Figure 1

What does a cube look like in higher dimensions?
Well, we can extrapolate by looking at lower dimensions.
A 0dimensional cube is a point, a vertex.
A 1dimensional "cube" is a line segment, with 2 vertices
at either end. It is obtained from a 0dimensional cube
by thickening it in one dimension.
A 2dimensional "cube" is square, with 4 vertices, obtained
by thickening up the line segment in a second dimension.
A 3dimensional "cube" is a cube, with 8 vertices, obtained
from the square by thickening it in a third dimension.
So, by extrapolation the 4dimensional "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 3D. 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.
How to Cite this Page:
Su, Francis E., et al. "Tesseract."
Math Fun Facts.
<http://www.math.hmc.edu/funfacts>.
