines indicating an emergency lane, meet at a point on the horizon, outline a possible visual interpertation of the real projective space. The projective space is constructed out of the many circles with an additional circle at infinity.

The Real Projective Plane (R'P²) is the 2 dimensional Real Projective Space. The Real Projective Plane has no edges, so the surface never intersects itself. The real projective plane cannot be shown in three space without it passes through itself somewhere, so, it is a four-dimensional figure. More importantly, the real projective plane is non-orientable.

Anna's part It is, according to mathematicians, a very unique non-orientable surface with Euler Characteristic 1. You already know that non-orientable means that it is impossible to define top-and-bottom-directions throughout a specific space. However, have we not yet explained the Euler Characteristic of a surface.

The Euler Characteristic is calculated using a Triangulation, simply the division of a surface into triangles. However these triangles have the following restrictions: The intersection of any two triangles must be:

A single point that is the vertex of each triangle
A single edge that is a side of each of the triangles

The images below can be found on Cornell's website for mathematics. These are examples of triangulations:

Triangulation plays a major role in the Euler Characteristic, which is equal to the number of vertices minus the number of edges plus the number of triangles in the triangulation. This is shown algebraically as:

$\boldsymbol{\chi}=v-e+f$ where $\boldsymbol{\chi}$ is the greek letter chi and where $f$ stands for the number of triangles.

Next, we come to embedding. It is important to understand this term because Boy's Surface is an immersion of the real projective plane embedded in 3-dimensional space. Luckily, unlike some of the previous terms, the definition is straight forward. An embedding is the instance of one topological object, such as a manifold or graph, inside another topological object in such a way that certain properties are preserved. In topological spaces, an embedding specifically preserves open sets.

Finally we come to our last major term, Immersion. I will first give you the mathematical definition according to WolframMathWorld:

A special nonsingular map from one manifold to another such that at every point in the domain of the map, the derivative is an injective linear transformation.

In topology, a map is a continuous function, meaning that each input has only one output. Below is an example of a mapping of continuity and non-continuity:

Continuity directly relates to Injective Linear Transformation such that for injective functions, there are always an equal number of outputs for the number of inputs (shown above). This can be visualized algebraically as well. For example, say you have a function such that $f(x)=x^2$ with inputs $3$ and $-3$ . When substituting these into the function, both result with $9$ . This function therefore is not continuous because it is not one-to-one, meaning that for each input there is not exactly one output.

Constructing Boy's Surface

Boy's Surface is most easily constructed and visualized by extending the axes of a 3-dimensional graph (with coordinates $X, Y, Z$ ).

By going far enough along any axis, you will eventually end up back at the origin of the three axes. This can be difficult to imagine, so I will explain it in depth: The situation of returning to your starting point can be compared to traveling on the earth's surface. As mentioned above, the earth seems locally flat, but if you travel along it far enough and long enough, you will eventually return to your starting point. Like the graph, that seems as though it has distinct endpoints, it is similar to the example with the earth: if you go far enough, most likely to infinity, you are going to return to your place of origin.

By applying this reasoning to the 3D graph, the positive x-axis can be connected to the negative y-axis, the positive y-axis to the negative z-axis and the positive z-axis to the negative x-axis.

Boy's Surface was discovered in 1901 by German mathematician Werner Boy when he was asked by his advisor, David Hilbert, to prove that an immersion of the projective plane in 3-space was impossible. Today, a large model of Boy's Surface is displayed outside of the Mathematical Research Institute of Oberwolfach in Oberwolfach, Germany. The model was constructed as well as donated by Mercedes-Benz.

Parametrization

There are several ways in which Boy's Surface can be parametrized, the most famous is Rob Kusner and Robert Bryant's. First, I will explain what parametrization is. It is writing a function so that all coordinates are expressed using the same variable. For example: If you have a function where $a=f[x,y]$ we could use a parameter $t$ to represent all the coordinates. The $y$ - coordinate is represented as $b(t)$ and the $x$ - coordinate as $w(t)$ and lastly $a$ is represented as $a[t]$ . So no we have $[a(t), b(t), w(t)]$ . Now you have only one input variable, which will make working with the function much easier.

Below is a parametrization of Boy's surface discovered by Rob Kusner and Robert Bryant.

Given a complex number z with a magnitude less than or equal to one, let

$g_1$	$=$	$-\frac{3}{2}Im(\frac{z(1-z^4)}{z^6+\sqrt{5}z^3-1})$
$g_2$	$=$	$-\frac{3}{2}Re(\frac{z(1+z^4}{z^6+\sqrt{5}z^3-1})$
$g_3$	$=$	$Im(\frac{1+z^6}{z^6+\sqrt{5}z^3-1})-\frac{1}{2}$
$g$	$=$	$x_1^2+x_2^2+x_3^2$

So that

$X$	$=$	$\frac{g_1}{g}$
$Y$	$=$	$\frac{g_2}{g}$
$Z$	$=$	$\frac{g_3}{g}$

$X$ , $Y$ , and $Z$ are the Cartesian coordinates of a point on the surface.

Teaching Materials

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References

[1] jacr. The Math of Non-Orientable Surfaces. Retrieved from \http://www.math.osu.edu/~fiedorow/math655/yale/math.htm

http://www.learner.org/courses/mathilluminated/units/4/textbook/06.php Accessed June 24 2011. Section 17.7 Surface Integrals “Integrating Functions over Arbitrary Surfaces" faculty.up.edu/wootton/Calc3/Section17.7.pdf Accessed June 24 2011

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Boy's Surface

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Contents

Basic Description

Immersion, The Real Projective Plane, and Embedding

Constructing Boy's Surface

Parametrization

Teaching Materials

References

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