Infinite Dimensional Infinitesimal

[Russell E. Rierson]

Any smooth connected 1 dimensional manifold is diffeomorphic either to the circle, or to some interval of real numbers.

Take a line segment of length 1. It is one dimensional.

A-------B

Find the midpoint of the line segment and rotate it into 2 dimensions

A
|
|
|------B


Each leg is 1/2

Rotate into 3 dimensions, and each leg is 1/3

A
|
|
|------C
|
|
B



rotate into N dimensions and each leg is 1/N

Continue this process as a limit

N---->oo

By the above process, an infinite dimensional universe is a point.

 

[ranyart]

Originally posted by Russell E. Rierson
Any smooth connected 1 dimensional manifold is diffeomorphic either to the circle, or to some interval of real numbers.

Take a line segment of length 1. It is one dimensional.

A-------B

Find the midpoint of the line segment and rotate it into 2 dimensions

A
|
|
|------B


Each leg is 1/2

Rotate into 3 dimensions, and each leg is 1/3

A
|
|
|------C
|
|
B



rotate into N dimensions and each leg is 1/N

Continue this process as a limit

N---->oo

By the above process, an infinite dimensional universe is a point.

A
I
I
I
D------I------B
I
I
I
I
C

So this is four-dimenisional?..how do you account for the fact there is only 26 letters in the alphabet?

The problem is that a Four dimensional space has inner products made from A+B..A Rotation does not constitute a added Dimension?