March 30, 2006

"minimal surfaces" (FAPP video)
Transforming surfaces: "turning the sphere inside out." video

Non-orientable Surfaces and the fourth dimension

Thinking about a fourth (and higher) 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.

(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 interest in surfaces: Examples
Ways to think of surfaces : cross-sections/ projections/moving curves/ using color to see another dimension.   ChromaDepthTM 3D;    CD Image gallery
How do 3d glasses work? |
Generalization of surfaces are called "manifolds".  cross sections / projections/ moving surfaces-solids.

Looking at the Torus and the Klein Bottle using four dimensions:
A torus as a circle in space that cycles about a central axis.

A Klein bottle: A circle in space that cycles without "rotation" A javaview visualization of the Klein bottle

On Line Article: Imaging maths - Inside the Klein bottle

Video on similarity.
Next class: Similarity, Magnification, and Looking at the very small and the very large.  How we  see the infinite.Microscopes and Telescopes.