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In the book A Wrinkle in Time, we see the beginning of a progression that leads to the tesseract. We start with a point, which moves in one direction to trace out a segment. If the point moved off in two different directions, it would determine a square.
If the segment moves off in one direction, it will trace out a square, and if it moves in two directions, it will determine a cube. If a square moves in one direction it traces out a cube, but if it moves off in two directions ("squaring the square"), it determines something very new, similar to what would happen if a cube could move off in some direction, completely outside of our familiar space. This new object is a Hypercube, also called a Tesseract.
How can we recognize a tesseract if we see one? (This is a question important to anyone who wants to go off through other dimensions on a rescue mission.) If we follow the dimensional signposts on the poster, we go from 0 dimensions, with one point, to 1 dimension with two corner points, to 2 dimensions with four endpoints, to 3 dimensions with eight corner points. The numerical progression is clear; 1, 2, 4, 8, . . . We expect a hypercube to have sixteen corner points, even though we have never seen one!
The most common representation of a tesseract is what is called a "perspective picture". Some of the square faces have their familiar shape, in the inner cube and the outer cube, and other faces are distorted into trapezoids. Seeing one representation of this object is already impressive and somewhat puzzling. Even more curious and mysterious is what happens when a tesseract "rotates" in four-dimensional space, appearing to turn inside out (see the interactive demonstrations). We can imagine Meg, Calvin, and Charles Wallace being drawn into this motion as they travel with their guides, Mrs. Which, Mrs. Who, and Mrs. What, through higher space.
For a thorough discussion of the mathematics in A Wrinkle in Time, see the essay by Jessica Weare in AWIT Final Project. A number of images and movie clips (with discussions) that might help you understand the hypercube better are available in Davide Cervone's materials for his course on "Visualizing the Fourth Dimension".