A Graph, Wall, Tome Project Proposal
Gauge theory could be the only language that you need in order to communicate almost every idea in physics. Big ideas in physics were difficult to penetrate when physicists introduced them in the beginning. Maybe because the set of connections that brought the evidence together used to be very small. Or, maybe they did not understand them well enough to explain. For whatever reason, it changed after waves of collaborations between mathematicians and physicists. Taking the work of James Simons and C.N. Yang in the 70s for example, where they both came up with fiber bundle descriptions of reality. Later when technology had advanced enough, you could visualize some of those fiber bundle descriptions, for example in the Planet Hopf animation due to Dror Bar-Natan. In the same trend of bundle descriptions, Edward Witten published Physics and Geometry in 1987 just to sum up our knowledge of physics in a single impressive paragraph. In 2004, the ultimate mathematical artist and now the Nobel laureate Sir Roger Penrose published his popular book: the Road to Reality. In 2010, Dr. Nina Douglas proposed the idea of an iconic Wall, on which one can see every great work of mathematicians and physicists. Fortunately, James who since his academic days had made lots of money, paid for the construction of arguably the most important installation art ever. The artist Christian White then carved the equations and shapes into stone by hand, which is up for display at the Stony Brook University in New York. However, there is work to do for addressing the connections between different layers of abstraction. The natural language of the Graph, the visual realm of the Wall and the search for truth in the Tome. So we have the Graph, the Wall and the Tome to inform and inspire scientists and artists alike.
As much as you try to break the limits in the visual realm your tools need to catch up to the level of sophistication. Even though the tricks that we used have been helpful in creating high-dimensional shapes, it is only a starter pack. Watching how masters bring to life abstract ideas in mathematics, we believe that we can invent new methods If we want to. If you watch seven-manifolds visualization by Niles Johnson, then you see that you are limited by your imagination and also by your understanding of fiber bundles. It inspires me how London Tsai uses tasteful color palettes in order to draw the attention of the viewer to a specific dimension among many. For example, in Gadget No. 5, latitudinal change in the base space affects hues in the tori. After Eric Weinstein went on Joe Rogan Experience and tried to explain gauge symmetry using the Hopf fibration, we knew that we had to explore the subject. He knows well how to monetize the accomplishments of the Physics community. Soon after, The Portal community took shape and Eric shared with us the idea of Graph-Wall-Tome for the first time. On our very first adventure we created the Porta.jl project for replicating some of London Tsai’s prints. On the Ides of March 2020, we introduced Porta over a call on The Portal Unofficial Discord server, during the worldwide self-quarantine due to COVID-19. Back then, Discord servers could handle no more than one hundred people on the same call before the audio started to break. Fast forward to now, in Porta there are abstract vector spaces, Clifford/Study algebras, spheres, stereographic projection, symmetry groups along with their representations, planetary-scale data input and audio signal processing. These tools appeared out of necessity by building cool mathematical objects. The input controls aside, a model is an information processing pipeline: (object space -> world space -> camera space -> screen space). And the medium of the models has been computer graphics imagery distributed on social media channels. At the moment of this writing, it has been forked 5 times on GitHub. So, it should not be very resource intensive to develop Porta. We took what was out there, built on it, and we did not have to limit our work to the industry standard tools.
Every potential thing requires work to become real, and this task, particularly so, imposes heavy demands. Mathematicians can develop a final picture so that you can see it in your mind’s eye as well. Because mathematical constructions follow a string of definitions, propositions, theorems and corollaries that are the same everywhere. We should first look at the greatest achievements in 20th century Mathematics and Physics. But, to take a Physics state of mind as the main framework here, we know that there must be something wrong with our interpretations. And also there must be latent connections between different concepts yet to be discovered. So, it is important to learn how to be less wrong. Then, we study the tools: Lie groups, Lie algebras, group actions, fiber bundles, connections, curvature, and spinors. However, we should be careful not to get carried away by the beauty of the maths. As the subset related to a near-future fundamental theory of nature is likely to be small. Thus it is important to organize the project’s activities and aim for efficiency in spending our energy in order to accomplish this complex goal.
The result is a sphere inscribed inside a translucent icosahedron and shows you how particles play on a stage made of fiber bundles. It is also a reminder that the universe is not flat. In order to get a mental picture, please find Hopf Flowers on YouTube. There, you will find three-dimensional shadows of a couple of Hopf flowers that are rotating in the four-dimensional space that they are embedded in. Each has a different number of petals. And we have placed each one on top of a different vertex of an icosahedron, so 12 flowers in total. To make for an otherworldly view, we have inscribed the Moon inside the icosahedron. During the animation, the camera rotates in an orbit around the Moon. On a vertex, we make a geometrical shape by first finding the related symmetry groups in a physical law, then constructing manifolds isomorphic to those groups and then animating selected variables to reproduce conservation laws and other interesting properties. The icosahedron will play the role of manifold X while the Moon will be spacetime manifold M, per the Graph description. Here, M should be at the bottom and X at the top, because it makes more sense to have the fiber manifold on top of the base. Then, we have to connect them with lines to elaborate the idea of a portal. Next, place objects related to general relativity down on M, displaying the curvature using Penrose stairs for example, and then, put the rest on X. Plus, the framework that we develop should work with most of the equations taken from the Wall. Because this is the whole point of creating an entirety. An engineering challenge will be the compartmentalization of computation to have a smooth transition as the camera moves from model to model. We know that there will be new designs and discoveries, but we do not know what they will be yet. So an incremental development plan fits the solution in a perfect way. The final picture should look like a giant icosahedron made of animated shapes.
Now, to present such a non flat Wall it is a good idea to take inspiration from two projects that The Portal community has done in the past. One of them is a clickable Wall, a web UI with a narrator Graph at the bottom of the screen. There, you can find arrows that will guide you through different parts of the Wall according to the story. The user can click and focus on an equation or a shape. It displays links to pages from the Tome as well as other online references to lead you to the search. We like this idea and we think that we should use it as a model for our user interface. To improve things, adding a nested view to move in the phase space inside the shapes, as well as the manifolds on the Wall might be a good idea. And adding control sliders too, to make it reactive and also maybe an audio player. With this kind of interactive UI, you should get a better feel of a physical law and its inner workings. Building our previous models has involved a lot of activities: participating book club weekly meetings to study and discuss the Tome, engaging in quests, listening in on conversations between real mathematicians, physicists and artists on The Portal podcast, building and testing versatile tools, gaining beautiful geometrical insights, collaborating with scientists and artists from various disciplines, and feeding the imagination of friends and family on the social media. Therefore, we wish to keep going down this rabbit hole and blaze a trail to include others in such transformational experiences.
The Graph, Wall, Tome project will produce an unbroken picture of reality. Fiber bundles are non-trivial products between two manifolds, called base and fiber. Like when we draw function graphs on the x-y plane, and sometimes they pass through the horizontal zero section. Now, principal bundles are a special type that have Lie groups as fibers. Also, vector bundles are another kind that have vector spaces as fibers. Think of the Standard Model as a system on a stage of principal bundles and vector bundles, on top of every point in the spacetime manifold. These objects are connected by basic laws of Physics. And for every symbol arrangement on a piece of paper there is a corresponding geometrical image. So, looking at the Standard model, we should hope that creating an image close to reality is a possibility.
The Renaissance painter Giotto brought about the much needed change by paying attention to depth in nature. He painted three-dimensional scenes on flat canvases. He saw the big configuration space of objects in a scene, and how the reality we perceive might be merely a single orbit in that space. Imagine a crowd of people scattered around, with each person wearing their own style of clothes, having their own unique posture, the inside of their head filled with a combination of emotions such as sadness/joy, despair/hope and bitterness/warmth, and not everyone is oriented towards the same thing, showing their personalities through the looks on their faces and how they connect to each other. Already a single work of art shows a space of infinite dimensions. And also contrary to the majority of the contemporaries, we know Giotto for his rigour in representing structures as one can find them in real life. For the people at the time, the Dark Ages, that picture was a sign that there could be a way out of a world of diseases and wars. Thus Giotto opened a portal to show how one can see a new whole picture through a higher-dimensional perspective.
Either it is the science communication or we have been in the dark for nearly half a century. This is important because physicists do important work. They gave us: electricity, telescopes, radiation therapy, and the web. All that and more have reduced the amount of suffering in the world. In fact, that is why many people turned from religion to science in the first place. To borrow a metaphor from Hans Moravec, science and art are the only two higher mountaintops in the landscape of human intelligence. Despite that, so much great work has been done in the 20th century that people are not aware of. One problem is, physics has become so advanced that only a small number of people understand its depth and beauty. And a metaphor of Eric Weinstein’s likens the physics literature to the sheet music of the great musicians, only, almost nobody seems to know what they sound like! In addition, theorists tend to get stuck in their ways and that makes it harder to innovate. The evolution and communication of physics appear to be not so overwhelming and so it calls for revolutionary ideas coming from an alternative energy source.
An interactive shape can be the extension of one’s body to help explore the platonic mathematical world. Say you want to visualize the idea behind spontaneous symmetry breaking in particle physics. The electroweak theory states that in the early universe, when things were so dense and hot, the Higgs field used to have the symmetry group G = SU(2)×U(1). After millions of years, things spread out and cooled down. Now, we believe that G has broken into two subgroups, namely SU(2) and U(1). The theory means that the Higgs field has lost a part of its symmetry. It was observed to be true up to a very good precision at CERN in 1983. We built a prototype to explore this idea and found an initial good question. For a vacuum vector p in the complex plane what does the action of the Lie group G on the Higgs field look like? On the one hand, SU(2) is three-dimensional and U(1) is one. On the other hand, the field is scalar, so it is one dimensional. That is why we need a few tricks to visualize this four-dimensional object. One can compress/blow out things to a standard S³ in order to be able to use stereographic projection. Then, use different hues and size scales to compensate for the effect of this particular trick. We could paint the orbit by measuring the distance between p and its new position under the G-action. Also to account for the field value, the object size could depend on how far p is from the origin. The orbit of p under the action turned out to be a solid ball, diffeomorphic to a 3-sphere. And finally we used slider controls for an SU(2) basis and the U(1) angle in order to get an idea of what the isotropy subgroup, the unbroken subgroup, might look like. Following an incremental development plan, we tried to use a theory and build a shape in order to understand it better.
We build our way forward by going on many adventures. Once we find an important problem an adventure down the rabbit hole begins, and it takes a few weeks in general. The problem should give us a knowledge buzz and capture our imaginations. Prototypes spring up from an adventure and either create a new model, or enhance/validate an existing one. The following workflow grew out of our own experiences in an organic way. First, we think how it is possible to implement a model of it in code. Secondly, start with data or the easiest part and program a scaffolding until we run out of ideas. We ask, what is the 20% of the task that could output 80% of the result? Thirdly, play with the model in interactive mode to get more insight into the problem. Using the new information, come up with a new design in the fourth place. Fifthly, realize that it is going to take a lot of time to build a good enough prototype, and so the sooner we start the sooner it finishes. Finally, run the code and generate visual results to tell a story of the adventure. Building prototypes helps us get to know both ourselves and nature. It will take a few prototypes to acquire the taste and necessary tools for an important problem.
The Graph-Wall-Tome project can serve as an abstract class of ideas, instantiations of which will drive change of dynamics in civilization and culture. What if we went beyond the weird topics on quantum reality? What if we instead talked about the deep and beautiful ocean of objects, in which particles and waves travel? And how gauge transformations can make the age-old dream of alchemists come true, by transforming fundamental particles into one another. In addition, there are causal relationships between these geometric objects that if enough people could see, it would help change the face of public discourse in modern politics too. Because we do not like to see another big collider project get too political. And we believe that art and science have the potential to go hand-in-hand to inspire those who are curious to explore the foundational materials. Although the most important outcome is going to be opening access to physics for a larger group of people. GWT bootstraps science and art in a generative way for the purpose of sharing ideas, evolving them and feeding them back into the community.
Please click on the image to watch the statement video!
Thanks to Aardvark and Tim (Worth/Mirth)less Swagman for reading drafts of this.