Where Do GROUPS Come From? | VISUAL Abstract Algebra | E1 #SoME2

Описание

We motivate the definition of an abstract group by looking at compositions of symmetries of geometric objects. We discover all symmetries of an equilateral triangle and compute a Cayley table of all their compositions. We then show that this results in a structure of a group, and that this group can be generated by one rotation and one reflection. We provide a glimpse into generators and relations and explain how they can be used to define the structure of a group.

This my submission for the Summer of Math Exposition 2 competition organized by @3blue1brown

This video was created in collaboration with Dr. Matthew Macauley from Clemson University, an author of the forthcoming book Visual Algebra.
Web: http://www.math.clemson.edu/~macaule/
Twitter: @VisualAlgebra
YouTube: https://www.youtube.com/c/ProfessorMacauley

CHAPTERS:
0:00 Intro
0:46 Introduction to symmetries
1:25 Symmetries of an equilateral triangle
3:13 Composing symmetries and Cayley table
8:33 Motivation for the definition of a group
10:10 What is a group?
10:30 The group of symmetries of a triangle
11:58 Rotations vs reflections
13:09 Let's kick it up another notch!
14:36 Using generators and relations to reconstruct the structure of a group
21:13 What's next and a glimpse into Cayley graphs

#mathflipped #manim #SoME2