An introduction to the fourth dimension:

A progression: Point and segment on a line, line segment and square in a plane (2-dim), square and a cube in space (3-dim), cube and a "hypercube" in hyperspace (4-dim)

The Hypercube and coordinates:

What do we measure? How does this determine "dimension?"

For a Line segment we can use one number to indicate distance and direction from a single point: 0 .... 1

For a Square we use two "coordinates" and we can identify the vertices of the square:

(0,0,1,0), (1,0,1,0), (0,1,1,0), (1,1,1,0)

(0,0,1,1), (1,0,1,1), (0,1,1,1), (1,1,1,1)

Another four dimensional object:

The hyper simplex!

point

line segment

triangle

tetrahedron ("simplex")

Cards and the fourth dimension.

(clubs,diamonds,hearts,spades)

(1,1,1,1) (0,0,0,0)

(1,1,0,1) (0,0,1,0)

(0,1,0,1) (1,0,1,0)

(0,0,0,1) (1,1,1,0)

(0,0,0,0) (1,1,1,1)

Hamiltonian Tour: move through each vertex once and only once.

13 cards : (5,3,0,5) (4,2,6,1)

Other ways to think about the hypercube:

video