522 CHAPTER 8 Geometry Imagine trying to paint a Klein bottle. You start on the “outside” of the large part and work your way down the narrowing neck. When you cross the self-intersection, you have to pretend temporarily that it is not there, so you continue to follow the neck, which is now inside the bulb. As the neck opens up, to rejoin the bulb, you find that you are now painting the inside of the bulb! What appear to be the inside and outside of a Klein bottle connect together seamlessly, since it is one-sided. If a Klein bottle is cut along a curve, the results are two Möbius strips; see Fig. 8.92. Thus, a Klein bottle could also be made by gluing together two Möbius strips along the edges. Figure 8.92 Two Möbius strips result from cutting a Klein bottle along a curve. Maps Mapmakers have known for a long time that regardless of the complexity of the map and whether it is drawn on a flat surface or a sphere, only four colors are needed to differentiate each country (or state) from its immediate neighbors. Thus, every map can be drawn by using only four colors, and no two countries with a common border will have the same color. Regions that meet at only one point (such as the states of Arizona, Colorado, Utah, and New Mexico) are not considered to have a common border. In Fig. 8.93(a), no two states with a common border are marked with the same color. Illinois Missouri Indiana Ohio Kentucky Tennessee West Va. Virginia Virginia (a) (b) Figure 8.93 The four-color problem was first suggested by a student of Augustus De Morgan in 1852. In 1976, Kenneth Appel and Wolfgang Haken of the University of Illinois— using their ingenuity, logic, and 1200 hours of computer time—succeeded in proving that only four colors are needed to draw a map. They solved the four-color map problem by reducing any map to a series of points and connecting line segments. They replaced each country with a point. They connected two countries having a common border with a straight line; see Fig. 8.93(b). They then showed that the points of any graph in the plane could be colored by using only four colors in such a way that no two points connected by the same line were the same color. The four-color problem is now referred to as the four-color theorem . Mathematicians have shown that, on different surfaces, more than four colors may be needed to draw a map. For example, a map drawn on a Möbius strip requires a maximum of six colors, as in Fig. 8.94(a). A map drawn on a torus (the shape of a doughnut) requires a maximum of seven colors, as in Fig. 8.94(b). MATHEMATICS TODAY Geometry and the Human Brain The human brain is an extremely complex structure that is very difficult for scientists to study. Now biologists, mathematicians, and computer scientists have combined efforts to create a model of brain activity that involves using 11-dimensional geometry. The Blue Brain Project is a Swiss research project aimed at creating a digital model of the human brain. Using a branch of mathematics known as algebraic topology along with an IBM Blue Gene supercomputer, scientists have created a digital model of brain tissue. The model includes representations of 55 distinct types of brain cells, called neurons, and 36 million nerve connections, called synapses. Within the model, two connected neurons are represented by a geodesic, a line that sits on a curved surface. Three connected neurons are represented by a filled-in triangle. Four connected neurons are represented by a solid pyramid. More connections of neurons are represented by higherdimensional shapes that are difficult for us to imagine, but the mathematics can be used to describe them. While the model is still considered a primitive approximation of the real human brain, researchers hope to be able to use the model to predict how the brain will react to various medical treatments. Why This Is Important Mathematics is often used to solve very complex problems. By using mathematics to model the human brain, researchers hope to better understand the causes and perhaps the cures for many diseases of the human brain including Alzheimer’s disease, Parkinson’s disease, and other causes of dementia. Ian Allenden/123RF
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