Tuesday,  October 21

More on 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.

(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

Other ways to use coordinates:
The Tower of Hanoi

The general problem: (illustrated with three objects)
Move objects that have an order (size) from one place to another using only a third place for "storage". No larger object can be placed on top of a smaller object during the move. Move only one object at a time!

Solution of the 3 Tower of Hanoi Puzzle.
(Using playing cards 1,2,3)

Card- Post    Changes to cards   0-1 Changes to cards
(0, 0, 0)            (0, 0, 0)

1.  1  →    B    (1, 0, 0)            (1, 0, 0)
2.  2  →    C    (1, 1, 0)            (1, 1, 0)
3.  1  →    C    (2, 1, 0)            (0, 1, 0)
4.  3  →    B    (2, 1, 1)            (0, 1, 1)
5.  1  →    A    (3, 1, 1)            (1, 1, 1)
6.  2  →    B    (3, 2, 1)            (1, 0, 1)
7.  1  →    B    (4, 2, 1)            (0, 0, 1)

Record your moves. Assume that 1 represents the ace and the posts are labelled A, B and C.
[Use the seven moves below from the 3 tower puzzle as a start.] Record also the coordinates in 4 dimensional space for the number of changes made to the 4 cards and the 0-1 switches.

Solution of the 4 Tower of Hanoi Puzzle.

Card→Post        Changes to cards    0-1 Switches to cards
(0, 0, 0, 0)            (0, 0, 0, 0)
1.  1 → B        (1, 0, 0, 0)            (1, 0, 0, 0)
2.  2 → C        (1, 1, 0, 0)            (1, 1, 0, 0)
3.  1 → C        (2, 1, 0, 0)            (0, 1, 0, 0)
4.  3 → B        (2, 1, 1, 0)            (0, 1, 1, 0)
5.  1 → A        (3, 1, 1, 0)            (1, 1, 1, 0)
6.  2 → B        (3, 2, 1, 0)            (1, 0, 1, 0)
7.  1 → B        (4, 2, 1, 0)            (0, 0, 1, 0)
8.
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Maps
Coordinates for "earth" - the sphere
Coordinates for the torus!