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3.3.1
Goniometricus: A Metabolic Grammar of Sine and Cosine
In this section we will outline a method to develop MP approximations of sine and
cosine functions. This will be done by defining an MP model, called
Goniometri-
cus
, which has two substances,
S
and
C
, approximating in time the evolution of the
functions sine and cosine respectively (apart from a constant offset).
We start from two time series of the same length, related to the two target func-
tions
sin
and
cos
along two oscillations (from 0 to 4
) with a step of 10
−
3
.As
depicted in Fig. 3.5, each value has been increased by 3 units to make all values
positive:
π
cos
(
0
)+
3
,
cos
(
0
.
001
)+
3
,
cos
(
0
.
002
)+
3
, ...
cos
(
4
π
)+
3
sin
(
0
)+
3
,
sin
(
0
.
001
)+
3
,
sin
(
0
.
002
)+
3
, ... ,
sin
(
4
π
)+
3
.
Fig. 3.5
The plot of the time series
T
C
and
T
S
When we start with angle 0, then cosine is 1 and sine is 0. Going from 0 to
2co-
sine decreases and sine increases. Therefore, we can assume a transformation from
cosine to sine. Moreover, initially the sum of sine and cosine is 1, but, when angle
π
/
π
/
4 is reached, the sum of sine and cosine values is
√
2, which is greater than 1. This
means that rules of increase and decrease of “matter” in the system are appropriate.
This intuition suggests the MP grammar given in Table 3.10, and depicted by the MP
graph of Fig. 3.6, which allows us to find, by means of Least Square Evaluation, the
right constants providing a good approximation of the required dynamics.
Ta b l e 3 . 1 0
A form for Goniometricus's MP grammar
r
1
:0
→
C
ϕ
1
=
c
1
+
c
2
C
r
2
:
C
→
S
ϕ
2
=
c
3
S
+
c
4
C
r
3
:
S
→
0
ϕ
3
=
c
5
+
c
6
S
