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© 1999-2010 by Francis Edward Su

From the Fun Fact files, here is a Fun Fact at the Medium level:

Tesseract

 Figure 1

What does a cube look like in higher dimensions?

Well, we can extrapolate by looking at lower dimensions.
A 0-dimensional cube is a point, a vertex.
A 1-dimensional "cube" is a line segment, with 2 vertices at either end. It is obtained from a 0-dimensional cube by thickening it in one dimension.
A 2-dimensional "cube" is square, with 4 vertices, obtained by thickening up the line segment in a second dimension.
A 3-dimensional "cube" is a cube, with 8 vertices, obtained from the square by thickening it in a third dimension.

So, by extrapolation the 4-dimensional "cube", also called a tesseract or hypercube, should have 16 vertices, and is obtained from a cube by thickening it up in a fourth dimension. Since we cannot easily visualize this, there are a number of ways we can understand this object by viewing projections, or "shadows" of it in 3-D. See Figure 1.

Presentation Suggestions:
See if students can guess by extrapolation how many vertices the tesseract should have.

The Math Behind the Fact:
Thinking in four dimensions is not easy, and takes practice. However, a number of science fiction books have been written around this idea and explain it and possible applications quite well; see the reference for one notable example.

Su, Francis E., et al. "Tesseract." Math Fun Facts. <http://www.math.hmc.edu/funfacts>.

References:

Keywords:    four dimensions, 4-space
Subjects:    geometry
Level:    Medium
Fun Fact suggested by:   Francis Su
Suggestions? Use this form.
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