Thursday,  March 24

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

For a Cube
we use three "coordinates" and we can identify the vertices of the cube with qualities such as "left..right", "up... down", and "front ... back":
(0,0,0) , (1,0,0), (0,1,0),(1,1,0)

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

For a Hypercube
....
we use four "coordinates" and we can identify the vertices of the hypercube with qualities such as "left..right", "up... down", and "front ... back" and "inside... outside": (0,0,0,0) , (1,0,0,0), (0,1,0,0),(1,1,0,0)

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

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

(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.