add a lot of things

5 files changed, 173 insertions, 8 deletions

diff --git a/mindmap/duality.org b/mindmap/duality.org
index a47e951..cef6faa 100644
--- a/mindmap/duality.org
+++ b/mindmap/duality.org
@@ -130,9 +130,6 @@ prime can be represented by a smaller factor of that number. For example, 4-alit
 
 What's interesting is that one is a factor of everything. This represents the "null filter", or "anti filter", which doesn't filter any data and
 simply represents it all as one thing. Very interesting.
-*** P-ality, where P is Prime
-I've yet to experiment with other P-alities, but I'm sure they work too. For now, I will say that they probably work, but won't be as elegant as
-a duality, for other reasons:
-** Two is small, and Close to e
-The last thing that makes two unique is simply that it is the smallest prime number, which means thinking about the concept is relatively easy. It is
-also an approximation of the constant e, or euler's constant, which has implications in computer science and storing data as well.
+** Two is small and not One
+The number two is also the smallest natural number that is not one. This means it is the simplest way to subdivide any particular object. This makes
+it more elegant compared to some other modalities.
diff --git a/mindmap/electrostatics.org b/mindmap/electrostatics.org
new file mode 100644
index 0000000..e71a05e
--- /dev/null
+++ b/mindmap/electrostatics.org
@@ -0,0 +1,97 @@
+:PROPERTIES:
+:ID:       32f0b8b1-17bc-4c91-a824-2f2a3bbbdbd1
+:END:
+#+title: electrostatics
+#+author: Preston Pan
+#+html_head: <link rel="stylesheet" type="text/css" href="../style.css" />
+#+html_head: <script src="https://polyfill.io/v3/polyfill.min.js?features=es6"></script>
+#+html_head: <script id="MathJax-script" async src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-mml-chtml.js"></script>
+#+options: broken-links:t
+
+* What is Electricity?
+Because this is an introduction and not a lesson in quantum mechanics, I will say that electricity is broadly
+defined by the charge that an object has that corresponds the force that it both feels and also gives
+to other objects. Charge is measured in coulombs and can be negative or positive, which leads us to the man himself:
+
+* Coulomb's Law
+In order to define the phenomena of electric force in the real world, we use
+this experimentally verified law known as Coulomb's Law. Let \( \vec{r_{1}} \) be the displacement
+of a charge \( Q \), and let \( \vec{r_{2}} \) be the displacement of a charge \( q \), where these two charges are named \( P_{1}\) and \( P_{2} \) respectively.
+Then let \( \vec{r} = \vec{r_{1}} - \vec{r_{2}} \) be the distance between the charges. For simplicity, we assume
+these charges have no mass or volume; we call these /point charges/. The equation for the force
+on \( P_{1} \) is as follows:
+
+\begin{align*}
+\vec{F(\vec{r})} = \frac{1}{4\pi\epsilon_{0}}\frac{qQ}{r^{2}} \hat{r}.
+\end{align*}
+
+Where \( \hat{r} \) is the unit vector pointing in the direction of \( P_{2} \). Note that there are a
+couple of interesting things about this force. First, it is an inverse square law, and the formula looks a lot like the one for gravitation,
+only charge can be negative and mass cannot. Second, it is symmetrical,
+in the sense that the force felt by \( P_{2} \) is going to be the same, only \( \hat{r} \)
+is pointing in the other direction. Also, note that due to linearity, this force calculation follows the /superposition principle/.
+That is, if we have different electrostatic forces acting on one particle:
+
+\begin{align*}
+\vec{F_{tot}} = \vec{F_{1}} + \vec{F_{2}} + … = \sum_{i=1}^{n} \vec{F_{i}}.
+\end{align*}
+
+Wait, where does the \( \frac{1}{4\pi\epsilon_{0}} \) term come from? Well, the surface area of a sphere
+is \( 4\pi r^{2}\) , which explains both the inverse square law and this \( 4\pi \) term in the denominator,
+but what about \( \epsilon_{0} \), what does it even mean?
+
+Well, it is simply a conversion of units from /speed of light/ terms to /SI unit terms/. If you
+think of it like that, you will never need to know what the units actually are, although I'm
+sure you can find that online. Just know that it is called the permeability of free space, and
+it is defined in terms of the speed of light and a constant relating to magnetism:
+
+\begin{align*}
+\epsilon_{0}\mu_{0} = \frac{1}{c^{2}}
+\end{align*}
+
+but since this is electrostatics and not electrodynamics, you will not have to worry about
+magnetic constants. Again, it is just a shift from speed of light units to our mortal units.
+
+** Electric Field
+Okay, now we can continue to defining the /electric field/ of a particle. Let's call \( P_{1} \) our
+/test charge/, and \( P_{2} \) our /source charge/. If we now want to measure the force on \( P_{1} \),
+our equation is going to be the same. However, we can define a field \( \vec{E(\vec{r})} \) such that:
+
+\begin{align*}
+\vec{F} = Q\vec{E}
+\end{align*}
+
+Where:
+
+\begin{align*}
+\vec{E} = \frac{\vec{F}}{Q}
+\end{align*}
+
+Therefore, the value of \( \vec{E} \) for a point charge must be:
+
+\begin{align*}
+\vec{E} := \frac{1}{4\pi\epsilon_{0}}\frac{q}{r^{2}}\hat{r}.
+\end{align*}
+
+The result is we find a way to express force in a /test charge independent way/. This is useful
+because we often want to find the force if an arbitrary object with an arbitrary charge is next
+to the particle in question, instead of focusing specifically on two charges.
+
+Note that it is trivial to prove that \( \vec{E} \) also follows the superposition principle.
+
+
+** Continuous Charge Distributions
+Now that we have a working definition of \( \vec{E} \), we can now find the electric field of an object
+that has a continuous charge distribution. Note that there aren't actually infinite charges in real
+world objects which is what we are assuming by taking an integral over some space of charge, but
+it's close enough because there are so many individual charges in real world objects. Assuming we are in
+three dimensions:
+
+\begin{align*}
+\vec{E(\vec{r})} = \frac{1}{4\pi\epsilon_{0}} \int_{space} \frac{\sigma(\vec{r_{2}})}{r^{2}}\hat{r}d\tau
+\end{align*}
+
+Where \( \tau \) is the patch of volume we are integrating over, and \( \sigma \) is the charge density function,
+which takes a position vector and returns the charge at that vector. Of course, surface and line integrals
+have their own analogues -- simply replace \( d\tau \) with \( da \) or \( dl \), and make sure your charge
+distribution is in the correct amount of dimensions.
diff --git a/mindmap/emergence.org b/mindmap/emergence.org
index 9a146cf..9c975a5 100644
--- a/mindmap/emergence.org
+++ b/mindmap/emergence.org
@@ -6,4 +6,27 @@
 #+html_head: <link rel="stylesheet" type="text/css" href="style.css" />
 
 * Emergence systems are…
-Many things.
+Existent in many forms and at many levels. They are the fundamental building
+block of everything that exists, I argue, and nothing in this theory of
+everything will contradict this statement!
+
+** What is Emergence?
+Emergence can be broadly defined as a system whose components organize themselves
+into a larger structure via simpler rules that each component follows.
+
+Entire systems such as [[id:a6bc601a-7910-44bb-afd5-dffa5bc869b1][mathematics]] can be explained in terms of a couple axioms,
+where all the theorems arising from those axioms are emergent from those axioms.
+At least, that is a relatively simple explanation, and does not capture the full
+beauty of emergent systems. Therefore, I call apoun an example from daily life:
+
+*** Societies
+At every scale, societies exhibit properties of emergence. For example, families
+and small communities comprise small areas of town, which comprise cities,
+which then make up provinces, and finally countries. In this particular example,
+we self organize into self-similar [[id:8f265f93-e5fd-4150-a845-a60ab7063164][recursive]] hierarchies. This is for a good reason;
+in order for societies to scale, there need to be abstractions. Each level in the
+hierarchy conveys more but less exact information, until we get to the national
+level which deals the most with aggregates.
+
+*** Markets Emergent
+To use a particular hierarchy example, markets are emergent from barter in goods.
diff --git a/mindmap/mathematics.org b/mindmap/mathematics.org
index 55c3dc5..d6d7065 100644
--- a/mindmap/mathematics.org
+++ b/mindmap/mathematics.org
@@ -12,4 +12,16 @@ With a couple of set theory axioms. One might
 describe it as an extrapolation framework
 without grounding (i.e. a set of implications;
 if p then q, but never specifying if p is a property
-of a real system or not).
+of a real system or not). Therefore, mathematics
+is suitable for modeling other things if we believe
+those other things have some rules and are logically
+consistent.
+
+There are many subfields in math, ranging from group theory
+to complex analysis. However, much of the time we are able to find
+morphisms between these different fields in mathematics, which we
+model using category theory.
+
+** Mathematical Models
+*** [[id:ece8bf94-4e3c-4939-a77a-9949c1ec0dc6][Physics]]
+This is a stub.
diff --git a/mindmap/physics.org b/mindmap/physics.org
new file mode 100644
index 0000000..b70f829
--- /dev/null
+++ b/mindmap/physics.org
@@ -0,0 +1,36 @@
+:PROPERTIES:
+:ID:       ece8bf94-4e3c-4939-a77a-9949c1ec0dc6
+:END:
+#+title: physics
+#+author: Preston Pan
+#+html_head: <link rel="stylesheet" type="text/css" href="../style.css" />
+#+html_head: <script src="https://polyfill.io/v3/polyfill.min.js?features=es6"></script>
+#+html_head: <script id="MathJax-script" async src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-mml-chtml.js"></script>
+#+options: broken-links:t
+
+* Laws of Nature
+By engaging in physics modeling, we are postulating that there are rules that
+"nature" follows, and that these rules are logically consistent. At least,
+mathematical models postulate this, and non-mathematical models of the universe
+have been relegated to the study of metaphysics.
+
+* Fields in Physics
+** Classical Mechanics
+Classical mechanics deals with fields of physics that do not deal with time dilation or quantum
+weirdness. It is called classical mechanics because most of these theories were invented before
+their quantum or relativistic counterparts, and usually serve as a baseline understanding
+for the later topics.
+*** Newtonian Mechanics
+It all started when Newton created three (but really two) fundamental laws of the universe that
+governed all things.
+*** Lagrangian Mechanics
+This is a different strain of classical mechanics. Instead of looking at direction and vectors,
+it looks at a single fundamental principle, even more fundamental than inertia and conservation of momentum:
+optimization. We believe that nature always optimizes for some parameters in all physical phenomena.
+*** Classical Electrodynamics
+Classical electrodynamics attempts to explain the electromagnetic force from a classical perspective. What is
+light? How does electricity actually work? All these questions you will find (half) answered in the study
+of electrodynamics.
+**** [[id:32f0b8b1-17bc-4c91-a824-2f2a3bbbdbd1][Electrostatics]]
+This is a study of the force of particles with charge on another in a motionless context. It is the baseline
+for studying more complicated electromagnetic theory.