Nina del Ser - Mathematics
octopus |
21
The maxima due to Β and Γ are shown as red and purple respectively (please note that the red lines hide any
purple lines underneath them). Also, it is not necessary to include the lines due to pn-qm because we know
that this value will be a multiple of the highest common factor, for the following reason:
Β and Γ can be re-written as multiples of their higher common factor, which we shall call k. Therefore,
Β mk, Γ nk, where n,mΕ Z
Β-Γ (m-n)k
the term pn-qm is simply a multiple of the highest common factor and does not need to be considered.
Conclusion
We have shown that it is possible to determine the rotational symmetry of any curve generated by a Dodeca-
pus, provided we are given the speeds of its rotors and that these speeds are rational numbers. The method we
have used to do this could also be extended further and applied to any Multipus (an Octopus with any number
of extra arms). Below are some figures generated by Multipuses; you can see that finding the highest common
factor of the numerators of the speeds yields the rotational symmetry of the curves:
8
figure 62: Hexadecapus with Ω 3, Γ 6, ∆ 9 figure 63: Duodecapus Ω 4, Γ , ∆ 12, Ε 16
figure 64: Duotetradecapus Ω 8, Γ 16, ∆ 24, Ε 40, Ζ 32 figure 65: Duooctadecapus Ω 6, Γ 9, ∆ 18, Ε 15, Ζ , Η 12
9
3
5
125