From Math Images
The purpose of this page is to provide the reader with vocabulary essential in understanding the construction and explanation of Boy's Surface
In geometry, we have many shapes with specific names, dimensions and properties. For instance, we are all familiar with 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.
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.
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. 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.
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:
where is the greek letter chi and where stands for the number of triangles.
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 with inputs and . When substituting these into the function, both result with . This function therefore is not continuous because it is not one-to-one, meaning that for each input there is not exactly one output.
A surface is a 2 dimensional manifold.
Orientability is a property of manifolds that describes whether directions, or orientations, can be consistently defined in the surface. Directions can be consistently defined on an orientable manifold, but not on a non-orientable manifold.
In the case of surfaces, a 2 mandifold is orientable if every closed path preserves the orientation of an entity that travels along it. A 2 mandifold is non-orientable if there exists at least one closed path that reverses the orientation of something traveling along it.
For instance, say that rotation in the counter-clockwise direction is considered to be positive, and clockwise, negative. If an object spinning counter-clockwise moves along a closed path in an orientable surface, it will return to its starting point spinning counter-clockwise. Yet, if the surface is non-orientable, then it will twist in such a way that, on certain paths, the object will return spinning clockwise, though the motion of its rotation never changed. Equivalently, in non-orientable surfaces, an object within the surface can travel along a path that will lead it back to its start point, but with its right and left sides flipped.
A good demonstration of non-orientability on the Mobius Strip is visible on this page from Plus Magazine. When viewing this, remember that the face is within the surface, not merely pasted on one of the sides.
Orientability and non-orientability are most often discussed in relation to surfaces. Nevertheless, the properties are descriptive of higher dimensional manifolds as well. One can consider a 3 dimensional space where certain paths would lead back to their start points, but flip the right and left sides of things that traveled along them.
Here is a more technical explanation: In a non-orientable manifold, there exists at least one path around the manifold such that, if we take a set of basis vectors for the manifold and move them along this path, they will arrive at their starting point with the following result. The determinant of the matrix composed of these basis vectors will have the opposite sign (positive/ negative) when it arrives back at the starting point as it did when it left.
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.
Not all image pages will discuss in depth how non-orientability arises for a particular surface; the phenomenon is discussed, however, for the Mobius Strip and Real Projective Plane, with images to explain the phenomenon.
Real Projective Plane
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.