Cross-cap

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Cross-cap and Cross-capped Disk
Fields: Topology and Geometry
Image Created By: Unknown
Website: Wikipedia

Cross-cap and Cross-capped Disk

The cross-capped disk is one 3 dimensional model of the Real Projective Plane. The cross-capped disk is a 2 dimensional surface that is non-orientable and has only one side. The Real Projective Plane is best represented using 4 spacial dimensions, rather than 3.


Contents

Basic Description

There are two things that are commonly referred to as a Cross-cap, however, only the first is actually a Cross-cap. This is a surface that is a model of a Mobius strip. The second is a disk fused with this first surface, which is more correctly referred to as a cross-capped disk. The Basic Description section discuss the cross-capped disk, while both the Mobius strip Cross-cap and cross-capped disk will be covered in the More Mathematical Explanation.


The cross-capped disk is a one-sided, non-orientable surface. Unlike the Mobius strip, the cross-capped disk has no edges, or more technically, boundary. As a surface without boundary, the cross-capped disk is a true 2 manifold. The presence of an edge in the Mobius strip allows sections of the shape to pass alongside each other, thus avoiding self intersection. Because the cross-capped disk has no edges there is no way for the surface to go around itself when it needs to close up. As a result, 3 dimensional models of it intersect, or pass through, themselves. As mentioned, the cross-capped disk is made by gluing a disk to the edge of a Mobius strip/ Cross-cap, thereby closing it up. The construction of the cross-capped disk is discussed in detail later, in the More Mathematical Explanation.

Image 1. A cross-capped disk that has been sliced open to better reveal the nature of the surface.
Image 1. A cross-capped disk that has been sliced open to better reveal the nature of the surface.


Image 1 shows a cross-capped disk that has been sliced open to better reveal the nature of the surface. The figure is labeled for clarification in the discussion that follows. In the image, one can see that the upper surface of the outside of the cross-capped disk continues to the inside of the surface. Hence, the top of the cross-capped disk on the outside is the same surface as the floor on the inside. Likewise, the bottom of the cross-capped disk on the outside is the same surface as the ceiling on the inside.


Here is a scenario that illustrates this notion. Notice that, starting at the top and walking along the outside of the cross-capped disk, one will arrive at the section where the surface seems to intersect itself. If one passes through this part (as though it were a hologram) and continues walking, they will now find themselves on the inside of the surface. Continuing on, one will again come to the intersection, and, passing through it, will arrive back on the outside, up top.


So the top of the cross-capped disk is the same surface as the inside floor, but how does it connect with the opposite side (the bottom and ceiling) to make the figure one sided? Image 1 shows a cross-capped disk that has been cut open; however, the cross-capped disk is actually closed-up, as in the image at the top of the page. Accordingly, the top and bottom of the figure are connected, like in a sphere. Thus: the floor on the inside is the same surface as the top on the outside; the top connects to the bottom of the cross-capped disk, which, in turn, is the same surface as the ceiling on the inside.


Next we will explain why the cross-capped disk seems to intersect itself. To better understand topological constructs like the cross-capped disk, we must become comfortable, or at least familiar, with objects existing in multiple dimensions. Recall that the cross-capped disk is a model of the Real Projective Plane. The self intersection in the cross-capped disk is a result of making a 3 dimensional model of the Real Projective Plane. If considered in 4 dimensions, the cross-capped disk no longer intersects itself. At the spots where it seems to intersect itself, the points in one part of the figure appear to be in the same place as the points in an other part, making it look like it passes through itself. These points seem to be in the same place when we look at them in our usual 3 dimensions because they indeed do have three of the same coordinates. Yet, they have a different fourth coordinate, and so, are actually different points at different positions. Because we live in 3 dimensions, we can only see three coordinates at once. As a result, we don't realize that the points are different, because we can only see the components of the points' locations that are the same.

Image 2. A red ball located at (8,7,0) and a blue ball directly in front of it at (8,7,6).
Image 2. A red ball located at (8,7,0) and a blue ball directly in front of it at (8,7,6).
Image 3. A different view of the same balls in Image 2.
Image 3. A different view of the same balls in Image 2.

This can be thought of as such: When viewed side-on, as in Image 2, the two balls, one behind the other, appear to be at the same point. Yet, when considered in 3 dimensions, as in Image 3, we know that they are at different places. This occurs because, when viewed from side-on, we can only see the part of the balls' locations that is the same. It is the same for the points where the Cross-cap passes through itself. Viewed in 3 dimensions, they appear to be at the same point. Yet, when considered in 4 dimensions, they can be shown to indeed be at different places.


Thus, while all of the 3 dimensional models of the Real Projective Plane, including the cross-capped disk, do intersect themselves, 4 dimensional models can be conceived of, and they don't self intersect. And one of the most available ways to conceptualize of a 4 dimensional model of the cross-capped disk is to think about the 3D model in the way described above. The Projective Plane is said to be 4 embeddable because 4 is the fewest number of dimensions needed to represent the Real Projective Plane without it intersecting itself.

A More Mathematical Explanation



[[Image:Crosscap open2.jpg|Image 4. The Cross-cap [...]



Image 4. The Cross-cap by itself. The vertical line is the region of self-intersection, and the gray ovals define the cross-section.
Image 4. The Cross-cap by itself. The vertical line is the region of self-intersection, and the gray ovals define the cross-section.

As previously mentioned, the object pictured at the top of the page is technically a cross-cap glued to a sphere after removing an open disk from the latter[1]. The Cross-cap itself is the part of the shape where the surface passes through itself, and is a model of a Mobius strip. Image 4 depicts a cross-cap by itself. The Cross-cap rises from a plane and intersects itself at the vertical crease. When sewn to a disk, as in the Images at the top of the page, the Cross-cap is a model of the projective plane with singularities constructed in 3 dimensional space.

Image 5. The Cross-cap/ Mobius strip represented via a fundamental polygon.
Image 5. The Cross-cap/ Mobius strip represented via a fundamental polygon.
Image 6. A disk represented via a fundamental polygon.
Image 6. A disk represented via a fundamental polygon.
Image 7. The Real Projective Plane represented via a fundamental polygon.
Image 7. The Real Projective Plane represented via a fundamental polygon.


Image 8. The construction of the Cross-cap and the Mobius strip from the Real Projective Plane with a disk removed from it.
Image 8. The construction of the Cross-cap and the Mobius strip from the Real Projective Plane with a disk removed from it.

The Real Projective Plane, and thus the cross-capped disk, can be made from the fundamental polygon Image 5 if the corresponding edges are joined so that the arrows line up. This requires twisting the polygon, and can be confusing to imagine. As previously mentioned, the Real Projective Plane can be made by fusing a Mobius strip and a disk. In order to achieve this, the disk must be distorted. By the same token, the Mobius strip and Cross-cap can be made by removing a disk from the Real Projective Plane, a process depicted in Image 8.


Constructing the Mobius Strip

First we will construct the Mobius strip, then the Cross-cap. Both processes begin with the fundamental polygon model of the Real Projective Plane at the green arrow.

Relabeling the fundamental polygon will offer better clarity as we construct the Mobius strip. Corresponding edges are now labeled with the same letter, with a single arrow indicating the orientation in which they are adjoined. We follow the arrows up the image to arrive at the next step.


In preparation for cutting the polygon down the middle, we will need to relabel the right and left sides. The sides of the polygon are made of many points. Notice that, going from the top to the middle, points on the left side successively match up to those on the right side from bottom right to middle. And it is the same for the bottom left and top right, as illustrated in Image 9, below. This symmetry across the middle allows us to split our label for the the sides in two, one label for the top halves, one for the bottom. Hence,the top half of the right side and the bottom half of the left are labeled b in Image 8, while the bottom of the right to the top of the left are labeled a.


Image 9. Looking at the edges, the top right half maps to the bottom left, and the bottom right to the top left. Thus, we can label the corresponding halves the same, just as we would label corresponding sides the same.
Image 9. Looking at the edges, the top right half maps to the bottom left, and the bottom right to the top left. Thus, we can label the corresponding halves the same, just as we would label corresponding sides the same.


Now we cut a disk out of the fundamental polygon, the key step in constructing the Mobius strip from the Real Projective Plane. At the top of Image 8, we cut the shape along the dotted line across its center, and label the corresponding edges that are created where the dotted line used to be.


Now heading down the left side of Image 8, we separate the two halves of the shape and place them back-to-back. Then, we flip the top half over (pancake style) so that the arrows on the c sides match, and we glue them together. Notice that we can now pinch the b sections together, as well as the a sections on the opposite side, arriving at the football-shaped figure.


Next, we make our the figure more square-shaped by shrinking the bulges and stretching the concave parts. To make the diagram less cluttered, we can do away with the labels for the a, b, and c sides since each have already been joined together.


In a similar way to how we relabeled the right and left sides in the beginning, we can now simplify them by considering the d and e sections of each edge as two halves of the same sides. Now we can join the left and right edges of the polygon, an, giving it a half-twist so that the arrows match, we arrive at a Mobius strip.


Constructing the Cross-cap

Following the down arrows, we begin to construct the Cross-cap by rotating the fundamental polygon so that it looks like a diamond. Next, we begin to close the polygon, bringing its edges toward each other and inflating the interior of the shape. Now we remove the bottom of the shape; notice that this bottom section is a slightly concave disk. Were we not to remove this disk, we would end up with the cross-capped disk shown at the top of the page.


Having removed this disk, the original, colored edges of the polygon are made to meet in the middle, each matching the orientation of the other side it corresponds to. The self-intersection region characteristic of the Cross-cap appears when we fuse the two corresponding pairs of sides together through each other. This is detailed in Image 10which depicts the self-intersection region of the Cross-cap. The colors correspond to the appropriate edges in Image 8, while the surface itself is not shown. The blue edge connects to the green one, and the pink to the red.


Additionally, by stretching the sides of the Cross-cap at the bottom of Image 8, it can be made to look like the blue one in Image 4.


Image 10. A detailed image of the self- intersection region of the Cross-cap.
Image 10. A detailed image of the self- intersection region of the Cross-cap.


As we just demonstrated, both the Mobius strip and the Cross-cap can be made by cutting a disk out of the Real Projective Plane. The significant difference between the two is that the Mobius strip does not intersect itself in 3 dimensions, while the Cross-cap does. As a result of the self intersection, the process of crating the Cross-cap from the Real Projective Plane cannot be smoothly reversed—the points at the self intersection get moved to two different places when the steps are reversed. This does not occur for the Mobius strip, because it is free of self intersection. As such, the Mobius strip can be turned into the Cross-cap, but the reverse cannot be done. This fact means that the Cross-cap is simply a model of the Mobius strip, rather than homeomorphic to the Mobius strip.



Constructing the Cross-cap Directly From the Mobius Strip

Image 11. Construction of the Cross-cap directly from the Mobius strip.
Image 11. Construction of the Cross-cap directly from the Mobius strip.


Image 11 depicts the construction of the Cross-cap directly from the Mobius strip. We start by cutting the Mobius strip so it is represented by the fundamental polygon in the second step. Next, the polygon is rotated and then stretched.


Now, just as we did in Image 8, we divide the right and left sides and label their corresponding halves in preparation for cutting the figure. Next we cut the figure across the middle, and label the two new edges with a the same color and arrow, in this case, green.


Again, we take the two new edges and divide them, labeling their corresponding halves. Now we halve a green and purple half. Rotating the top section of the figure, we end up with the red edges on the right and the blue on the left. Taking the top section again, we rotate it and move it so that the two red arrows match up.


Merging the red edges and the blue edges in the proper orientation, we arrive at a cylinder. Now we begin folding the green and purple edges inward, as in Image 8. Finally we merge the purple edges and green edges in the correct orientation, passing them through each other and creating the self intersection region on the Cross-cap. Interactive applets of the figures appear to the right of the equations that parametrize them. I recommend right clicking on them, and opening the control panel, which will let you set transparency of the figure so that you can see the internal structure.




Parametrization of the Cross-capped Disk

The cross-capped disk can be parametrized using the following equations [2], where the r represents an arbitrary scaling constant that governs the proportions of the function. The scaling constant can be employed to stretch or flatten the cross-capped disk in different ways to make it better presentable. As parameters, u and v govern the 2 dimensional surface itself (as opposed to the surface's position in 3 dimensional space). For instance, in the first parametrization, u spans the floor/ ceiling part of the surface while v produces the wall that encloses the figure.


Across from each set of equations is an interactive applet[3] of the surface that the equations parametrize. For a better look at the models, right click them and open the Control Panel. This will allow you to make the surfaces transparent, better revealing their structures. The applets may take take up to a minute to load depending on your internet connection.


  • x(u,v)=r \cos{(u)} \sin{(2v)}
  • y(u,v)=r \sin{(2u)} \sin^2{(v)}
  • z(u,v)=r \cos{(2u)} sin^2{(v)}


For u=[0,2\pi) and v=[0,\tfrac{\pi}{2}].



Or

  • x(u,v)=\cos{(u)} \sin{(2 v)}
  •  y(u,v)=\sin{(u)} \sin{(2 v)}
  •  z(u,v)=\left ( \cos^2{(v)} - \cos^2{(u)} \sin^2{(v)} \right )


For u=[0,2\pi) and v=[0,\tfrac{\pi}{2}].



Both sets of equations generate the same shape, they merely construct it in a different orientation. The following parametrization generates the cross-capped disk in a different way from the first two, producing a model slightly more similar to the one at the top of the page[4]. The figures that all of these equations generate are homeomorphic to each other.

  • x(u,v)= r \left ( 1 + \cos{(v)} \right ) \cos{(u)}


  • y(u,v)= r \left ( 1 + \cos{(v)} \right ) \sin{(u)}


  • z(u,v)= - \tanh {(u - \pi)}~r \sin({v)}


For u~\text{and}~v=[0,2\pi]



Image 1 can be generated by the following set of equations, which is merely a slight alteration from the first set. Most crucially, v now ranges from 0 to \tfrac{\pi}{2.6}, rather than \tfrac{\pi}{2}, meaning the shape is truncated before it can meet itself to close.

  • x(u,v)=\tfrac{7}{10}~r \cos{(u)} \sin{(2v)}
  • y(u,v)=r \sin{(2u)} \sin^2{(v)}
  • z(u,v)=r \cos{(2u)} \sin^2{(v)}


For u=[0,2\pi) and v=\lbrack 0,\tfrac{\pi}{2.6} \rbrack.




Switching the cos(u) in the x(u,v) equation with Sin(u), merely creates a mirror image of the cross-capped disk.


Why It's Interesting

The cross-capped disk is a 3 dimensional model of the Real Projective Plane. Most notably this means that the cross-capped disk is non-orientable and one-sided, as is the Cross-cap. If a 2 dimensional animal were living within the surface , it would be able to travel around the Cross-cap and arrive back at it's starting point with its right and left sides flipped.


Though there are many manifolds that have these properties, very few, if any, occur naturally. This means that we usually don't encounter them in our daily life, and their unusual properties can evoke anything from amusement to fascination.


Furthermore, the Real Projective Plane itself has many interesting aspects. In addition to its non-orientability, the Real Projective Plane is easily used to model human vision, and is one of three basic building blocks for all surfaces.


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Additional Resources

References

  1. Edelsbrunner. H. (2006). II Surfaces: II.1 Two-dimensional Manifolds [PDF document]. Retrieved from CPS296.1: Computational Topology: http://www.cs.duke.edu/courses/fall06/cps296.1/
  2. Wolfram Alpha LLC. 2010. Wolfram|Alpha knowledgebase. http://www.wolframalpha.com/entities/surfaces/cross-cap/gd/qi/t7/. Accessed: June 2, 2011
  3. Figures created by Htasoff using Mathematica. Applets created by Md. Alimoor Reza.
  4. Cross-cap. August 19, 2010. Wikipedia, The Free Encyclopedia. http://en.wikipedia.org/w/index.php?title=Cross-cap&oldid=379792499. Accessed: June 8, 2011

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