Nina del Ser - Mathematics
4
octopus |
13
4
figure 21: Ω 9 , t 4Π
figure 22: Ω 9 , t 8Π
4
4
figure 23: Ω 9 , t 18Π
figure 24: Ω 9 , t 25Π
The curve has rotational symmetry order 4 and as predicted, it does not evolve anymore after when t>18Π
because it is just going back over its original path.
What happens when Ω is irrational?
Intuitively I believe that the full area between the external and internal circles will be covered and
given enough time and rotations, every point in the that area will be reached. All of the points (x,y) would be
located in a “donut” area between two circles so that
1 b x 2 y 2 < 1 + b.
However, this proof is outside the bounds of this paper.
Also, Π may be written as a continued fraction and thus approximated as a rational number (the most
common one being
22
).
7
In order to increase the accuracy we need to increase the size of both the numerator
and denominator, so that other fractions which are better at approximating Π are
places) and
355
(accurate
113
333
(accurate
106
to 5 decimal
to 6 decimal places). The next rational which is accurate to more than 6 decimal
places has... a denominator greater than 30,000. A time period as big as this would be too much for Mathemat-
ica to handle! It also probably would not bring us any closer to the actual value of Π. Figures 25-27 show the
progression of the Octopus when Ω Π:
figure 25: Ω Π , t 10Π
figure 26: Ω Π , t 35Π
117
figure 27: Ω Π , t 150Π