120°+120°=90° in Dynkin Diagrams. Teamwork in Creating Learning Paths. (What is Geometry?)
Channel: Math 4 Wisdom
Date: 2023-03-03

I present and explore the value of teamwork in creating learning paths with participants in the Math 4 Wisdom community. I am philosopher Andrius Kulikauskas (Eiciunai, Lithuania) and I interact with mathematician John Harland (Carlsbad, California, USA), physicist Thomas Gajdosik (Vilnius, Lithuania), technical writer and facilitator Jon Brett (Auckland, New Zealand).  I also include excerpts from videos by educator and visionary Kirby Urner (Portland, Oregon, USA).

Specifically, I take up the challenge of investigating, understanding and conveying the equation 120°+120°=90° in three dimensions as the central fact that links together Dynkin diagrams and thus simple Lie algebras and simple Lie groups, and in that sense, multidimensional space.

I act out two ways of thinking - a tetrahedral-minded Martian who thinks in terms of root systems and a cubic-minded Earthling  - and thus reference Kirby's vision of how two such civilizations might struggle to communicate. Kirby's Digital Math has its own learning paths and so I want to discover how we can work together. Kirby is inspired by Buckminster Fuller's Synergetics.

My main theme is that I can't create learning materials in isolation.  I need our help - I need your help - so that I can understand what you understand - and vice versa - so we can create videos that reflect that - and thereby grow in our understanding.  Jon emphasized to me that narration is key and I think that explains my impulse to act out characters.  

From Jon's paper model I now realize that 120°+120°+90°=0° and thus 120°+120°=-90° better says what I mean to say. With Jon I also realize that I need to emphasize that math lets us think not in two or three or four but multiple dimensions. I am learning that the Dynkin diagrams are modeling the assumptions of counting, which is that counting is higly symmetric - we can simply relabel everything - but that ultimately counting is sequential and thus chains together dimensions. We'll explore that further, more and more as a team, in future videos in this series, What is Geometry?

0:00 Andrius introduces the topic of teamwork in creating learning paths
1:00 Kirby talks about his participation in Andrius's Math 4 Wisdom
2:00 Andrius about 120°+120°=90° as an entry way into a Math 4 Wisdom learning path
2:50 Martian shows 120° rotation in A2 root system and shows 120°+120°=90° in A3 root system
4:15 Kirby describes Martian math as a context for presenting the geometry of Synergetics
4:55 Martian reads Dynkin diagram as 120°+120°=90° 
5:55 Andrius distinguishes Math for Speculation from Math for Wisdom
6:40 Jon presents the thinking tool Emergently http://www.emergently.net
8:45 Andrius on his goal of expressing cognitive frameworks with advanced math
9:40 Exact sequences model divisions of everything. Grad-curl-div models the fivesome.
10:45 Math 4 Wisdom learning paths lead to Bott periodicity and the grounds for 4 geometries (affine, projective, conformal, symplectic)
11:45 Kirby's learning paths: Digital Math
12:05 Kirby about introducing nonliteral truth into STEM
12:55 Andrius's challenge of working together with different "religious" perspectives on truth. John's thoughts on the monumental nature.
14:20 Key fact: 120°+120°=90°
15:00 Earthling Cowboy illustrates this with a cube
15:55 Earthling with cube shows 3 x 120° = 360°
16:45 How to show it is 90°? Earthling and Martian.
18:10 Earthling shows 120°+120°=90° with cube
19:05 Validating by calculating angles with Law of Cosines
19:50 Jon's paper circle
21:25 Dynkin diagrams as chains. Earthling talks of root systems of the complex simple Lie algebras, emphasizes the compact classical groups and the quest for 4 geometries
23:05 Dynkin diagrams, root systems, special linear Lie algebra. A1, A2, A3. 
25:10 Earthling organizes the 12 midpoints of a cube's edges
26:15 Andrius organizes the A3 roots
26:40 Martian organizes the A3 roots
27:50 A2 within A3. Martian explains
30:00 Andrius explains
30:40 Earthlings models cube with two tetrahedrons
32:20 Confusion about what is meant by dimensions with Dynkin diagrams. End of Zoom session.
34:00 Visualizing the A3 root system within a cube using Math3D.org https://www.math3d.org
44:40 Earthling and Martian are friends
45:15 Kirby on the need for social engineering
45:30 THANK YOU!

Join discussion at https://www.freelists.org/list/math4wisdom

Thank you to BigKahuna360 for the electric shocks! https://freesound.org/people/BigKahuna360/sounds/160421/

Kirby's Martian Math https://www.youtube.com/watch?v=lyQDz-zmnRM&list=PLeBIlkJm4xgmpE0eona7dpOJHm7yWivSk

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Public Domain, 2023

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