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.

(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)

"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:

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 interest in surfaces: Examples

Ways to think of surfaces : cross-sections/ projections/moving curves/ using color to see another dimension. ChromaDepth

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 javaview visualization of 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.