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

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. Twitter: @VisualAlgebra 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&kick it up another notch! 14:36 Using generators and relations to reconstruct the structure of a group 21:13 What&next and a glimpse into Cayley graphs #SoME2