Drag the end points of the line segment a direction "perpendicular" to the line segment to make a square.
Drag the corner points of the square a direction "perpendicular" to the square to make a cube.
Drag the corner points of the cube a direction "perpendicular" to the cube to make a 4-D Cube.
Of course the secret to performing these steps is figuring out what "perpendicular" means.
You'll learn that when you study Linear Algebra (what I call Matrix Algebra)
in college and study the concept of "orthogonal."
QUESTIONS
How many "points" does a 4-D hypercube have?
[Note how at each iteration we grab all the points and drag them into a new dimension,
doubling the number of points.]
How many "lines" (edges) does a 4-D hypercube have?
How many "planes" (faces) does a 4-D hypercube have?
[Concept: Instead of dragging all the "points" at each iteration, think about
grabbing all the "lines" at each iteration and dragging all the "lines" into a
new dimension. As you drag each line it creates a new plane (or face). For example,
to create a 2-D square we grab the 1-D line and drag it into a new dimension to make a square.]
How many "cubes" does a 4-D hypercube have?
[Concept: Instead of dragging all the "points" or "lines," think about grabbing all
the "planes" (faces) and dragging them into a new direction. For example, to create
a 3-D cube we grab the 2-D plane (face) and drag it into a new dimension creating a cube.]
FURTHER RESEARCH
Search Google
for "hypercube"
The concept of fractional dimensions is covered under the topic of fractals.
Flatland
by Edwin A. Abbott