Nina del Ser - Mathematics
12
| octopus
r 2 =cos 2 t sin 2 t b 2 cos 2 1 Ω t b 2 sin 2 1 Ω t 2 bcostcos 1 Ω t 2 bsintsin 1 Ω t
r
2
1 b 2 2 bcos Ωt
From (6) we may conclude that |r| is periodic with a time period of
(6)
2 Π
,
Ω
which implies that any curve gener-
ated with positive integer values of Ω will also have Ω axes of symmetry. It is also interesting to note the
difference in the shapes of corresponding values of Ω: for instance, figure 13 has four axes of symmetry, just
like figure 9, but looks “inverted”.
Rational values of Ω
The following curves were all generated using rational values of Ω.
3 figure 18: Ω 9 ,
4
58 figure 20: Ω
figure 17: Ω 2
figure 19: Ω 25
13
4
Once again, a pattern emerges from these curves-the numetor of Ω corresponds exactly to the number of axes
of symmetry in a given curve. The method used to obtain (6) need only be changed slightly to be extended to
all rational values of Ω:
Ω
p
q
p, qΕ Z, q0, p and q are in their lowest terms
r 2 =cos 2 t sin 2 t b 2 cos 2 1 q t b 2 sin 2 1 q t 2 bcostcos 1 q t 2 bsintsin 1 q t
p
r
p
2
1 b 2 2 bcos
p
p
q
p
t
(7)
From (7), we conclude that for any curve generated by a rational value of Ω, |r| will have a time period of
2 Πq
p
seconds which means that it will have p different, evenly spaced axes of symmetry and that it will take
2Πq seconds for the seat to complete one entire cycle. For example, a seat revolving at Ω
8
rad/s
35
will have
rotational symmetry of order 8 and take 35·2Π219 s to complete one full cycle. Figures 21-24 shows the
4
evolution of the curve trace by a seat with Ω 9 rad/s over time:
116