Nina del Ser - Mathematics
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| octopus
figure 53: Ω 3, Γ 4 figure 54: Ω 5, Γ 13 figure 55: Ω 5, Γ 19
figure 56: Ω 2, Γ 12 figure 57: Ω 6, Γ 36 figure 58: Ω 3 , Γ 4
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Although these shapes look rather more complicated than those generated by the Octopus, we can utilize the
same trick as that used to obtain (7), extending it only to include the tertiary rotor, to understand how the
symmetries in these curves arise:
p
Ω q , Γ
m
n
p, q,m,n Ε Z, q,n0, p, q and m, n are in their lowest terms
r 2 =cos 2 t sin 2 t b 2 cos 2 1 t b 2 sin 2 1 t c 2 cos 2 1
2 bsintsin 1 t
p p m
q q n
t c 2 sin 2 1
m
n
t 2 bcostcos 1 t
p
q
p
q
2 c costcos 1
r
2
m
n
t 2 c sintsin 1
1 b 2 c 2 2 b cos
p
q
m
n
t 2 bc cos
t c cos
m
n
p
q
t cos
m
n
t+2 bc sin
t bc cos
p
q
p
q
m
n
t sin
t
m
n
t
(12)
As can be seen from (12), the maximum r occurs when all of the cosines are equal to 1 simultaneously. We
have also established that the number of axes of symmetry for one cosine function depends only on the
numerator of the fraction, whilst the denominator defines the time period, T, for the curve. For there to be a
maximum on a curve generated by the Dodecapus all three cosines need to coincide with their maximum
values, so that the value which defines the number of axes of symmetry is the highest common factor of the
three numerators: p, q and pn-qm. Figures 59-61 illustrate this principle :
figure 59: Β 4, Γ 6
figure 60: Β 5, Γ 10
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9
figure 61: Β 8 , Γ 4
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:
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