The Portal Wiki

Decoding the Graph-Wall-Tome Connection: Difference between revisions

Page Discussion

Decoding the Graph-Wall-Tome Connection (view source)

Revision as of 01:11, 12 April 2023

9,616 bytes added ,  12 April 2023
m
Protected "Decoding the Graph-Wall-Tome Connection" ([Edit=Allow only autoconfirmed users] (indefinite) [Move=Allow only autoconfirmed users] (indefinite))
No edit summary
m (Protected "Decoding the Graph-Wall-Tome Connection" ([Edit=Allow only autoconfirmed users] (indefinite) [Move=Allow only autoconfirmed users] (indefinite)))
 
(49 intermediate revisions by 6 users not shown)
Line 1: Line 1:
{{InfoboxProject
|project=Graph-Wall-Tome Connection
|image=[[File:Ascending and Descending.jpg]]
|topic=[[Graph, Wall, Tome]]
|leader=Aardvark
|startdate=21 April 2020
|customlabel1=New Graph-Wall Mind Map
|customdata1=[https://drive.google.com/file/d/1ZMT8nDyF6qT5nh6hiCjrHyX4giKKmGSA/view?usp=sharing Link]
|customlabel2=Graph Mind Map
|customdata2=[https://drive.google.com/file/d/16r60-56mhYCx4KKBvZv_6HDLOLw43X7y/view?usp=sharing Link]
|customlabel3=Google Drive
|customdata3=[https://drive.google.com/drive/folders/1706CBEJQEMppV60OU8OtcXxicluk2T3Y?usp=sharing Drive]
|customlabel4=Master Planning
|customdata4=[https://docs.google.com/document/d/1t9AvvFZzODw1WiGRZwRsFFZdPdBzYVJGLHiqWNrMtIA/edit?usp=sharing Doc]
|link1title=Website
|link1=[https://graphwalltome.com/ Homepage]
|link2title=
|link2=
|link3title=
|link3=
|link4title=Discord
|link4=[https://discord.gg/Z3u3pPm Invite]
}}


An important aspect of the [[Graph,_Wall,_Tome#Eric_Weinstein.27s_Prompt|promt]] is that neither the Graph, nor the Wall or the Tome are that important.  
An important aspect of the [[Graph, Wall, Tome#Prime Directive|prompt]] is that neither the [[Graph, Wall, Tome#Graph|Graph]], nor the [[Graph, Wall, Tome#Wall|Wall]] or the [[Graph, Wall, Tome#Tome|Tome]] are that important on their own. Only together does their explanatory power exceed the limits of any part alone. The common threads that run through all of them are what matter: what is common among them is what should be expanded, and what is particular to one should be promoted and developed if it is a part of fundamental physics toolkit, or removed if it isn't.


''What really matters are the common threads that run through all of them. ''
The goals of this project are to:
* Identify the common threads (the "unifying idea") in the Graph, Wall, and Tome. These will indicate how to [[Editing the Graph|Edit the Graph]] and [[Defacing the Wall|Deface the Wall]].
* Create and collect resources that make it easy to understand them, thus contributing to [[Rewriting the Tome]].  


The goals of this project are to:
== Guiding Questions and Comments by Eric Weinstein ==
1. Identify the common threads in the Graph, Wall, and Tome.
<div class="toccolours mw-collapsible mw-collapsed" style="width:400px; overflow:auto;">
2. Create and collect resources that make it easy to understand them.
<div style="font-weight:bold;line-height:1.6;">What is <math>F_A</math> geometrically?</div>
<div class="mw-collapsible-content">
<math>F_A</math> is the curvature tensor associated with the connection or vector potential <math>A</math>.
</div></div>


== Guiding Questions by Eric Weinstein ==


<div class="toccolours mw-collapsible mw-collapsed" style="width:400px; overflow:auto;">
<div class="toccolours mw-collapsible mw-collapsed" style="width:400px; overflow:auto;">
<div style="font-weight:bold;line-height:1.6;">What is $$F_A$$ geometrically?</div>
<div style="font-weight:bold;line-height:1.6;">What are <math>R_{\mu v}</math> and <math>R</math> geometrically?</div>
<div class="mw-collapsible-content">
<div class="mw-collapsible-content">
$$F_A$$ is the curvature tensor associated to the connection or vector potential $$A$$.
[https://www.youtube.com/watch?v=UfThVvBWZxM&t=12m6s Explanation of <math>R</math>]
 
<math>R</math> is a scalar value, describing the "curvature of the spacetime manifold" at each point along the manifold. It's based on a concept of 'parallel transport', where you move a vector around some path on the manifold.
 
<math>R</math> can be computed at each point on the manifold, and describes the difference in the vector's angle after following an infinitesimally small path around the neighborhood of that point, vs. what it was originally. The video does a great job of visualizing when and why that vector angle change would happen, with flat vs. curved manifolds.
 
In the video, they focus first on the curvature of space. Hopefully they incorporate back in curvature in time, because that's less obvious.
 
The same video then proceeds to explain <math>R_{\mu v}</math>. It progresses through some concepts.
</div></div>
</div></div>




<div class="toccolours mw-collapsible mw-collapsed" style="width:400px; overflow:auto;">
<div class="toccolours mw-collapsible mw-collapsed" style="width:400px; overflow:auto;">
<div style="font-weight:bold;line-height:1.6;">What are $$R_{\mu v}$$ and $$R$$ geometrically?</div>
<div style="font-weight:bold;line-height:1.6;">Further thoughts on the meaning of R</div>
<div class="mw-collapsible-content">
<div class="mw-collapsible-content">
Computing length in non-orthogonal bases
First, just describing the length of a vector on a curved space is hard. It is given by:
<math>Length^{squared} = g_{11}dX^{1}dX^{1} + g_{12}dX^{1}dX^{2} + g_{21}dX^{2}dX^{1} + g_{22}dX^{2}dX^{2}</math>
Some notes:
* This is not Pythagorean theorem, because <math>dX^{1}</math> and <math>dX^{2}</math> are not perpendicular.
* Instead, looks like a formula to get the diagonal from two opposite vertices in a parallelogram.
* If <math>dX^{1}</math> and <math>dX^{2}</math> are perpendicular, then <math>g_{12}</math> and <math>g_{21}</math> would be 0, and we would get <math>Length^{squared} = g_{11}(dX^{1})^{2} + g_{22}(dX^{2})^{2}</math>
* See: [https://www.youtube.com/watch?v=UfThVvBWZxM&t=14m27s the video @ 14m27s]
Computing vector rotation due to parallel transport
Then, they show parallel transport when following a parallelogram, but over a curved 3D manifold. To compute the vector rotation by components, they show:
<math>dV^{1} = dX^{1}dX^{2} (V^{1}R^{1}_{112} + V^{2}R^{1}_{212} + V^{3}R^{1}_{312})</math>
<math>dV^{2} = dX^{1}dX^{2} (V^{1}R^{2}_{112} + V^{2}R^{2}_{212} + V^{3}R^{2}_{312})</math>
<math>dV^{3} = dX^{1}dX^{2} (V^{1}R^{3}_{112} + V^{2}R^{3}_{212} + V^{3}R^{3}_{312})</math>
or, using <math>i</math> to summarize across all 3 components (difference vectors):
<math>dV^{i} = dX^{1}dX^{2} (V^{1}R^{i}_{112} + V^{2}R^{i}_{212} + V^{3}R^{i}_{312})</math>
or , using <math>j</math> to index over all 3 components (original vector):
<math>dV^{i} = dX^{1}dX^{2} \Sigma_{j} [(V^{j}R^{i}_{j12}]</math>
See: [https://www.youtube.com/watch?v=UfThVvBWZxM&t=19m33s the video @ 19m33s]
Open questions:
* Why a parallelogram?
* How to properly overlay the parallelogram onto the 3d manifold, in order to do the parallel transport?
* How does this relate to the length computation above?
Putting it all together
Now, moving to 4D, we can compute <math>R_{\mu v}</math> as:
<math>R_{00} = R^{0}_{000} + R^{1}_{010} + R^{2}_{020} + R^{3}_{030}</math>
<math>R_{10} = R^{0}_{100} + R^{1}_{110} + R^{2}_{120} + R^{3}_{130}</math>
<math>R_{01} = R^{0}_{001} + R^{1}_{011} + R^{2}_{021} + R^{3}_{030}</math>
etc.
Indexing i over all 4 component vectors / dimensions, we get:
<math>R_{00} = \Sigma_{i} R^{i}_{0i0}</math>
<math>R_{10} = \Sigma_{i} R^{i}_{1i0}</math>


[https://www.youtube.com/watch?v=UfThVvBWZxM&t=12m6s Explanation of $$R$$]
<math>R_{01} = \Sigma_{i} R^{i}_{0i1}</math>


$$R$$ is a scalar value, describing the "curvature of the spacetime manifold" at each point along the manifold. It's based on a concept of 'parallel transport', where you move a vector around some path on the manifold.  
etc.


$$R$$ can be computed at each point on the manifold, and describes the difference in the vector's angle after following an infinitesimally small path around the neighborhood of that point, vs. what it was originally. The video does a great job of visualizing when and why that vector angle change would happen, with flat vs. curved manifolds.
Summarizing on <math>\mu</math>, we get:


In the video, they focus first on the curvature of space. Hopefully they incorporate back in curvature in time, because that's less obvious.
<math>R_{\mu 0} = \Sigma_{i} R^{i}_{\mu i0}</math>
 
<math>R_{\mu 1} = \Sigma_{i} R^{i}_{\mu i1}</math>
 
etc
 
Summarizing on <math>v</math>, we get:
 
<math>R_{\mu v} = \Sigma_{i} R^{i}_{\mu iv}</math>


The same video then proceeds to explain $$R_{\mu v}$$. It progresses through some concepts.


[[Further thoughts on the meaning of R]]
Open questions:


* If we hadn't moved from 3D to 4D, what would this all have looked like?
* What does this have to do with the parallelogram?
* Why are there two indices?
</div></div>
</div></div>


Line 68: Line 171:
</blockquote>
</blockquote>
</div></div>
</div></div>
<blockquote>"The source code of the universe is overwhelmingly likely to determine a purely geometric operating system written in a uniform programming language." - Eric Weinstein </blockquote>
* Another valuable resource is the [[Talk:Graph,_Wall,_Tome#EricRWeinstein_2020-02-02_at_1:31_PM|comments Eric made regarding how the Wall should be modified]].
== Direct Connections between the Graph, the Wall, and the Tome ==
=== Connections between the Graph and the Wall===
<div class="toccolours mw-collapsible" style="width:800px; overflow:auto;">
<div style="font-weight:bold;line-height:1.6;">Graph->Wall connections suggested by Eric Weinstein</div>
<div class="mw-collapsible-content">
[[File:Graph To Wall eric.png|800px|center]]
</div></div>
<div class="toccolours mw-collapsible mw-collapsed" style="width:800px; overflow:auto;">
<div style="font-weight:bold;line-height:1.6;">Alternative representation of the Graph->Wall connections suggested by Eric Weinstein</div>
<div class="mw-collapsible-content">
[[File:Bad Version of Graph to Wall.png|800px|center]]
</div></div>
<div class="toccolours mw-collapsible mw-collapsed" style="width:800px; overflow:auto;">
<div style="font-weight:bold;line-height:1.6;">Alternative Graph->Wall Connections</div>
<div class="mw-collapsible-content">
The following figure connects the Graph to elements on the Wall. There are certain holes since there aren't visualizations or formulas for all concepts mentioned in the Graph.
[[File:Graph To Wall original.png|800px|center]]
</div></div>
<div class="toccolours mw-collapsible mw-collapsed" style="width:800px; overflow:auto;">
<div style="font-weight:bold;line-height:1.6;">Alternative representation of the connections between the Graph and the Wall</div>
<div class="mw-collapsible-content">
The following figure connects the Graph to elements on the Wall. There are certain holes since there aren't visualizations or formulas for all concepts mentioned in the Graph.
[[File:Graph To Wall original.png|800px|center]]
</div></div>
<div class="toccolours mw-collapsible mw-collapsed" style="width:800px; overflow:auto;">
<div style="font-weight:bold;line-height:1.6;">Alternative representation of the connections between the Graph and the Wall</div>
<div class="mw-collapsible-content">
On the left-hand side, we have the [[Graph,_Wall,_Tome#The_Graph|Graph]]. On the right-hand side, there is a list of all elements that appear on the [[Graph,_Wall,_Tome#The_Wall|Wall]].
[[File:Graph-Wall connections.png|center]]
</div></div>
<div class="toccolours mw-collapsible mw-collapsed" style="width:800px; overflow:auto;">
<div style="font-weight:bold;line-height:1.6;">Interactive representation of the connections between the Graph and the Wall</div>
<div class="mw-collapsible-content">
An interactive version of the Wall that shows direct connections to the graph is available [https://dev.theportal.dev/wall/ here].
</div></div>
<div class="toccolours mw-collapsible mw-collapsed" style="width:800px; overflow:auto;">
<div style="font-weight:bold;line-height:1.6;">Connections between an updated Graph and updated Wall</div>
<div class="mw-collapsible-content">
In the following figure, the holes in the wall have been filled with appropriate illustrations and equations.
[[File:Graph To Wall.png|800px|center]]
</div></div>
=== Connections between the Wall and the Tome ===
<div class="toccolours mw-collapsible mw-collapsed" style="width:800px; overflow:auto;">
<div style="font-weight:bold;line-height:1.6;">Connections between Wall and Tome</div>
<div class="mw-collapsible-content">
On the left-hand side, we have a list of all elements that appear on the [[Graph,_Wall,_Tome#The_Wall|Wall]] and on the right-hand side we have the table of contents of the [[Graph,_Wall,_Tome#The_Tome|Tome]]. Only connections between concepts that are also mentioned in the Graph are shown.
[[File:Wall-Tome Connections.png|center]]
</div></div>
=== Connections between the Graph and the Tome ===
<div class="toccolours mw-collapsible mw-collapsed" style="width:800px; overflow:auto;">
<div style="font-weight:bold;line-height:1.6;">Connections between Graph and Tome</div>
<div class="mw-collapsible-content">
On the left-hand side, we have the [[Graph,_Wall,_Tome#The_Graph|Graph]] and on the right-hand side we have the table of contents of the [[Graph,_Wall,_Tome#The_Tome|Tome]].
[[File:Graph-Tome Connection.png|center]]
</div></div>
== Resources ==
* [https://drive.google.com/file/d/1ZMT8nDyF6qT5nh6hiCjrHyX4giKKmGSA/view?usp=sharing New Graph-Wall Mind Map]
* [https://drive.google.com/file/d/16r60-56mhYCx4KKBvZv_6HDLOLw43X7y/view?usp=sharing Map of Witten's Physics and Geometry Essay]
* [https://docs.google.com/document/d/1t9AvvFZzODw1WiGRZwRsFFZdPdBzYVJGLHiqWNrMtIA/edit?usp=sharing Master Planning Doc]
* [https://drive.google.com/drive/folders/1706CBEJQEMppV60OU8OtcXxicluk2T3Y?usp=sharing Drive Folder]
* [https://docs.google.com/document/d/18rN-zfv41xeH3WFNOrZNb7Clz-yu1dgulP4bcEPKbcY/edit?usp=sharing List of Reference Material]
[[Category:Graph, Wall, Tome]]
[[Category:Projects]]
Retrieved from "https://theportal.wiki/wiki/Special:MobileDiff/3917...13175"