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Chapter 4. Part 2

What Became of Area?

I can probably answer that. In Chapter 3, please recall that the final area measurement was an approximation made by a statistical method called Monte Carlo. Should we be satisfied with an approximation? If the function which describes the boundary is not too complicated, we could achieve the ideal solution through integration, but keep two things in mind. The first is that nature's problems are very complex, and the second is that, if we are measuring some physical region bounded by real matter, Heisenberg's electrons on that boundary won't be still and allow us to measure exactly. Approximation is the way to go.

We also have rigor in the Monte Carlo technique. The properties of a metric space have been delineated [1]. Measure has been given an abstract definition to accompany our intuitive ideas [2]. Bernoulli's Theory of Large numbers guarantees that with more trials comes more accuracy, and we have an error estimate which is a function of the number of trials [3]. We need not have the boundary described by a function, as long as we can determine whether our random points fall within or outside of the region. In truth though, the complexity in the boundaries of fractal regions presents some difficulty in this determination.

The simplest example of an area measured with this technique is that of a circular region. Many computer routines have been written to perform this simulation. Center a unit circle at the origin and circumscribe a square about it. Find random pairs of numbers in the interval (-1, 1), use them for x and y-coordinates, and you get random points inside the square. The distance from the origin decides whether points are inside the circle as well. After 10,000 trials or so, use the percentage of points in the circle to estimate its area. The result will also yield an estimate for pi because theoretically the ratio of the areas is pi to 4.

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One question which was raised in my mind a few years ago centered on the space that the circle takes up in its square as compared with that of the sphere snug within a cube. We can extend our Monte Carlo routine to three dimensions and get a volume estimate for the sphere. We also have the advantage of knowing the ideal formulas: edge cubed for the cube, and (pi/6) times the edge cubed for the sphere. Using either method we learn that the sphere takes up about 52% of the cube's volume. That's a lot of wasted space in a basketball's box, isn't it... and it is more space proportionally than the circle wastes. Hmmm...

I wondered if inscribed spheres of higher and higher dimensions would waste more and more space. Would the percent go to zero, meaning that n-dimensional balls would be taking up little or no space inside n-cubes, or would the percent approach a minimum asymptotically? Or would something else happen? And what of the "ideal"? What are the formulas for the volumes of n-balls, and how is pi involved? If you consider area to be a particular case, a 2-dimensional volume let's say, perhaps you would allow that these questions attempt to generalize the problem of measuring the capacity of certain regions. I succeeded in finding the n-ball solutions.

The easy part was extending the computer simulation to higher dimensions. Points merely pick up a fourth, fifth, and sixth coordinate, and so on. In this type of metric space, the distance formula based on Pythagoras also extends naturally.

d(<x>, <y>) = SQRT [ (x1 - y1)^2 + (x2 - y2)^2 + (x3 - y3)^2 + (x4 - y4)^2 +...]

So data was generated to make the estimates. I found the ratio which compared the capacities of the hypersphere and the hypercube to be about 30.5%. Even more waste! For each higher dimension the n-ball's proportion continued to shrink by at least a factor of two.

Could I verify these results with theory? Fortunately, a unit n-sphere has one of the simplest formulas in radial coordinates, namely r = 1. There is, however, the complication of multiple integrations of the Jacobian of the transformation equations from the Cartesian system. There is also the conceptual problem of choosing the appropriate integration limits of each of the angular variables. Nevertheless, I was able to derive the volume formulas for dimensions four through eight, and in general. For the four- and five-balls respectively, those formulas are:

(1/2)(pi^2)(r^4) and (8/15)(pi^2)(r^5),

and they are consistent with my statistical results. The exponent of pi increases by one, but only when you increase to an even dimension. This means that the formulas for the sixth- and seventh-dimensional spheres involve the third power of pi, and so on. I hope to get the full report in shape to present on this website in the near future. I'd like to thank Professors Keith Ferland of Plymouth State College, and Thomas Banschoff of Brown University for their assistance with this problem. Dr. Banschoff maintains a website gallery of 4-D art in addition to his research projects. I also include a link to an interactive page dedicated to Math Awareness Month, and focusing on "dimension."

Link to Banschoff

What do the results mean? I have a hard time myself envisioning four- dimensional space. I can tell you what the results do not mean. No one should have a picture of a tiny ball rolling around in a big box. Remember that these are inscribed spheres in all cases, meaning that they are tangent to all faces of the n-cubes. Second, if we take a slice of a hypersphere, we get a sphere. If we slice it on an equator of the hypersphere, we get a sphere identical to the 3-D ball, and yet the strange thing is, the 4-D measures of these cross-sections are all zero, just as the 2-D circular disk has no 3-D volume.

Here's my fanciful interpretation. As n increases, the n-cube gains more and more compartments and yet has remained near the origin and maintained a unit volume (albeit, an n-dimensional unit volume). There is infinitely more space in a box of volume one than in a square of area one, so there is much space for things to be happening inside these hyper-hypercubes. The inscribed n-balls still require a radius sufficient to reach the outer faces, so they haven't shrunk, but I see much more unused space inside the n-box. Why is this fanciful? I like to think that the space in the universe has a dimension higher than three. This would create the space needed for awesome ideas such as parallel universes, worm holes, and time-travel to be physical possibilities. It's something to think on.

Chapter Four Summary

In the modern era mathematicians generalized the abstract properties of concepts such as area and measure. They are now more difficult to understand, but they now apply to a broader range of real problems than ever before. Many practical problems were attempted which overtaxed even the powerful multi-variate calculus. As a result, new analytical methods were devised. There are now tools for analyzing complicated dynamical systems and solving more realistic problems.

Something else has changed which will influence the future forms that mathematical study takes. Technology has grown so powerful that it not only supports the new systemic analysis and modern statistical techniques, it is encouraging a new era of experimentation. Some experts complain about the danger of moving away from rigorous, logical proof, and in taking on problems that no human could begin to verify the absolute truth of. To those I would say to keep in mind that it took hundreds of years of experimentation to provide the Greeks with the raw material of ideas around which they built their logical edifice. Perhaps the processes of abstraction and generalization have dried up the idea bank, and it is time for an age of playing "what if" and utilizing the technological power to generate new paths for formal investigation.

Project Conclusion

Mathematics continues to evolve through individual and team efforts to solve problems. The solutions often require the invention of new methods and the discovery of changes or additions to the body of mathematical theory. This "new math" inevitably gives rise to new questions and problems to solve. This cycle has sustained our discipline and kept it the essential foundation for the other sciences.

In true scientific fashion, mathematicians have discarded the ideas proven inadequate or false. Without this flexibility in thinking, mathematics would become irrelevant. Teachers of mathematics need to cultivate this flexibility in their students, who should not be given the impression that the rules of mathematics are a closed set, and that there is no room for further inquiry. Those students need to have that message at a young age so that our future general populace can understand just how complex our universe is, and appreciate the triumphs and frustrations experienced by our scientific leaders.

Mitchell Feigenbaum, a leader in field of Chaos Theory, once remarked, "Somehow the wondrous promise of the earth is that there are things beautiful in it, things wondrous and alluring, and by virtue of your trade you want to understand them" [4]. Isn't it interesting that after 2500 years, what the Greeks were after, remains a major human endeavor?

Endnotes

  1. Eves, Great Moments in Mathematics After 1650. Mathematical Association of America, 1983. pp. 151-152.
  2. Rota, "Geometric Probability." The Mathematical Intelligencer 20:4, Springer-Verlag, New York, 1998, pp. 11-12.
  3. Dunham, The Mathematical Universe. Wiley, New York, 1994. pp. 16-22.
  4. Gleick, Chaos: The Making of a New Science. Penguin, New York, 1987. p. 187.

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