Insights into Mathematics | Platonic solids and their symmetries | Dynamics on Graphs 10 | Wild Egg Maths @njwildberger | Uploaded 2 years ago | Updated 8 hours ago
We tabulate the familiar Platonic solids and their symmetry groups, looking both at rigid motions (which in this case are all rotations) and also those including reflections. This is a stepping stone to understanding a lot of exceptional structures in mathematics and physics.

The 4 dimensional versions of these regular solids have many enticing connections to the standard model in particle physics and string theory and we'll eventually see there are also interesting algebraic objects connected to these, such as Coxeter groups, root systems, Lie algebras, the division algebras of the quaternions and octonions, Jordan algebras, hypergroups and additional combinatorial lattices and posets with deep and remarkable properties.

Our approach to all these exceptional objects is to try to obtain them from relatively simple dynamical games on graphs -- in particular the Mutation Game, and Mozes' Numbers Game. So we will want to connect these symmetries to graph theory; but we will also want to extend this understanding to the 4 dimensional regular solids. The Platonic solids are a good starting point.

This is part of a series of Explore Research Level Mathematics video lectures designed for motivated amateur, or professional, mathematicians wanting to delve into pure maths research without the abstraction and complexity that currently besets much of the subject. We will aim to always connect our abstract musings to computational reality by focusing on examples and explicit, concrete calculations --- naturally without any pretense of being "able to do an infinite number of things"!

To access more Explore Research Level Mathematics, please join Norman's Wild Egg Maths channel at youtube.com/channel/UCriFv3G22iOUidUhkIGXuhw, or become a Patron at patreon.com/njwildberger . There are already dozens of videos there, and many more will appear over the next months and years. One additional advantage: you get to access pdf summaries of some of the videos along with GeoGebra worksheets.

***********************
Here are all the Insights into Mathematics Playlists:

Elementary Mathematics (K-6) Explained: youtube.com/playlist?
list=PL8403C2F0C89B1333
Year 9 Maths: youtube.com/playlist?list=PLIljB45xT85CcGpZpO542YLPeDIf1jqXK
Ancient Mathematics: youtube.com/playlist?list=PLIljB45xT85Aqe2b4FBWUGJdYROT6-o4e
Wild West Banking: youtube.com/playlist?list=PLIljB45xT85DB7CzoFWvA920NES3g8tJH
Sociology and Pure Mathematics: youtube.com/playlist?list=PLIljB45xT85A-qCypcmZqRvaS1pGXpTua
Old Babylonian Mathematics (with Daniel Mansfield): youtube.com/playlist?
list=PLIljB45xT85CdeBmQZ2QiCEnPQn5KQ6ov
Math History: youtube.com/playlist?list=PL55C7C83781CF4316
Wild Trig: Intro to Rational Trigonometry: youtube.com/playlist?list=PL3C58498718451C47
MathFoundations: youtube.com/playlist?list=PL5A714C94D40392AB
Wild Linear Algebra: youtube.com/playlist?list=PLIljB45xT85BhzJ-oWNug1YtUjfWp1qAp
Famous Math Problems: youtube.com/playlist?list=PLIljB45xT85Bfc-S4WHvTIM7E-ir3nAOf
Probability and Statistics: An Introduction: youtube.com/playlist?list=PLIljB45xT85AMigTyprOuf__daeklnLse
Boole's Logic and Circuit Analysis: youtube.com/playlist?list=PLIljB45xT85CnIGIWb7tH1F_S2PyOC8rb
Universal Hyperbolic Geometry: youtube.com/playlist?list=PLIljB45xT85CN9oJ4gYkuSQQhAtpIucuI
Differential Geometry: youtube.com/playlist?list=PLIljB45xT85DWUiFYYGqJVtfnkUFWkKtP
Algebraic Topology: youtube.com/playlist?list=PL6763F57A61FE6FE8
Math Seminars: youtube.com/playlist?list=PLBF39AFBBC3FB30AF
************************

And here are the Wild Egg Maths Playlists:

Solving Polynomial Equations: youtube.com/playlist?list=PLzdiPTrEWyz7hk_Kzj4zDF_kUXBCtiGn6
Exceptional Structures in Maths and Physics via Dynamics on Graphs" youtube.com/playlist?list=PLzdiPTrEWyz5HBT_Yo1G4DfeqUfI9zkKM
Triangle Centres: youtube.com/watch?v=iLBGXDSUohM&list=PLzdiPTrEWyz6VcJQ5xcuqY6g4DWjvpmjM
Six: An elementary course in pure mathematics: youtube.com/playlist?list=PLzdiPTrEWyz4KD007Ge10dfrDVc4YwlYS
Algebraic Calculus One: youtube.com/playlist?list=PLzdiPTrEWyz4rKFN541wFKvKPSg5Ea6XB
Algebraic Calculus Two: youtube.com/playlist?list=PLzdiPTrEWyz5VLVr-0LPPgm4T1mtU_DG-