c a t e g o r y   e x c h a n g e s   i n   s p a c e t i m e

By Henryk Szubinski

 

 

using a flat based platform for a flying car by applications of generators for lift generations where the vectors are indicated as being in interactions with the angle of bending at the crucial point

One way to represent the Möbius strip as a subset of R3 is using the parametrization:

x(u,v)= \textstyle \left(1+\frac{1}{2}v \cos \frac{1}{2}u\right)\cos u
y(u,v)= \textstyle \left(1+\frac{1}{2}v\cos\frac{1}{2}u\right)\sin u
z(u,v)= \textstyle \frac{1}{2}v\sin \frac{1}{2}u

where 0 ≤ u < 2π and −1 ≤ v ≤ 1. This creates a Möbius strip of width 1 whose center circle has radius 1, lies in the xy plane and is centered at (0, 0, 0). The parameter u runs around the strip while v moves from one edge to the other.

IMG_0013

In cylindrical polar coordinates (r, θ, z), an unbounded version of the Möbius strip can be represented by the equation:

\textstyle \log(r)\sin\left(\frac{1}{2}\theta\right)=z\cos\left(\frac{1}{2}\theta\right).

Topologically, the Möbius strip can be defined as the square [0,1] × [0,1] with its top and bottom sides identified by the relation (x, 0) ~ (1 − x, 1) for 0 ≤ x ≤ 1, as in the diagram on the right.

The Möbius strip is a two-dimensional compact manifold (i.e. a surface) with boundary. It is a standard example of a surface which is not orientable. The Möbius strip is also a standard example used to illustrate the mathematical concept of a fiber bundle. Specifically, it is a nontrivial bundle over the circle S1 with a fiber the unit interval, I = [0,1]. Looking only at the edge of the Möbius strip gives a nontrivial two point (or Z2) bundle over S1.

because the point in motion around the mobius strip has a accellerative effect THE USAGE OF A DELAY BREAK AT THE NOSE SECTION WOULD REDEFINE STABILITY EVEN AS SUCH THE COLLECTIVE COMPLEXITY OF MOBIUS MULTI DELAYS CAN BE PULLED OUT OF SHAPE AT THE TAIL SECTIONS AS A TYPE AREA format: when used in 2 vector combinations next to each other:

 

IMG_0014

A simple construction of the Möbius strip which can be used to portray it in computer graphics or modeling packages is as follows :

exchanges of

CATEGORY = FUNCTION

objects 1,2,3,4,5,6

FORMS =VOLUME ; displacement, quantals, Areas / polymers

DATA SYSTEMS STORE ( artificial intelligence memory)

USAGE : MOTIVATIONS

interactive states of LINKAGES

FORCES : weak energy, electro magnetism, gravity…….

IMG_0002

The edge of a Möbius strip is topologically equivalent to the circle. Under the usual embeddings of the strip in Euclidean space, as above, this edge is not an ordinary (flat) circle. It is possible to embed a Möbius strip in three dimensions so that the edge is a circle, and the resulting figure is called the Sudanese Möbius Band.

To see this, first consider such an embedding into the 3-sphere S3 regarded as a subset of R4. A parametrization for this embedding is given by

z_1 = \sin\eta\,e^{i\varphi}
z_2 = \cos\eta\,e^{i\varphi/2}.

Here we have used complex notation and regarded R4 as C2. The parameter η runs from 0 to π and φ runs from 0 to 2π. Since  z1  the embedded surface lies entirely on S3. The boundary of the strip is given by | z2 | = 1 (corresponding to η = 0, π), which is clearly a circle on the 3-sphere.

BY USING 2 SIDE SECTIONS OF THE MOBIUS STRIP COMPRESSED SHAPEN INTO A RELATIONSHIP OF TYPE CATAMARRAN FLYING CAR SECTIONS AT EITHER SIDE:

IMG_0005