The Math Book by Clifford A. Pickover

Author:Clifford A. Pickover
Language: eng
Format: epub
Publisher: Sterling Publishing Co., Inc
Published: 2011-02-07T16:00:00+00:00

The cover of Flatland, 6th Edition, by Edwin Abbott Abbott. Notice that “My Wife” is portrayed as a line within the pentagonal house. In Flatland, women could be particularly dangerous, due to their sharp ends.

Tesseract

1888

Charles Howard Hinton (1853–1907)

I know of no subject in mathematics that has intrigued both children and adults as much as the idea of a fourth dimension, a spatial direction different from all the directions of our everyday three-dimensional space. Theologians have speculated that the afterlife, heaven, hell, angels, and our souls could reside in a fourth dimension. Mathematicians and physicists frequently use the fourth dimension in their calculations. It’s part of important theories that describe the very fabric of our universe.

The tesseract is the four-dimensional analog of the ordinary cube. The term hypercube is used more generally when referring to cube analogues in other dimensions. Just as a cube can be visualized by dragging a square into the third dimension and watching the shape that the square traces through space, a tesseract is produced by the trail of a cube moving into the fourth dimension. Although it is difficult to visualize a cube being shifted a distance in a direction perpendicular to all three of its axes, computer graphics often help mathematicians develop a better intuition for higher-dimensional objects. Note that a cube is bounded by square faces and a tesseract by cubical faces. We can write down the number of corners, edges, faces, and solids for these kinds of higher-dimensional objects:

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