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| octopus
Nina del Ser - Mathematics
Introduction
This project was originally inspired by a homework problem, in which a series of questions were
posed about the motion of a funfair ride known as “the Octopus”. The Octopus is composed of a total of five
cross-shaped rotors: one primary rotor (shown in red in the graphic below), with a radius of 1 meter, which
rotates anticlockwise at speed of 1 rad/s and four smaller rotors, each 0.5 meters in radius (in blue), spinning
in the clockwise direction at 2 rad/s about each of the four ends of the primary rotor. The passenger seats
(depicted as orange squares) are located at the ends of the smaller rotors.
Figure 1, which was drawn using Mathematica, shows the basic setup of the Octopus:
0.5 metres
1 metre
S
figure 1
The question originally posed was to plot the path of one of these seats over time. Once the path
performed by one seat in particular was known (in this case, the seat S was chosen), its behaviour could be
generalised to the rest of the points because, by symmetry, they should hypothetically all form the same curve.
I became particularly interested in this problem when I started experimenting with changing parame-
ters such as the lengths of the rotors and their angular speeds, and the first aspect of the resulting observations
that drew me in was the aesthetic beauty and sheer diversity of curves that could be generated using such a
simple set-up. Further modifications, such as adding on extra pairs of rotors on the ends of the arms of the
secondary rotors led to even more intricate and interesting curves.
Another interesting aspect of this topic is that it is not just confined to amusement parks, but also
manifests itself in other fields. For example its application is evident in astronomy, where a mechanism very
similar to that which operates the Octopus can be used to model the motion of moons revolving around
planets, themselves revolving around the sun.
The principal aim of this project is to gain better understanding of the curves generated (eg. being able
to predict their rotational symmetry based on information about the rotors and their speeds). Throughout the
investigation, illustrations of different types of curves generated by the Octopus will be used very extensively
both to gain a better intuitive understanding and to test any hypotheses, once they have been formulated.
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