Example of a Simulation: The Moebius band

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.

Figure 1.3: The figure shows a simple Möbiusband in perspective 3D-projection. The three function boxes contain the parameter representations for the space coordinates. The variables

p

and

q

vary in the range

- π

π

. The parameter

a

that determines the number of twists (here

1 ∕ 2

) can be changed with a slider, as well as the parameter

b

that controls the height of the band. The

z

-component can be periodically modulated for

v > 0

with the angular velocity

v

(play button) in order to create the impression of a rotating band. Using a check box the

x y

-symmetry plane can be shown or suppressed.

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.

Figure 1.4: Examples from the simulation in Figure 1.3; on the left a simple loop with

a = 1

, on the right Möbiusband with an additional twist (

a = 1.5

)

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.

Figure 1.5: Description window of the simulation. Here it contains 4 pages, of which the visualization page is opened. Please test the possibilities after opening the simulation!

End of chapter 1