April 7, 2014

Video on Erdos Problem for n=2,3.

Coding the Torus, and the Sphere: The Torus:
Start at A for coding of edges-   +T, -S,-T, +S where + indicates clockwise and - indicates counterclockwise for the edge and how the edges are to be attached.

The sphere:

Start at A for coding of edges-   +T, +S,-S, -T

The Tower of Hanoi "puzzle" Coding Activity

More thinking about a fourth (and higher) dimension and coding:
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.