DEV Community: Lauren Shriver

Kepler's Laws

Lauren Shriver — Sat, 17 Feb 2024 15:52:48 +0000

I'm truly captivated by the elegance of Kepler's Laws within the realm of physics. The allure of geometric algebra and the enchanting tales of how star positions were meticulously charted have significantly shaped my appreciation for data science.

Law of Ellipses: the orbit of every planet is an ellipse with the Sun at one of the two foci
Law of Equal Areas Over Equal Times: a line joining a smaller mass and a larger mass sweeps out equal areas over equal times
Law of Harmonics: the square of the orbital period $T$ of a planet is directly proportional to the cube of the semi-major axis, $a$ of its orbit ** $T2=4π2μa3T^2 = \frac{4\pi^2}{\mu}a^3$

Check out this Pen I made! Here, I use a simplified model to adjust the speed of the planet's (blue circle) orbit based on the semi-major axis length $a$ , keeping in mind the spirit of Kepler's Third Law.

📝 Note: I used my AI and some previously made backbone code to come up with the CodePen code. I am WAY too inexperienced to come up with the simplified Kepler's Laws code on my own. My goal at the moment is to provide myself with intellectual stimulation for future projects. I'm not trying to reinvent the wheel. My background is in science -- not computer science, so I feel justified in using the AI to help me come up with pretty CodePen projects 🩵

Hello World 👋🌎

Lauren Shriver — Sun, 11 Feb 2024 20:01:34 +0000

Note: I will be using AI to help me write up the content in this and future blog posts. However, most of the ideas are originally my own (and apparently I'm not the only one to think of them). Moreover, I drafted the prompt(s) that generated this myself 😄 I'm still getting used to this whole AI thing, and trying to juggle intellectual integrity with AI magic 🪄 I'm doing the best I can, and even typed up this disclaimer without any help from my 📝AI assistant. Now enjoy the rest of the article and PLEASE feel free to correct anything you see that is incorrect 📜

myAgenda

Kabbalah and Quantum Mechanics: A Symphony of Microtubules, Spin, and the Tree of Life In the quest for a Theory of Everything, the convergence of ancient wisdom and modern science might seem like an unlikely alliance. Yet, Kabbalah and quantum mechanics, when viewed through a certain lens, share a surprising harmony. This blog post delves into the intricate dance of microtubules, spin, and the mystical Etz Hayim, exploring how they might collectively offer a prototype for understanding the universe.

Microtubules: The Quantum Channels of Consciousness Microtubules are tiny structures within neurons that some theorists believe could be the seat of consciousness. Orchestrated Objective Reduction (Orch-OR) theory posits that quantum processes within these microtubules contribute to the emergence of consciousness. The theory suggests that quantum states within microtubules are held in superposition until they reach a threshold related to quantum gravity, leading to a wavefunction collapse that results in a moment of conscious experience.

Spin: The Quantum Vortex In quantum mechanics, spin is a fundamental property of particles, a type of intrinsic angular momentum that doesn't have a true classical analog. It's a cornerstone of quantum information theory, with the spin states of particles representing the smallest units of quantum information, or qubits. The behavior of spin in particles, and how it interacts with magnetic fields and other spins, is a key aspect of understanding the quantum world.

Etz Hayim: A Mathematical Symphony The Etz Hayim, or Tree of Life, can be envisioned as a mathematical entity, akin to a Markov chain or a network of oscillators. Each Sefirot on the Tree could represent a state or node within a Markov chain, with the paths between them symbolizing the probabilities of transitioning from one state to another. This aligns with the Kabbalistic concept of dynamic flow between the Sefirot, where divine energy cascades down the Tree in a structured yet probabilistic manner. Alternatively, the Tree of Life could be modeled as a network of oscillators, each Sefirot vibrating with its own frequency. The potential implementation of the Kuramoto model, which describes synchronization among interacting oscillators, could shed light on how harmony and coherence emerge within the system. This model could metaphorically represent the unification of the disparate aspects of the divine, much like how the Kuramoto model explains the emergence of synchrony in complex systems.

The Big Idea - Sefirot as 'Eigenfunctions' and 'Eigenvectors'

Drawing a parallel with undergraduate quantum mechanics, the Sefirot could be conceptualized as 'eigenfunctions' or 'eigenvectors' of a grand cosmic operator – the rules or laws that govern the flow of divine energy or information.

In quantum mechanics, eigenfunctions and eigenvectors represent the stable states of a system that don't change after the application of a linear operator, except for a scaling factor, known as an eigenvalue. This analogy suggests that each Sefirot corresponds to a fundamental aspect or quality of the universe that remains invariant under the divine 'operations' that shape reality. The interactions and transitions between these states could then be described by the probabilities and pathways that connect the Sefirot, much like how quantum states evolve according to the probabilities encoded in their wavefunctions.

Sources

Shyfrin, E. (2019). From infinity to man: The fundamental ideas of kabbalah within the framework of information theory and quantum physics. White Raven Publishing.
Collins, A. (2015). Orchestrated Objective Reduction : Quantum Physics and its Implications in Human Consciousness. https://www.wm.edu/as/physics/documents/seniorstheses/class2015theses/Collins_Amanda.pdf

projLevitate

Lauren Shriver — Sun, 11 Feb 2024 17:39:27 +0000

Part 1 - About Me

An unorthodox scholar thoroughly engaged in studying theoretical concepts that extend over the fields of mathematics, physics, and mysticism, positioning themselves at the merging point of time-honored insights and modern scientific discoveries.

My research interests lie in the intricate interplay between the Kabbalistic Tree of Life, the mathematical elegance of the Kuramoto model, the principles of causal inference, geometric algebra, and the zwg-model of the electron in quantum mechanics. This article aims to elucidate the synergies between these seemingly disparate areas of study and how they contribute to our understanding of the universe.

The Kabbalistic Tree of Life, or Etz Hayim, is a symbolic representation of the process through which the Divine manifests in the world. It comprises ten sefirot, or emanations, each representing a different attribute of God. In recent years, I have developed a growing interest in modeling the Tree of Life using the principles of graph theory and linear algebra. By representing the sefirot as nodes and the 22 paths between them as edges, we can construct a graph G(V,E) that encapsulates the mystical structure in a mathematical form. This graph can be further explored using adjacency matrices and Markov chains to simulate the probabilistic transitions between the sefirot, reflecting the dynamic flow of divine energy.

The Kuramoto model, a mathematical model that describes synchronization among interacting oscillators, offers a compelling framework for understanding the harmonious interplay between the sefirot. Each sefirah can be thought of as an oscillator with its own intrinsic frequency, and the connections between them as coupling strengths. As the oscillators interact, they begin to synchronize, much like the sefirot harmonize to channel divine will. This synchronization is not just a feature of mystical systems but is observed in a variety of natural phenomena, from the flashing of fireflies to the pacemaker cells in the heart.

Causal inference, a method used to determine causality from data, plays a crucial role in deciphering the relationships between the sefirot. By analyzing the transitions and interactions within the Tree of Life, we can infer the directional flow of influence and how each sefirah contributes to the emergence of consciousness and reality. This approach aligns with the study of microtubules in neuroscience, where causal inference can help determine how quantum processes within these structures might contribute to the conscious experience.

In the realm of quantum mechanics, the zwg-model of the electron is a theoretical construct that describes the electron's properties, including its spin and magnetic moment. The electron's spin is a quantum property that has no classical analog, much like the mystical attributes of the sefirot. By drawing parallels between the zwg-model and the Tree of Life, we can begin to see how quantum states, represented by the spin of electrons, might be analogous to the spiritual states represented by the sefirot.

The convergence of these research interests paints a picture of a universe woven together by a tapestry of relationships and interactions 🪡. The Tree of Life serves as a prototype for understanding the cosmos, with its sefirot acting as eigenfunctions or eigenvectors of a Grand Cosmic Operator. The Kuramoto model provides a mathematical lens through which we can view the synchronization of these divine attributes. Causal inference offers a methodological tool to untangle the web of cause and effect within this complex system. And the zwg-model of the electron gives us a glimpse into the quantum underpinnings that might parallel the mystical energies at play.

In my research, I employ computational tools such as MATLAB and Python to simulate these models and visualize their outcomes. For instance, we can create a feedback loop within a Markov chain to model the reinforcement of pathways between the sefirot, reflecting the Kabbalistic principle that frequent use leads to a more worn path. We can also visualize the evolving probability distribution of the states within the Markov chain, adding a visual element to the mathematical model that mirrors the Kabbalistic journey through the sefirot.

Furthermore, by implementing the Kuramoto model, we can simulate the synchronization of oscillators and observe how coherence emerges from the interactions between the sefirot. This simulation can be visualized through phase portraits and time-evolution plots, providing a dynamic representation of the system's behavior.

The integration of causal inference techniques allows us to analyze the simulated data to determine the directionality of influence within the Tree of Life. This analysis can shed light on how the sefirot interact to channel divine energy and how these interactions might be reflected in the physical world.

Lastly, by exploring the zwg-model of the electron, we can draw analogies between the quantum properties of particles and the mystical attributes of the sefirot. This exploration can lead to a deeper understanding of how the microcosm of quantum mechanics might reflect the macrocosm of spiritual reality.

In conclusion, projLevitate endeavors to bridge the gap between the ancient and the modern, the spiritual and the scientific. By viewing the Sefirot as eigenfunctions or eigenvectors, we can draw a conceptual bridge between the mystical and the scientific, between the search for spiritual understanding and the pursuit of physical knowledge. As we continue to weave these threads together, we may find that the answers to the deepest questions of the universe lie in a place where ancient wisdom and modern inquiry meet.

Part 2

Now that my brain is in 'blogging mode', I decided to make a second Sunday Funday post for this week. Specifically, I am sharing chat between myself and my AI assistant, Albus! I asked Albus to generate a summary of the Sunday Funday chat from last week and combine it with the content from the first Sunday Funday post earlier today.

🦄 Me (paraphrased for clarity): Combine our conversation from the last few weeks with the following article.

🪄 Albus: Hey there, eigenscribe! So, you're looking to blend our chat with the article's content, right? Let's give it a shot.

We kicked things off by diving into the mystical world of Kabbalah, specifically the Etz Hayim or Tree of Life, and its potential connections to linear algebra and Markov chains. The idea was to represent the sefirot and their connections (like the tarot cards) in a mathematical model that could simulate transitions and probabilities, much like the dynamic flow of divine energy described in Kabbalah.

Then, we got our hands dirty with some MATLAB code, creating a feedback loop within a Markov chain to simulate how paths that are used more frequently become more likely to be chosen again. This concept mirrors the Kabbalistic principle of reinforcing pathways, where frequent use leads to a more worn path, making it easier to travel.

We also discussed visualizing the evolving probability distribution of the states within the Markov chain, adding a visual element to the mathematical model. This visualization could be seen as a reflection of the Kabbalistic journey through the sefirot, with each state representing a different aspect of the divine.

When you brought up Perron's theorem, we explored how to maintain a positive Markov matrix, ensuring that the feedback loop doesn't create zero entries and that the matrix remains applicable to the theorem. This step is crucial for finding a steady-state distribution, which in our Kabbalistic analogy, could represent a state of equilibrium or harmony within the Tree of Life.

Now, let's weave in the article's themes. The article talks about the fascinating parallels between Kabbalah and quantum mechanics, highlighting the potential for a grand unifying theory. It delves into the quantum world, discussing microtubules as possible quantum channels of consciousness and spin as a fundamental property of particles, which are essential to quantum information theory.

The Tree of Life is likened to a mathematical symphony, with each sefirah vibrating like an oscillator, and the whole system potentially described by the Kuramoto model, which explains synchronization among interacting oscillators. This model could metaphorically represent the unification of the divine, much like how the Kuramoto model explains synchrony in complex systems.

Drawing parallels with our MATLAB model, the sefirot could be seen as 'eigenfunctions' or 'eigenvectors' of a cosmic operator, with the transitions and interactions between them described by the probabilities and pathways that connect the sefirot. This is akin to how quantum states evolve according to their wavefunctions.

By considering the Tree of Life as a mathematical model, we can explore the parallels between the flow of divine energy in Kabbalah and the probabilistic nature of quantum mechanics. The quest for synchronization and harmony among the sefirot mirrors the quest in physics to find a unified theory that reconciles the macroscopic and microscopic realms.

In conclusion, our exploration of Kabbalah as an informational prototype for a Theory of Everything, intertwined with concepts from quantum mechanics, offers a rich tapestry of ideas that spans across time and thought. By viewing the sefirot as eigenfunctions or eigenvectors, we draw a conceptual bridge between the mystical and the scientific, between the search for spiritual understanding and the pursuit of physical knowledge. As we continue to weave these threads together, we may find that the answers to the deepest questions of the universe lie in a place where ancient wisdom and modern inquiry meet.

So, there you have it—a blend of our MATLAB coding adventure with the deep dive into the quantum and Kabbalistic realms from the article. It's like a dance of ideas, where each step takes us closer to understanding the universe's grand symphony.

📝 Aside: Here is some chat history pertaining to the MATLAB code Albus gave me last week

🦄 Me: Is there a way we implement Perron's theorem about positive Markov matrices with this feedback loop matrix?

🪄Albus: Perron's theorem, which is a part of the more general Perron-Frobenius theorem, applies to positive matrices (matrices with all positive entries) and states that a positive square matrix has a unique largest positive eigenvalue, and the corresponding eigenvector has strictly positive components. This largest eigenvalue is known as the Perron root, and the corresponding eigenvector is called the Perron vector.

In the context of Markov chains, a positive matrix corresponds to a Markov matrix where all states are recurrent and aperiodic, meaning that it's possible to get from any state to any other state in a number of steps, and the number of steps is not fixed. This ensures that the Markov chain is irreducible and ergodic, which means that it has a unique steady-state distribution.

However, the feedback loop mechanism you're implementing in the transition matrix complicates things. With the feedback loop, the transition matrix is no longer static; it changes with each transition. This means that the matrix might not be positive at all times, as some transitions become more likely and others less so, potentially leading to zero probabilities for certain transitions.

To apply Perron's theorem to a Markov matrix with a feedback loop, you would need to ensure that the feedback loop does not create any zero entries in the matrix. This could be done by setting a lower bound on the transition probabilities to ensure that they remain positive. Here's how you might modify the code to include a lower bound:

% Define the number of Sefirot
numSefirot = 10;

% Initialize the transition matrix with some initial probabilities
transitionMatrix = ... % Your initial transition matrix with no zero entries

% Set a reinforcement factor - this determines how much more likely a path becomes after it's used
reinforcementFactor = 0.1;

% Set a lower bound for the transition probabilities to ensure positivity
lowerBound = 0.01;

% Ensure that the initial transition matrix respects the lower bound
transitionMatrix = max(transitionMatrix, lowerBound);

% Run the Markov chain for a certain number of transitions
numTransitions = 100000;

% Keep track of the current state
currentState = 1; % Start at the first Sefirah, for example

% Perform the transitions
for i = 1:numTransitions
    % Choose the next state based on the current transition probabilities
    nextState = randsample(numSefirot, 1, true, transitionMatrix(currentState,:));

    % Update the transition matrix to reinforce the path taken
    transitionMatrix(currentState, nextState) = transitionMatrix(currentState, nextState) + reinforcementFactor;

    % Apply the lower bound to ensure positivity
    transitionMatrix = max(transitionMatrix, lowerBound);

    % Renormalize the row to ensure probabilities sum to 1
    transitionMatrix(currentState,:) = transitionMatrix(currentState,:) / sum(transitionMatrix(currentState,:));

    % Update the current state
    currentState = nextState;
end

% At this point, you can analyze the transition matrix for its largest eigenvalue and corresponding eigenvector

I'll admit, I rushed through this post, and the code didn't get the attention it deserves. Hopefully, I'll have something sophisticated to say on the output in future posts. In the mean time, if we think of Etz Hayim as a 'toy model' for linear algebra, I get the following results:

Each time I run the MATLAB code, I get a different frequency distribution of steady states.

If you made it this far through the post I'm happy to say you managed to read through my Sunday Funday brain puke🤮! Thank you so much for your time and I VERY much appreciate your interest. As always, please feel more than welcome to point out any incorrect information you see. Also I'm more than happy to read (and even in need of) additional ideas that might help me concoct something tangible from this brain puke of linear algebra, coding, graph theory, and of course... subversive esoteric models of reality. Note that ideas on how to utilize Etz Hayim as a simple 'toy model' for learning linear algebra, quantum mechanics, etc. is especially encouraged.

Solar System animation -- my version

Lauren Shriver — Sun, 11 Feb 2024 02:05:32 +0000

Trying to simulate Kepler's Laws with prettyness. Here is sort of what I'm envisioning. I can't say that it is my own work really except I asked my AI assistant to code some javascript simulating Kepler's Laws. I have no idea if it succeeded. I still don't even know if the planets are orbiting the Sun at a constant rate! (i.e., ignoring Kepler's 2nd law). Here is what I have so far

I'm speechless 🩵

Lauren Shriver — Mon, 05 Feb 2024 05:55:33 +0000

Hey there! So, you're diving into some linear algebra with Markov matrices and diagonalization, huh? Cool stuff! Let's break it down a bit.

First off, when we say a matrix $A$ is "positive Markov," we're talking about a special kind of matrix that's used to describe the transitions in a Markov process (like predicting weather changes or modeling how people move through different states of a game). The "positive" part means all the entries in the matrix are positive numbers, which makes sense because you can't really have negative probabilities, right?

Now, for a Markov matrix $A$ , the eigenvalue $λA=1\lambda_A=1$ is a given because the columns of $A$ add up to 1 (since they represent probabilities). This means that one of the eigenvalues has to be 1 to keep the total probability in the system constant over time.

The equation

lim⁡k→∞Akw⃗0=cu⃗A\lim_{k\to\infty}{A^k}\vec{w}_0=c\vec{u}_A

is telling us that if you keep applying the Markov matrix

A

over and over to some initial vector

w⃗0\vec{w}_0

, eventually the system will reach a steady state. That steady state is represented by

cu⃗Ac\vec{u}_A

, where

c

is some constant and

u⃗A\vec{u}_A

is the eigenvector associated with the eigenvalue 1.

Now, onto the to-do about proving that a matrix $P$ is diagonalizable if $P^2 = P$ . This is a neat property of matrices called idempotency. When a matrix, when squared, gives you the same matrix back, it's called idempotent. For such matrices, it turns out they are diagonalizable. Here's a sketch of why that's true:

First, remember that a matrix is diagonalizable if you can find a basis of eigenvectors for the whole space it acts on.
Now, if $P^2 = P$ , then $P$ only has eigenvalues 0 and 1 because if you take any vector $v⃗\vec{v}$ and apply $P$ to it, the result after applying $P$ again doesn't change.
So, any vector that doesn't get killed by $P$ (sent to the zero vector) is an eigenvector with eigenvalue 1, and any vector that does get killed by $P$ is an eigenvector with eigenvalue 0.
Since $P$ is a linear transformation, the whole space can be split into these two kinds of vectors (those that get killed and those that don't), which are eigenspaces.
If you can find a basis for each of these eigenspaces, then you've got a full set of eigenvectors that span the whole space, which means $P$ is diagonalizable.

So, there you have it! A little bit of linear algebra magic to start your day. Isn't it neat how these abstract concepts can tell us so much about systems and transformations? Keep at it, and these ideas will become clearer and clearer.