### 1.4 Example of a Simulation: The Moebius band

As example for the possibilities of interactive simulations as they will be used in the
following. Figure 1.3 shows a rotating Möbiusband in three dimensional
projection. Among the closed bands in space the Möbiusband is characterized by
the fact, that it makes half a twist, and thus during one circulation both
sides are covered; it has “only one surface “. In the picture of the simulation
one sees the formulas for the three spatial coordinates with the variables
$p$ and
$p$, which contains
two parameters $a$ and
$b$, that can be changed with
sliders. The slider for $a$
changes the number of half twists , while the other one changes the height of
the band. If a non-integer number is chosen for the number of half twists
$a$ the
band can be cut, and rejoined with another number. If this number is even, one
obtains normal bands with 2 surfaces. It this number is odd, one obtains a Möbius
bands that has additional twists.

The formulas for the three space coordinates as well as the time dependent
animation component can be edited, i.e. the can be changed. Using the same
simulation arbitrary animated surfaces in space can be visualized. The ability to
edit opens a wide training field for the advanced understanding of functions
that describe three and four dimensional processes. Figure 1.4 shows two
examples from the simulation of Figure 1.3. On the left a simple band with a
full twist and on the right a Möbiusband with one and a half twists were
calculated.

The text pages of the simulation contain extensive descriptions, hints for many
alternatives of the 3D-projection and suggestions for experiments. Figure 1.5 shows
the description window, that appears next to the simulation when it is opened. For
this example it contains 4 pages:

Introduction with a description of the simulation and its controls,

Visualization with hints about the possibilities of the 3D-projection,

Functions for the discussions of the mathematical formalism,

Experiments with suggestions for experiments that make sense.

In the figure the page for Visualization is opened. It describes the easy
possibilities of different three dimensional presentations:

- Rotation
- Translation
- Zoom
- With or without perspective distortion
- Projections along one of the three axes

You are encouraged to use this example, to try the different means of experimentation,
before you start with the next chapter.

End of chapter 1