Hyperspace and Warped Space Help (page 2)
Introduction to Hyperspace and Warped Space
Some of the concepts in this chapter are among the most esoteric in all of mathematics, with far-reaching applications. Hyperspace (space of more than three dimensions) and warped space can be envisioned even by young children. Some people who think they are not mathematically inclined find non-Euclidean geometry interesting, perhaps because some of it resembles science fiction.
As we have seen, the rectangular (or Cartesian) coordinate plane is defined by two number lines that intersect perpendicularly at their zero points. The lines form axes, often called the x axis and the y axis. Points in such a system are identified by ordered pairs of the form ( x , y ). The point defined by (0,0) is called the origin. Cartesian three-space is defined by three number lines that intersect at a single point, corresponding to the zero point of each line, and such that each line is perpendicular to the plane determined by the other two lines. The lines form axes, representing variables such as x , y , and z . Points are defined by ordered triples of the form ( x , y , z ). The origin is the point defined by (0,0,0). What about Cartesian four-space ? Or five-space ? Or infinity-space ?
A system of rectangular coordinates in four dimensions—Cartesian four-space or 4D space—is defined by four number lines that intersect at a single point, corresponding to the zero point of each line, and such that each of the lines is perpendicular to the other three. The lines form axes, representing variables such as w , x , y , and z . Alternatively, the axes can be labeled x 1 , x 2 , x 3 , and x 4 . Points are identified by ordered quadruples of the form ( w , x , y , z ) or ( x 1 , x 2 , x 3 , x 4 ). The origin is defined by (0,0,0,0). As with the variables or numbers in ordered pairs and triples, there are no spaces after the commas. Everything is all scrunched together.
At first you might think, “Cartesian four-space isn’t difficult to imagine,” and draw an illustration such as Fig. 11-1 to illustrate it. But when you start trying to plot points in this system, you’ll find out there is a problem. You can’t define points in such a rendition of four-space without ambiguity. There are too many possible values of the ordered quadruple (w,x,y,z) , and not enough points in 3D space to accommodate them all. In 3D space as we know it, four number lines such as those shown in Fig. 11-1 cannot be oriented so they intersect at a single point with all four lines perpendicular to the other three.
Imagine the point in a room where the walls meet the floor. Unless the building has an unusual architecture or is sagging because of earth movement, there are three straight line segments defined by this intersection. One of the line segments runs up and down between the two walls, and the other two run horizontally between the two walls and the floor. The line segments are all mutually perpendicular at the point where they intersect. They are like the x , y , and z axes in Cartesian three-space. Now think of a fourth line segment that has one end at the intersection point of the existing three line segments, and that is perpendicular to them all. Such a line segment can’t exist in ordinary space! But in four dimensions, or hyperspace, it can. If you were a 4D creature, you would not be able to understand how 3D creatures could possibly have trouble envisioning four line segments all coming together at mutual right angles.
Mathematically, we can work with Cartesian four-space, even though it cannot be directly visualized. This makes 4D geometry a powerful mathematical tool. As it turns out, the universe we live in requires four or more dimensions in order to be fully described. Albert Einstein was one of the first scientists to put forth the idea that the “fourth dimension” exists.
You’ve seen time lines in history books. You’ve seen them in graphs of various quantities, such as temperature, barometric pressure, or the Dow Jones industrial average plotted as functions of time. Isaac Newton, one of the most renowned mathematicians in the history of the Western world, imagined time as flowing smoothly and unalterably. Time, according to Newtonian physics, does not depend on space, nor space on time.
Wherever you are, however fast or slow you travel, and no matter what else you do, the cosmic clock, according to Newtonian (or classical) physics, keeps ticking at the same absolute rate. In most practical scenarios, this model works quite well; its imperfections are not evident. It makes the time line a perfect candidate for a “fourth perpendicular axis.” Nowadays we know that Newton’s model represents an oversimplification; some folks might say it is conceptually flawed. But it is a good approximation of reality under most everyday circumstances.
Mathematically, we can envision a time line passing through 3D space, perpendicular to all three spatial axes such as the intersections between two walls and the floor of a room. The time axis passes through three-space at some chosen origin point, such as the point where two walls meet the floor in a room, or the center of the earth, or the center of the sun, or the center of the Milky Way galaxy.
In four-dimensional (4D) Cartesian time-space (or simply time-space) , each point follows its own time line. Assuming none of the points is in motion with respect to the origin, all the points follow time lines parallel to all the other time lines, and they are all constantly perpendicular to three-space. Dimensionally reduced, this sort of situation can be portrayed as shown in Fig. 11-2.
Position Vs Motion
Things get more interesting when we consider the paths of moving points in time-space. Suppose, for example, that we choose the center of the sun as the origin point for a Cartesian three-space coordinate system.
Imagine that the x and y axes lie in the plane of the earth’s orbit around the sun. Suppose the positive x axis runs from the sun through the earth’s position in space on March 21, and thence onward into deep space (roughly towards the constellation Virgo for you astronomy buffs). Then the negative x axis runs from the sun through the earth’s position on September 21 (roughly through Pisces), the positive y axis runs from the sun through the earth’s position on June 21 (roughly toward the constellation Sagittarius), and the negative y axis runs from the sun through the earth’s position on December 21 (roughly toward Gemini). The positive z axis runs from the sun toward the north celestial pole (in the direction of Polaris, the North Star), and the negative z axis runs from the sun toward the south celestial pole. Let each division on the coordinate axes represent one-quarter of an astronomical unit (AU), where 1 AU is defined as the mean distance of the earth from the sun (about 150,000,000 kilometers). Figure 11-3A shows this coordinate system, with the earth on the positive x axis, at a distance of 1 AU. The coordinates of the earth at this time are (1,0,0) in the xyz -space we have defined.
Of course, the earth doesn’t remain fixed. It orbits the sun. Let’s take away the z axis in Fig. 11-3A and replace it with a time axis called t . What will the earth’s path look like in xyt -space, if we let each increment on the t axis represent exactly one-quarter of a year? The earth’s path through this dimensionally-reduced time-space is not a straight line, but instead is a helix as shown in Fig. 11-3B. The earth’s distance from the t axis remains nearly constant, although it varies a little because the earth’s orbit around the sun is not a perfect circle. Every quarter of a year, the earth advances 90° around the helix.
Practice problems for these concepts can be found at: Hyperspace And Warped Space Practice Test.
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