last edited 13dec09 by firstname.lastname@example.org
Find this document at http://new.math.uiuc.edu/math198/serio2/
Can't Touch T H I S!
This project's goal is to reproduce three dimensional spatial objects along with the inclusion of a fourth dimension, color. This will be done in order to help visualize a higher dimension, and will briefly explain how the dimension of color can be applied.
Klein Bottle, With Color Dimension Exhibited
Change the source file tor1.py example in order to diagram a Figure 8 Klein Bottle with motion abilities.
Investigate the use of normal vectors to control the color scheme of an object.
Use the same machinery in order to reproduce other complex shapes and model the color continuum over these objects.
I have made progress towards the completion of my project by modifying tor1.py in order to model both the Figure 8 Klein Bottle and the normal Klein Bottle
I have used my knowledge of Calculus 3 in order to properly color the above shapes according to their normal vectors; however, I still need to decipher a way to model them as I wish.
I have since finished the proper coloring of the klein bottle, and will now be able to model how the 4th dimension is present in this figure.
Also, I have decided to use VPython to display the rudimentary shapes associated with nontrivial 4th dimension behavior, specifically the study of knots.
A Guide to the Color Dimension
The study of Euclidean spaces with more than three dimensions has its roots in the 19th century, when in 1827 August Mobiüs conjectured that a fourth dimension would allow a three dimensional object to be translated into its mirror image. The idea of higher dimensions grew in academic value with Bernhard Riemann's writings in 1854, in which he described a point as any series of coordinates, including points with more than 3 spatial references. In his 1888 book A New Era of Thought, Charles Hinton described the tesseract technique for visualizing a four dimensional cube, and also coined the terms "ana" and "kata" to describe this additional dimension. Leading into the 20th century, scholars such as Minkowski and Einstein became more concerned with spacetime, and science fiction books such as H.G. Wells' The Time Machine popularized the idea of time as the only fourth dimension. However, mathematicians today still study the Euclidean geometry of four dimensional space, and they are often found using the innovations of computer programming along with mathematical knowledge to more accurately predict how higher dimensions would appear.
The concept of using color as a technique for observing an additional (4th) spatial dimension can be understood by first looking at an analogy to 2 dimensional spaces and then extrapolating to an additional dimension. The best way to grasp the idea of color as a spatial dimension is to consider a 2-sphere (three spatial dimensions) of any radius that is about to pass through a two-dimensional surface, in this case a plane. If the sphere moves perpendicularly to the plane and proceeds to move through it, the sphere will appear as a series of cross sections in the plane, from a single point to a circle with the radius of the sphere. After reaching this maximum radius, the projection onto the plane will again reduce to a circle. From the perspective of an individual living within this two dimensional plane, the series of circles of varying radii will not appear to have an additional dimension without some indication of the dimensional shift, and instead will only appear as a "movie" of successively increasing and then decreasing one-toned circles. With the addition of the color dimension, each end of the sphere will be colored with opposing colors (black-white, red-blue), while each successive circle will be shaded according to its radius. A being living within the plane can "see" this spatial dimension by perceiving this color shift an analogy for another dimension. In other words, the person living in this two dimensional world now has the ability to visualize three spatial dimensions, namely the north-south, east-west, and red-blue dimensions.
This same concept can be extrapolated by imagining a sphere with four spatial dimensions, known as a hypersphere or 3-sphere, in the context of the Earth's three-dimensional plane. A hypersphere moving perpendicularly through the plane will appear as a similar "movie" to a being living within the surface, but instead of a series of circles, there will be a succession of spheres, varying from a point sphere to a sphere of a specific radius, each changing color successively according to the requirements of the color parameter. In this context, the four spatial dimensions will be north-south, east-west, up-down, and red-blue.
A few of the more interesting applications of color as an additional dimension include its employment upon knots and additionally with shapes that are nonorientable in three spatial dimensions but can be more easily visualized with color. Let's say that your friend has a knot in his shoe and is at his wit's end attempting to untie it. A knot can be defined as an overlapping linear curve that cannot be oriented, or "untangled," because two parts of the curve with always come into contact if the curve is not first separated. By adding an additional dimension to a knot, such as the trefoil knot, (Picture thanks to rdrop.com) the knot can be "untangled" by remembering this notion occupying the same space. To undo a knot through this technique, choose a color, such as red, for the entire knot. Then, push a portion of the knot near one of the three-dimensional overlaps into the fourth dimension, blue, making sure to maintain continuum by highlighting the area near this shift in successive shades of magenta. By doing this, one can pass the "blue" dimension piece of the knot "through" the "red" dimension part of the knot, resulting in a newly untangled curve. This motion is not the same as moving something "through" in the three dimensional sense, since the change in dimensionality actually causes the possible points of intersection in 3-D space to never interact. For example, point (1,1,1,red) will not intersect with point (1,1,1,blue), no matter how hard it tries. We can see this dimension shift because of the color denotation, but from an Earth perspective of this process, the piece of the knot that is undergoing untangling will actually "disappear" and then reappear once it returns to the same fourth spatial dimension as Earth. However, we now know that it doesn't really "disappear" at all, but rather uses another dimension to its advantage. Although your friend will probably be all for this idea and agree with this logic, humans do not currently have the ability to change an object's fourth spatial dimension in this manner the act itself will be fruitless. At that point, tell him to go find some scissors.
A final interesting application of color dimensionality is to model how the Klein bottle, a self-intersecting, nonorientable surface in three dimensions, does not actually intersect itself in four dimensions. The Klein bottle, determined by German mathematician Maxwell Klein in 1882, was originally named Kleinsche Fläch, meaning Klein surface in German. This was incorrectly assumed to mean Kleinsche Flasche, or Klein bottle, and the name stuck as so. The Klein bottle is built from a series of squares that have been topologically contorted into the proper shape. The shape is considered to be nonorientable because a picture on the surface that is moved continuously around it will end up displaying a mirror image when it revisits its initial position. In three dimensions, the Klein Bottle (www.kleinbottle.com) is forced to intersect itself because the bottle goes through itself at a 2-dimensional circle of intersection, known as the nexus. In the fourth dimension though, as can be seen through the color analogy, the Klein bottle does not intersect itself as long as the shift along this fourth dimension takes place a bit before the three dimensional nexus and continues through the remainder of the surface.
This is an example of what I will be presenting in my own project.
1. Primarily, the project focused upon the creation of the Klein bottle model within the Open GL extension of the Python Library. To edit the source file of tr1.py accordingly, I had to first reconfigure the vertices being drawn by the program which were subsequently being used to render a shape. After finding a set of parametric equations for the Klein bottle, I entered the equations into the script and ran the program. The coloring appeared quite odd at first, which I eventually realized was problematic because the improper normal vectors had been chosen. These normal vectors helped provide proper direction for the ambient light "shining" upon the surface. Using my recently acquired Calculus III knowledge, I determined the proper ambient light settings for my creation.
2. The second issue that I faced with my project was actually seeing the whole shape. I realized early on I was not able to see the entire Klein bottle, but instead only a few of the strips created when the vertices were connected. With the help of Professor Francis, I constructed a scaling mechanism, "zoom," that works with keyboard response for the amount of required scaling.
3. The third issue faced by the programmer was how to accurately color the object according to the plan. The program tr1.py originally used "cat" and "dog" settings to help determine the color of the shape. I decided to abandon these settings and instead consider the shape as a function shifting from blue to red using periodic function as matrix elements multiplied by the change along the v direction.
Song Copyright MC Hammer, 1990
Thanks to Liz Pyde for helping with the website's title.