Platonic solids and their symmetries
Insights into Mathematics
98.9K subscribers
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 https://www.youtube.com/channel/UCriF..., or become a Patron at https://www.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.
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Here are all the Insights into Mathematics Playlists:
Elementary Mathematics (K-6) Explained: https://www.youtube.com/playlist?
list=PL8403C2F0C89B1333
Year 9 Maths: https://www.youtube.com/playlist?list...
Ancient Mathematics: https://www.youtube.com/playlist?list...
Wild West Banking: https://www.youtube.com/playlist?list...
Sociology and Pure Mathematics: https://www.youtube.com/playlist?list...
Old Babylonian Mathematics (with Daniel Mansfield): https://www.youtube.com/playlist?
list=PLIljB45xT85CdeBmQZ2QiCEnPQn5KQ6ov
Math History: https://www.youtube.com/playlist?list...
Wild Trig: Intro to Rational Trigonometry: https://www.youtube.com/playlist?list...
MathFoundations: https://www.youtube.com/playlist?list...
Wild Linear Algebra: https://www.youtube.com/playlist?list...
Famous Math Problems: https://www.youtube.com/playlist?list...
Probability and Statistics: An Introduction: https://www.youtube.com/playlist?list...
Boole's Logic and Circuit Analysis: https://www.youtube.com/playlist?list...
Universal Hyperbolic Geometry: https://www.youtube.com/playlist?list...
Differential Geometry: https://www.youtube.com/playlist?list...
Algebraic Topology: https://www.youtube.com/playlist?list...
Math Seminars: https://www.youtube.com/playlist?list...
And here are the Wild Egg Maths Playlists:
Solving Polynomial Equations: https://www.youtube.com/playlist?list...
Exceptional Structures in Maths and Physics via Dynamics on Graphs" https://www.youtube.com/playlist?list...
Triangle Centres: https://www.youtube.com/watch?v=iLBGX...
Six: An elementary course in pure mathematics: https://www.youtube.com/playlist?list...
Algebraic Calculus One: https://www.youtube.com/playlist?list...
Algebraic Calculus Two: https://www.youtube.com/playlist?list...