### 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 $Math content$ and $Math content$, which contains two parameters $Math content$ and $Math content$, that can be changed with sliders. The slider for $Math content$ 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 $Math content$ 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