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Thomas Banchoff papers on the fourth dimension

Biographical Note

Although the concept of higher dimensions has existed since Descartes, not until the nineteenth century was it more fully explored. These early intellectual explorations took the forms of both mathematical and imaginative enquiries, such as Charles H. Hinton’s “scientific romance” titled “What Is the Fourth Dimension?” (1888) and Edwin Abbott Abbott’s novella Flatland (1884). Both Hinton and Abbott built their works on a mathematical foundation, subsequently developing them into moral and social commentaries that reflected a Victorian preoccupation with religion and social class. This Victorian “geometry of higher dimensions” referenced Plato’s shadows, Kant’s transcendental idealism, Gauss’s curved surfaces, and Zollner’s four-dimensional spirits.

In his article “From Flatland to Hypergraphics: Interacting with Higher Dimensions” (1990) Thomas Banchoff outlines the evolution of the idea of the fourth dimension expressed in Abbott’s Flatland. Edwin Abbott was a headmaster, educator, and theologian, but not a mathematician. He was certainly influenced by the contemporary concept of higher-dimensional ideas, but Banchoff links Abbott’s primary interest in the higher dimensions to his exposure to the writings of the younger scientist Charles Hinton. These writings gave Abbott the framework for his novella about "a two-dimensional universe inhabited by flat beings." But, according to Banchoff, Abbott “seems to have been the first to have developed it into a social allegory and the first to treat the possibility of an encounter between beings of different dimensions, with all its challenges and frustrations.” Abbott's fourth dimension suggests the divine space that challenges mortal understanding and expression.

Computers have expanded our ability to visualize the higher dimensions by way of both mathematics and popular culture. According to Professor Banchoff, “As we watch images moving on the screen of a graphics computer, we are faced with challenges like those of the first scientists to make use of telescopes or microscopes or X rays. We are seeing things now that have never been before, and we are just learning how to interpret these images. It is literally true that we are in the first stages of a new era when it comes to visualizing dimensions.”

Banchoff, Thomas F. “Introducing dimensions and scaling and measurement.” Chap. 1 in Beyond the third dimension: geometry, computer graphics, and higher dimensions. Scientific American Library, 1990-1996. http://www.mathaware.org/mam/00/master/essays/B3D/1/technology.html

Banchoff, Thomas F. “From Flatland to hypergraphics: interacting with higher dimensions.” Updated version of what was originally published in Interdisciplinary Science Reviews, 1990. http://www.math.brown.edu/~banchoff/gc/ISR/ISR.html