Boy's Surface

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Boy's Surface
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.

Contents

Basic Description

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

Image: XYZ Graph.jpeg

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 we 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. The Youtube video below shows this phenomenon.


A More Mathematical Explanation

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 that is an abstract version of geometry Topology is the study of properties that remain constant regardless how distorted the object is. For a more in-depth explanation of these topology terms, refer to the helper page Topology Glossary

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 tto 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.


This parametrization is extremely abstract, the one below is much more concise and can be followed more easily, it includes trigonometric functions.


f(x,y,z) = \frac{1}{2}[(2x^2-y^2-z^2)(x^2+y^2+z^2)+2yz(y^2-z^2)+xz(x^2-z^2)+xy(y^2-x^2)]
f(x,y,z) = \frac{1}{2}(\sqrt{3}[(y^2-z^2)(x^2+y^2+z^2)+zx(z^2-x^2)+xy(y^2-x^2)]
f(x,y,z) = \frac{1}{8}(x+y+z)[(x+y+z)^3+4(y-x)(z-y)(x-z)]


So that


x = cosusinv
y = sinusinv
z = cosv





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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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