Inside the Octagon: Dihedral Group D8 | Abstract Algebra | Group Theory | Dgmathic | Python - Manim

Описание: https://dogmathic.com/

Inside the octagon, we build the dihedral group D8: all 16 rigid symmetries of a regular octagon. We label vertices 1 through 8, encode rotations and reflections as permutations, then multiply symmetries by composing functions. Using the relations r^8=e, s^2=e, and srs=r^-1, we build the rules, read patterns in the Cayley table, and see exactly why D8 is nonabelian. Then we finish with subgroups, generators, and the geometric intuition behind the algebra.

This video was made using Manim, a Python-based animation engine.

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   • Group Theory  
   • Abstract Algebra  

PROPERTIES AND CONCEPTS USED
Rigid Symmetry
Regular Octagon
Dihedral Group D8
Rotation r
Reflection s
Vertex Labeling 1 Through 8
Permutation Representation
Function Composition
Closure
Associativity
Identity Element e
Inverses
Cyclic Subgroup ⟨r⟩
Conjugation
Group Presentation
Cayley Table
Nonabelian Group
Generators
Subgroups

CHAPTERS
00:00 Introduction
00:55 What We Will Build
01:40 Rotations As Permutations
02:30 Reflection Axes
03:20 Build All Reflections
04:10 List All 16 Elements
05:00 Composition Rule
06:20 Presentation Relations
07:10 Cayley Table Patterns
08:00 Nonabelian Order Matters
09:00 Subgroups And Generators
09:50 One Thing To Remember
10:35 Thanks For Watching


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