A Visit to Flatland

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Spacelander, welcome to Flatland! Meet Ms. Line, Reverend Circle, and A. Square, all citizens of a magical place first described by Edwin A. Abbott in his 1884 satirical novel Flatland, A Romance of Many Dimensions. Flatlanders live in only two dimensions. They see each other's edges only. You can get a feeling for their diminished point of view by comparing the lower, horizontal display with the more privileged bird's-eye view, which only you—as a visitor from Spaceland—can have hovering over Flatland. Drag the geometric shapes around or set them in motion by clicking the "animate" button. The real challenge for you after visiting Flatland is to imagine by analogy your three-dimensional world embedded in a larger four-dimensional (or greater) space. Ask yourself what would a four-dimensional Mr. Sphere look like visiting your three-dimensional world?

Contributed by: John Kiehl (March 2011)
Open content licensed under CC BY-NC-SA



For over 100 years, Flatland, A Romance of Many Dimensions has been used as a tool to introduce geometry students to higher dimensions. The strategy is clever yet straightforward: simply take a step backward to lower dimensions. Try to imagine the narrow-mindedness and confusion that a 2D creature would have if you explained the notion of an additional dimension. Perhaps you would materialize seemingly out of nowhere by dropping down into Flatland from above in an effort to get them thinking out of their 2D box. Or perhaps, you would perform open-heart surgery on them without cutting through their skins, since of course you would have access to their insides—again from above. Before losing your patience with their myopia try now to imagine a 4D surgeon repairing your heart without the need to cut through your skin and bones. Thomas Banchoff has been quoted as saying that after a lifetime of thinking about the fourth dimension, that only on two occasions, and only for seconds each time, did he feel like he actually experienced the fourth dimension.


[1] E. A. Abbott, Flatland, A Romance of Many Dimensions, London: Seeley and Co., Ltd., 1884.


[2] I. Stewart, The Annotated Flatland, Philadelphia: Perseus Books, 2002.

[3] D. Burger, Sphereland: A Fantasy about Curved Spaces and an Expanding Universe, New York: Sterling Publishing, 1965.

[4] A. K. Dewdney, The Planiverse: Computer Contact with a Two-Dimensional World, New York: Copernicus, 2001.

[5] R. Rucker, Spaceland, New York: Tor, 2002.

[6] R. Rucker, The Fourth Dimension: Toward a Geometry of Higher Reality, New York: Houghton-Mifflin, 1984.

[7] T. F. Banchoff, Beyond the Third Dimension, New York: Scientific American Library, 1990.

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