Inside the Octagon: Dihedral Group D8 | Abstract Algebra | Group Theory | Dgmathic | Python - Manim скачать в хорошем качестве

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

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. The Gateway to Group Theory: Groups in Under an Hour :    • The Gateway to Group Theory: Groups in Und...   The Triangle Conspiracy: Dihedral Group Mechanics:    • The Triangle Conspiracy: Dihedral Group Me...      • 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 #dogmathic #GroupTheory #AbstractAlgebra #DihedralGroup #D8 #Octagon #Symmetry #CayleyTable #Permutations #NonAbelian #Generators #Subgroups #python #manim