storytelling (systems thinking):
https://t.iss.one/storytellingsystemsthinking

demonstrators:
https://t.iss.one/systemsdemonstrators

sharing and reposts:
https://t.iss.one/systemssharing

talks and writings:
https://t.iss.one/stochastictheoryofcommunication

cybernetic culture:
https://t.iss.one/communicationsdevelopment

books and papers:
https://t.iss.one/booksonsysapproaches

basics and elements:
https://t.iss.one/elementsofsysapproaches

Sierpiński triangle and Games of Chaos with pygame

Glider Gun with pygame

Conway's Game of Life with pygame

Maxwell–Boltzmann
distribution
with
Matplotlib

#distribution #thermodynamics #complexsystems #multiagent

Overture for the Stochastic Theory of Communication with matplotlib widgets and matplotlib animations


Description:

State of a cell is represented as RGB. Number of possible states (state space size) for each cell is limited to alphabet_size^number_of_parameters = 3^3 = 27.

alphabet_size = 3 ({0, 1, 2})
number_of_parameters = 3 (R, G, B)

Base rule (connection): 1. none of two cells in the neighborhood can have same state. 2. if no possible states remain, cell keeps its state.

#entropy #communication #synergy #selforganization #multiagent

Not-self-intersecting trajectory
with
matplotlib

#trajectory #phasespace #projection #torus #matplotlib

Inspiration

This animation draws a curve

z(θ) = exp(iθ) + exp(i𝝅θ),

where θ ∈ ℝ, z ∈ ℂ.


1 Harmonics with an irrational ratio of frequencies:

exp(iθ) and exp(i𝝅θ) are periodic functions that draw cycles on the complex plane. But since the ratio of their frequencies 𝝅θ and θ is irrational 𝝅θ / θ = 𝝅, their phases relation never repeats. By the time the first harmonic exp(iθ) completes one full turn (θ = 2𝝅), the second harmonic exp(i𝝅θ) has been rotated by 2𝝅² radians.

This irrationality (transcendentality is not necessary actually) of frequencies' ratio (namely of 𝝅 in this example) explains why the curve z(θ) from the animation is quasiperiodic and never closes.


2 Self-intersecting projection of not-self-intersecting curve:

Observing that the curve z(θ) never closes, I thought that there might be some other curve mapped in z(θ) that lives in another space of higher dimension and never intersects itself.

(I remembered the Lorenz curve that is not-self-intersecting and thought that there might be some chaotic features found in a transcendentality of 𝝅. But there are not. Actually the behaviour observed in the animation is quasiperiodic and not chaotic.)

With the help of GPT I found out that the not-self-intersecting curve I was looking for is

θ↦(exp(iθ), exp(i𝝅θ)) ∈ ℂ²
and
z(θ) = exp(iθ) + exp(i𝝅θ) is its ℂ¹ projection.

In order to somehow visualise the curve (exp(iθ), exp(i𝝅θ)) I considered a torus that it wraps around. exp(iθ) and exp(i𝝅θ) trace out circles in two orthogonal complex planes. The first circle revolves with frequency θ, while the second - with 𝝅θ. These frequencies can then be used as angles for parameterization of torus in ℝ³ and it's visualisation.

Two-Box Bit Visualization. Energy Gradient Relaxation and Entropy Increase

Discussed with chatgpt connection between information theory and thermodynamics

The animation is a microscopic parable of how energy flow, probability bias, entropy reduction, and emergent order are inseparably linked—exactly the mechanism through which complex structures arise in nature according to Prigogine’s theory of dissipative systems.

Chapter 7 Thermodynamics of Complex Systems

#energy #probability #distribution #entropy #information #thermodynamics #complexsystems

Cellular automatons are
a classic example of models
in which complex behavior
arises from very simple rules
of state transition.

• The animation above draws
one dimensional automaton
evolving in time according to
one of Wolfram's rules of
transition which can be set in
interactive mode.
• The model considers mapping
between a cell and its
neighbourhood consisting of
the cell itself, the left neighbour
of the cell and the right one.
• While a state space of one cell
is {0, 1} and the neighborhood
consists of three cells, there are 8
possible configurations of the
collective state of the
neighborhood.
• These configurations are listed
in the first row of the table on
the bottom of the animation
window.
• And the second row of the table
is filled with cell's state at next
discrete moment of time
depending on the state of cell's
neighbourhood. This string of 8
bits is a binary representation of
rule number which can be set in
the box on the left bottom of the
window.
• Since there are 8 possible
configurations of neighbourhood
state, the rule is 8-bit number,
what means that there are 256
possible rules.
• Each rule cause a distinct type
of dynamical behaviour of the
automaton.

The animation was inspired by
Chapter 2 (Simple Rules) of
this Complex Systems book

Some related scripts and other resources are stored in this repository. Run animations, explore, develop, copy, and use algorithms and their implementations.

https://codeberg.org/mucoder/cmplx-sys-simulations