Dynkin diagrams: 120 degrees + 120 degrees = 90 degrees

[AndriusKulikauskas]

I am creating a YouTube video and possibly learning materials to help people appreciate and visualize the geometry by which Dynkin diagrams link dimensions. Consider the section Finite Dynkin diagrams in the Wikipedia article on Dynkin diagrams. You will see that the diagram A_3 is simply a chain with three nodes and two edges. The chain may be longer and shorter but it is the key feature of every single Dynkin diagram in this table. What does the chain mean?

Lie theory gives the answer and I am still in the process of learning it. I would say the subject takes a lot of study in graduate school to master. But that energy is spent on the abstract framework. The actual results are fascinating and very concrete. For example, a key role is the fact in three-dimensional geometry that you can turn a cube 120 degrees round one axis and then 120 degrees round another axis and the result will be that you have made a 90 degree reflection round a third axis.

So I want to start at the end - the concrete results - and understand and convey that, and work outwards, and learn only that abstract context which I personally need, which may be hardly none at all. My personal goal is to investigate why, cognitively, there are four classical Lie groups, and in what sense they may give the foundation for four different geometries (affine, projective, conformal, symplectic). With Math 4 Wisdom I am showing how I investigate such questions and cultivating a participatory community.

Dynkin diagrams are used in Lie theory to classify Lie groups, which is to say, groups of actions which are not discrete but continuous, such as rotations around a circle, rotations on a sphere, translations along a line, translations in a plane, and so on. Dynkin diagrams distinguish multidimensional spaces. Each node stands for the same space SL(2). Two nodes linked by an edge stands for SL(3). They are related to SU(2) and SU(3) which are used in physics to model the weak force and the strong force, respectively.

Geometrically, in the diagram, if two nodes are not directly linked, then they are independent, thus separated by 90 degrees. If they are linked, then it is by 120 degrees. The diagram shows that node A is linked by 120 degrees with its neighbor B, and B is linked by 120 degrees with its neighbor C. Yet A and C are not directly linked and so they differ by 90 degrees. This can be shown variously with a three-dimensional root system or with a cube. I have made a model:

This is a rather sophisticated thing to visualize and thus to present. Also, I still need to undertand better myself what this all means.

About the author

I am Andrius Kulikauskas, the host of Math 4 Wisdom, which relates the language of wisdom and the language of mathematics. I live in Lithuania. Here is a map of the learning paths I am developing.

Target audience

My target audience is self learners who are interested in how advanced mathematics can help clarify, develop and convey wisdom about the big questions and big answers in life, and also how wisdom can inform research in advanced mathematics.

I believe that the ideas for this video can be presented to motivated viewers with a knowledge of high school math (geometry, algebra, trigonometry, vectors, angle formula). But the interpretation will be, I think, novel even for graduate students of Lie theory.

Target medium

I am creating a YouTube video. Several people at the Math 4 Wisdom community are helping me. I appreciate a variety of help:

• from domain experts so that I understand the concepts better
• from learners from who I could see what is more or less understandable, interesting, watchable
• from animators, artists, programmers to visualize the three-dimensional rotations
• from actors to make this fun and engaging

I have a draft of a script but I need to rethink it and rework it.

Contact details

My email is math4wisdom AT-SIGN gmail DOT com

Public Domain

All creative work contributed by myself and others must be in the Public Domain, copyright-free, for all to use in their own best judgement. Likewise, this post and all replies are understood to be in the Public Domain or please delete them.

[AndriusKulikauskas]

I created a video "120°+120°=90° in Dynkin Diagrams. Teamwork in Creating Learning Paths. (What is Geometry?)" with the help of Kirby Urner, Jon Brett, John Harland, Thomas Gajdosik and Giedre Gajdosik.
https://www.youtube.com/watch?v=kJYFleyQiGo Of particular help was the online 3D graphing tool http://www.math3d.org which served my needs and allowed me to publish my 3D graphs in the Public Domain, copyright free, without restriction, unlike some other tools. I also made use of a sound from http://www.freesound.org I keep making more videos so please contact me if you are interested to help!