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Introducing Dimensions

As we watch through the lenses of a microscope, an amoeba goes about the course of its virtually two-dimensional life, confined to the narrow region between the slide and its coverslip. We observe from above as the amoeba moves around, encountering other creatures like itself, capturing food, and avoiding predators. Part of the cell membrane forms a line of defense entirely surrounding the amoeba and protecting its nucleus inside from threats by other creatures on the slide. But the words inside and surrounding do not mean the same to us in three-dimensional space as they do to the inhabitants of this nearly flat space. No amoeba in this space can ever come into direct contact with the nucleus of another. We, however, can look down from another direction entirely and see the very insides of the organism. Not only is the nucleus exposed to our view, but we can also poke it directly, a strange and disturbing event for the surprised amoeba. From our three-dimensional perspective, we visualize the world of the microscope slide in a totally different way than do its inhabitants.

The cover of the 1884 first edition of Flatland not only invited the readers into realms of new dimension, but into the two-dimensional house of the books narrator A square. Although A Square can see only one room at a time, his house is totally open to our view.

One hundred and six years ago, a brilliantly conceived book exploited this fundamental idea of interaction between creatures of different dimensions to encourage its readers to break the bonds of limited perspective and open their minds to new ways of perceiving. Its author, Edwin Abbott Abbott, was a clergyman and the headmaster of a school in Victorian England. As a leader in the movement to provide educational opportunities for young men and women of all social classes, he was often frustrated by prevailing social attitudes and by establishment views in education and religion. Of his fifty books, the one that still speaks clearly to our own day is his little masterpiece Flatland, simultaneously a social satire and an introduction to the idea of higher dimensions.

Flatland describes an entire race of beings who are two-dimensional, living on a flat plane, unaware of the existence of anything outside their universe. How they lived and interacted and communicated is a fascinating story, and the narrator, A Square, does an excellent job of interpreting his society and his world to us living in what he calls "Spaceland." His task is prodigious, because as difficult as it is for us to imagine how the flat world looks to its citizens, it is truly impossible for the two-dimensional narrator to appreciate the full reality of Spaceland. In particular, he cannot conceive the kind of total view of his existence that we possess. Like a technician watching the movements of the amoeba, we can observe the changing positions of the creatures in Flatland. We can see all parts of a house simultaneously and the contents of any room or any enclosure. From the Flatland point of view, we are omnivident, seeing everything. It is little wonder that A Square, on first hearing about this superior vision, supposes that anyone who possesses it must be divine.

A Square views the inhabitants of Lineland.

To help A Square understand the all-encompassing view from the third dimension, Abbott presents a dimensional analogy. He asks A Square to imagine what it would be like for him to observe Lineland, a one-dimensional universe populated by segments. A Square would be able to see all creatures in this world at the same time. The King of Lineland, a long segment, would be very surprised if A Square poked his inside without disturbing either of his extremities.

Just as a Flatland creature can view all of Lineland, we in space have a superior view of Flatland. In the story, the power of the analogy makes a great impression on A Square. He asks what it would be like for a being from a fourth dimension to "look down from on high" and see everything in three-space, even the very insides of solid humans. What about worlds of five or six dimensions, each one able to look down on its predecessor and each one open to the all-seeing scrutiny of the next?

Abbott used dimensional analogies to great effect in raising questions about the way we see the world, especially when we come into contact with the truly transcendental. For over a century, mathematicians and others have speculated about the nature of higher dimensions, and in our day the concept of dimension has begun to play a larger and larger role in our conception of a whole range of activities.

Mathematics Awareness Month is sponsored each year by the Joint Policy Board for Mathematics to recognize the importance of mathematics through written materials and an accompanying poster that highlight mathematical developments and applications in a particular area.